text: 26 ESCHER AND SEARLE 6. The Proof of Theorem A We ﬁrst give a brief outline of the proof of Theorem A. We have shown in Theorem 5.4 that an almost maximal action is actually maximal for this class of manifolds, so we only − need to prove Theorem A for maximal actions, that is, when the free rank is equal to 2k n. To prove the equivariant classiﬁcation we split the proof into two cases: Case (1), where the free dimension equals the free rank, and Case (2), where the subtorus corresponding to the free rank acts almost freely but not freely. In both Case (1) and Case (2) there are two further subcases: Subcase (a), where the rank of the action is equal to the number of facets of the orbit space Mn/Tk and Subcase (b), where the rank of the action is strictly less than the number of facets of the orbit space Mn/Tk. In Cases (1a) and (1b), we show that the quotient of Mn by the freely acting subtorus, − Mn/T2k n, is a torus manifold of non-negative curvature and the result will follow in each case by Corollary 6.2 and Corollary 6.4, respectively. For Case (2), the quotient of Mn is a torus orbifold of non-negative curvature. In Case (2a), we use Proposition 3.11 to show that Mn is equivariantly diﬀeomorphic to an n-dimensional manifold M¯ which admits a freely − − acting torus of rank 2k n, such that M¯ /T2k n is a torus manifold admitting a family of metrics with almost non-negative curvature and the quotient M¯ /Tk is weight-preserving − diﬀeomorphic to M/Tk. By Theorem 2.23, we see that the torus manifold M¯ /T2k n is locally standard and then we can once again use Theorem 6.2 to obtain the result. For Case (2b), one can show that there is a closed, principal Tl bundle over Mn, with simply-connected total space, Nn+l, admitting a Tl+k-invariant family of metrics of almost non-negative curvature and such that the rank of the torus action on N is equal to the number of facets of the orbit space Mn/Tk = Nn+l/Tk+l. We then appeal to Proposition − 3.11, as in Case (2a), to ﬁnd N¯ such that N¯/T2k+l n is a torus manifold admitting a family of metrics of almost non-negative curvature and proceed in a similar fashion as in Case (2a). Recall that we assume here that the free dimension of the torus action is equal to the free rank. As detailed above, we will break the proof into two cases as follows. Part (1) of Corollary 5.3 tells us that the quotient of M by the free action is a torus manifold, that is, M is a principal torus bundle over a torus manifold. We will show in this subsection that M is equivariantly diﬀeomorphic to the corresponding moment angle − − manifold of the simple polytope P = M2n 2k/Tn k = Mn/Tk. We will change super-indices for the following theorem to simplify its statement. We assume that the manifold N is of dimension 2n + p, admits a Tn+p action and is a Tp principal bundle over the torus manifold M2n. We will ﬁrst show in Theorem 6.1 that if the number of facets of the simple polytope Pn = N2n+p/Tn+p is equal to m = n + p, then Z N is equivariantly diﬀeomorphic to the moment angle manifold . P − Theorem 6.1. Let Nn+m be a simply connected principal Tm n-bundle over M2n where M2n has a locally standard, smooth Tn action with orbit space Pn, where Pn is an acyclic manifold with acyclic faces, with m facets. Then Nn+m is equivariantly diﬀeomorphic to Z the moment angle manifold . P − Proof. The proof is based on work by Davis [8]. Let Nn+m be a principal Tm n-bundle − over M2n. Such bundles are classiﬁed by homotopy classes of maps of M2n into BTm n,

text: ESCHER

text: AND

text: SEARLE

label: section

text: The

text: Proof

text: of

text: Theorem

text: A

text: We

text: ﬁrst

text: give

text: a

text: brief

text: outline

text: of

text: the

text: proof

text: of

text: Theorem

text: A.

text: We

text: have

text: shown

text: in

text: Theorem

text: that

text: an

text: almost

text: maximal

text: action

text: is

text: actually

text: maximal

text: for

text: this

text: class

text: of

text: manifolds,

text: so

text: we

text: only

text: need

text: to

text: prove

text: Theorem

text: A

text: for

text: maximal

text: actions,

text: that

text: is,

text: when

text: the

text: free

text: rank

text: is

text: equal

text: to

text: n.

text: To

text: prove

text: the

text: equivariant

text: classiﬁcation

text: we

text: split

text: the

text: proof

text: into

text: two

text: cases:

text: Case

text: where

text: the

text: free

text: dimension

text: equals

text: the

text: free

text: rank,

text: and

text: Case

text: where

text: the

text: subtorus

text: corresponding

text: to

text: the

text: free

text: rank

text: acts

text: almost

text: freely

text: but

text: not

text: freely.

text: In

text: both

text: Case

text: and

text: Case

text: there

text: are

text: two

text: further

text: subcases:

text: Subcase

text: (a),

text: where

text: the

text: rank

text: of

text: the

text: action

text: is

text: equal

text: to

text: the

text: number

text: of

text: facets

text: of

text: the

text: orbit

text: space

text: Mn/Tk

text: and

text: Subcase

text: (b),

text: where

text: the

text: rank

text: of

text: the

text: action

text: is

text: strictly

text: less

text: than

text: the

text: number

text: of

text: facets

text: of

text: the

text: orbit

text: space

text: Mn/Tk.

text: In

text: Cases

text: and

text: we

text: show

text: that

text: the

text: quotient

text: of

text: Mn

text: by

text: the

text: freely

text: acting

text: subtorus,

text: n,

text: is

text: a

text: torus

text: manifold

text: of

text: non-negative

text: curvature

text: and

text: the

text: result

text: will

text: follow

text: in

text: each

text: case

text: by

text: Corollary

text: and

text: Corollary

text: respectively.

text: For

text: Case

text: the

text: quotient

text: of

text: Mn

text: is

text: a

text: torus

text: orbifold

text: of

text: non-negative

text: curvature.

text: In

text: Case

text: we

text: use

text: Proposition

text: to

text: show

text: that

text: Mn

text: is

text: equivariantly

text: diﬀeomorphic

text: to

text: an

text: n-dimensional

text: manifold

text: M¯

text: which

text: admits

text: a

text: freely

text: acting

text: torus

text: of

text: rank

text: n,

text: such

text: that

text: M¯

text: n

text: is

text: a

text: torus

text: manifold

text: admitting

text: a

text: family

text: of

text: metrics

text: with

text: almost

text: non-negative

text: curvature

text: and

text: the

text: quotient

text: M¯

text: /Tk

text: is

text: weight-preserving

text: diﬀeomorphic

text: to

text: M/Tk.

text: By

text: Theorem

text: we

text: see

text: that

text: the

text: torus

text: manifold

text: M¯

text: n

text: is

text: locally

text: standard

text: and

text: then

text: we

text: can

text: once

text: again

text: use

text: Theorem

text: to

text: obtain

text: the

text: result.

text: For

text: Case

text: one

text: can

text: show

text: that

text: there

text: is

text: a

text: closed,

text: principal

text: Tl

text: bundle

text: over

text: Mn,

text: with

text: simply-connected

text: total

text: space,

text: Nn+l,

text: admitting

text: a

text: Tl+k-invariant

text: family

text: of

text: metrics

text: of

text: almost

text: non-negative

text: curvature

text: and

text: such

text: that

text: the

text: rank

text: of

text: the

text: torus

text: action

text: on

text: N

text: is

text: equal

text: to

text: the

text: number

text: of

text: facets

text: of

text: the

text: orbit

text: space

text: Mn/Tk

text: Nn+l/Tk+l.

text: We

text: then

text: appeal

text: to

text: Proposition

text: as

text: in

text: Case

text: to

text: ﬁnd

text: N¯

text: such

text: that

text: n

text: is

text: a

text: torus

text: manifold

text: admitting

text: a

text: family

text: of

text: metrics

text: of

text: almost

text: non-negative

text: curvature

text: and

text: proceed

text: in

text: a

text: similar

text: fashion

text: as

text: in

text: Case

text: Recall

text: that

text: we

text: assume

text: here

text: that

text: the

text: free

text: dimension

text: of

text: the

text: torus

text: action

text: is

text: equal

text: to

text: the

text: free

text: rank.

text: As

text: detailed

text: above,

text: we

text: will

text: break

text: the

text: proof

text: into

text: two

text: cases

text: as

text: follows.

text: Part

text: of

text: Corollary

text: tells

text: us

text: that

text: the

text: quotient

text: of

text: M

text: by

text: the

text: free

text: action

text: is

text: a

text: torus

text: manifold,

text: that

text: is,

text: M

text: is

text: a

text: principal

text: torus

text: bundle

text: over

text: a

text: torus

text: manifold.

text: We

text: will

text: show

text: in

text: this

text: subsection

text: that

text: M

text: is

text: equivariantly

text: diﬀeomorphic

text: to

text: the

text: corresponding

text: moment

text: angle

text: manifold

text: of

text: the

text: simple

text: polytope

text: P

text: k

text: Mn/Tk.

text: We

text: will

text: change

text: super-indices

text: for

text: the

text: following

text: theorem

text: to

text: simplify

text: its

text: statement.

text: We

text: assume

text: that

text: the

text: manifold

text: N

text: is

text: of

text: dimension

text: p,

text: admits

text: a

text: Tn+p

text: action

text: and

text: is

text: a

text: Tp

text: principal

text: bundle

text: over

text: the

text: torus

text: manifold

text: We

text: will

text: ﬁrst

text: show

text: in

text: Theorem

text: that

text: if

text: the

text: number

text: of

text: facets

text: of

text: the

text: simple

text: polytope

text: Pn

text: is

text: equal

text: to

text: m

text: n

text: p,

text: then

text: Z

text: N

text: is

text: equivariantly

text: diﬀeomorphic

text: to

text: the

text: moment

text: angle

text: manifold

text: P

text: Theorem

text: Let

text: Nn+m

text: be

text: a

text: simply

text: connected

text: principal

text: Tm

text: n-bundle

text: over

text: where

text: has

text: a

text: locally

text: standard,

text: smooth

text: Tn

text: action

text: with

text: orbit

text: space

text: Pn,

text: where

text: Pn

text: is

text: an

text: acyclic

text: manifold

text: with

text: acyclic

text: faces,

text: with

text: m

text: facets.

text: Then

text: Nn+m

text: is

text: equivariantly

text: diﬀeomorphic

text: to

text: Z

text: the

text: moment

text: angle

text: manifold

text: P

text: Proof.

text: The

text: proof

text: is

text: based

text: on

text: work

text: by

text: Davis

text: Let

text: Nn+m

text: be

text: a

text: principal

text: Tm

text: n-bundle

text: over

text: Such

text: bundles

text: are

text: classiﬁed

text: by

text: homotopy

text: classes

text: of

text: maps

text: of

text: into

text: BTm

text: n,


text: 28 ESCHER AND SEARLE Theorem 6.3. Let N2n+p be a simply connected principal Tp-bundle over M2n where M2n has a locally standard, smooth Tn action with orbit space Pn, where Pn is an acyclic − manifold with acyclic faces, with m facets and with p < m n. Then there is a smooth − − −→ principal Tm n p- bundle π : Y N2n+p and if Y is simply connected, it is Tm- equiv- ariantly diﬀeomorphic to the moment angle manifold Z . Hence N2n+p is diﬀeomorphic to Z − − P /Tm n p. P − − Proof. Let Y be a principal Tm n p-bundle over N. Such bundles are classiﬁed by homo- − − − − topy classes of maps of N into BTm n p, hence by [N, BTm n p]. But [N, BTm−n−p] ∼= [N, BS1] × ... × [N, BS1] ∼= ⊕m−n−pH2(N; Z). i=1 ∼ Z − Since N is a principal Tp-bundle over M2n, and H2(M2n) = m n, (see for example Theorem 7.4.35 in [5]), using the Leray-Serre spectral sequence or the homotopy sequence Z Z − − for the bundle we obtain that H2(N; ) contains m n p as a subgroup. We now use the − fact that Y is a Tm n-principal bundle over M2n since in the category of ﬁnite dimensional manifolds the composition of two principal bundles is again a principal bundle (see, for example, [40]). This means that we are now in the setting of Theorem 6.1 if Y is simply Z connected. Hence Y m+n is equivariantly diﬀeomorphic to the moment angle manifold P as claimed. The following diagram illustrates this case. Z n+m P − − /Tm n p (cid:118)(cid:118) N2n+p − /Tm n /Tp (cid:15)(cid:15) (cid:125)(cid:125) /Tn+p M2n /Tn (cid:31)(cid:31) (cid:15)(cid:15) Pn (cid:3) As in Case (1a), we now apply Theorem 6.3 to the case of a closed, simply-connected, principal Tp-bundle admitting a family of metrics of almost non-negative curvature over M2n, where M2n is a torus manifold with non-negatively curved orbit space Pn and this principal bundle satisﬁes Condition F. We obtain the following corollary of Theorem 6.3 by applying Theorem 4.4. Corollary 6.4. Let N2n+p be a closed, simply-connected, principal Tp-bundle admitting a family of metrics of almost non-negative curvature over M2n, where M2n is a torus manifold with non-negatively curved orbit space Pn as in Display (2.2), with m facets with − − − p < m n and N2n+p satisﬁes Condition F. Then there is a smooth principal Tm n p- −→ bundle π : Y N2n+p. Further, if Y is simply connected, then it is Tm-equivariantly Z diﬀeomorphic to the moment angle manifold , which is itself a product of spheres. Hence Z − − P N2n+p is diﬀeomorphic to /Tm n p. P ##LTLine## ##LTLine## ##LTLine## ##LTLine##

label: paragraph

text: ESCHER

text: AND

text: SEARLE

text: Theorem

text: Let

text: be

text: a

text: simply

text: connected

text: principal

text: Tp-bundle

text: over

text: where

text: has

text: a

text: locally

text: standard,

text: smooth

text: Tn

text: action

text: with

text: orbit

text: space

text: Pn,

text: where

text: Pn

text: is

text: an

text: acyclic

text: manifold

text: with

text: acyclic

text: faces,

text: with

text: m

text: facets

text: and

text: with

text: p

text: m

text: n.

text: Then

text: there

text: is

text: a

text: smooth

text: principal

text: Tm

text: n

text: p-

text: bundle

text: π

text: Y

text: and

text: if

text: Y

text: is

text: simply

text: connected,

text: it

text: is

text: Tm-

text: equiv-

text: ariantly

text: diﬀeomorphic

text: to

text: the

text: moment

text: angle

text: manifold

text: Z

text: Hence

text: is

text: diﬀeomorphic

text: to

text: Z

text: P

text: /Tm

text: n

text: p.

text: P

text: Proof.

text: Let

text: Y

text: be

text: a

text: principal

text: Tm

text: n

text: p-bundle

text: over

text: N.

text: Such

text: bundles

text: are

text: classiﬁed

text: by

text: homo-

text: topy

text: classes

text: of

text: maps

text: of

text: N

text: into

text: BTm

text: n

text: p,

text: hence

text: by

text: [N,

text: BTm

text: n

text: p].

text: But

text: [N,

text: BTm−n−p]

text: [N,

text: [N,

text: Z).

text: Z

text: Since

text: N

text: is

text: a

text: principal

text: Tp-bundle

text: over

text: and

text: m

text: n,

text: (see

text: for

text: example

text: Theorem

text: in

text: using

text: the

text: Leray-Serre

text: spectral

text: sequence

text: or

text: the

text: homotopy

text: sequence

text: Z

text: Z

text: for

text: the

text: bundle

text: we

text: obtain

text: that

text: contains

text: m

text: n

text: p

text: as

text: a

text: subgroup.

text: We

text: now

text: use

text: the

text: fact

text: that

text: Y

text: is

text: a

text: Tm

text: n-principal

text: bundle

text: over

text: since

text: in

text: the

text: category

text: of

text: ﬁnite

text: dimensional

text: manifolds

text: the

text: composition

text: of

text: two

text: principal

text: bundles

text: is

text: again

text: a

text: principal

text: bundle

text: (see,

text: for

text: example,

text: This

text: means

text: that

text: we

text: are

text: now

text: in

text: the

text: setting

text: of

text: Theorem

text: if

text: Y

text: is

text: simply

text: Z

text: connected.

text: Hence

text: Y

text: m+n

text: is

text: equivariantly

text: diﬀeomorphic

text: to

text: the

text: moment

text: angle

text: manifold

text: P

text: as

text: claimed.

text: The

text: following

text: diagram

text: illustrates

text: this

text: case.

text: Z

text: n+m

text: P

text: /Tm

text: n

text: p

text: /Tm

text: n

text: /Tp

text: /Tn+p

text: /Tn

text: Pn

text: As

text: in

text: Case

text: we

text: now

text: apply

text: Theorem

text: to

text: the

text: case

text: of

text: a

text: closed,

text: simply-connected,

text: principal

text: Tp-bundle

text: admitting

text: a

text: family

text: of

text: metrics

text: of

text: almost

text: non-negative

text: curvature

text: over

text: where

text: is

text: a

text: torus

text: manifold

text: with

text: non-negatively

text: curved

text: orbit

text: space

text: Pn

text: and

text: this

text: principal

text: bundle

text: satisﬁes

text: Condition

text: F.

text: We

text: obtain

text: the

text: following

text: corollary

text: of

text: Theorem

text: by

text: applying

text: Theorem

text: Corollary

text: Let

text: be

text: a

text: closed,

text: simply-connected,

text: principal

text: Tp-bundle

text: admitting

text: a

text: family

text: of

text: metrics

text: of

text: almost

text: non-negative

text: curvature

text: over

text: where

text: is

text: a

text: torus

text: manifold

text: with

text: non-negatively

text: curved

text: orbit

text: space

text: Pn

text: as

text: in

text: Display

text: with

text: m

text: facets

text: with

text: p

text: m

text: n

text: and

text: satisﬁes

text: Condition

text: F.

text: Then

text: there

text: is

text: a

text: smooth

text: principal

text: Tm

text: n

text: p-

text: bundle

text: π

text: Y

text: Further,

text: if

text: Y

text: is

text: simply

text: connected,

text: then

text: it

text: is

text: Tm-equivariantly

text: Z

text: diﬀeomorphic

text: to

text: the

text: moment

text: angle

text: manifold

text: which

text: is

text: itself

text: a

text: product

text: of

text: spheres.

text: Hence

text: Z

text: P

text: is

text: diﬀeomorphic

text: to

text: /Tm

text: n

text: p.

text: P

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: 6 ESCHER AND SEARLE ARCs grant #150353 and partial support from a Simons Foundation grant #355508. This material is based in part upon work supported by the National Science Foundation under Grant No. DMS-1440140 while the authors were in residence at the Mathematical Sciences Research Institute in Berkeley, California, during the Spring 2016 semester. 2. Preliminaries In this section we will gather basic results and facts about transformation groups, torus actions, torus manifolds, torus orbifolds, as well as results concerning G-invariant manifolds of non-negative and almost non-negative sectional curvature. { ∈ Let G be a compact Lie group acting on a smooth manifold M. We denote by G = g } ∈ { ∈ } (cid:39) x G : gx = x the isotropy group at x M and by G(x) = gx : g G G/G the orbit x of x. Orbits will be principal, exceptional or singular, depending on the relative size of their isotropy subgroups; namely, principal orbits correspond to those orbits with the smallest possible isotropy subgroup, an orbit is called exceptional when its isotropy subgroup is a ﬁnite extension of the principal isotropy subgroup and singular when its isotropy subgroup is of strictly larger dimension than that of the principal isotropy subgroup. ∩ The ineﬀective kernel of the action is the subgroup K = ∈ G . We say that G acts x M x eﬀectively on M if K is trivial. The action is called almost eﬀective if K is ﬁnite. The action is free if every isotropy group is trivial and almost free if every isotropy group is ﬁnite. As mentioned in the Introduction, the free rank of an action is the rank of the maximal subtorus that acts almost freely. In order to further distinguish between the case when the free rank corresponds to a free action and the case when it corresponds to an almost free action, we make the following deﬁnition. Deﬁnition 2.1 (Free Dimension). Suppose that the free rank of a Tk-action is equal to r, then we say that the free dimension is equal to the dimension of the largest subtorus of Tr that acts freely. { ∈ ∈ } We will sometimes denote the ﬁxed point set MG = x M : gx = x, g G of the G-action by Fix(M; G) and deﬁne its dimension as { } dim(Fix(M; G)) = max dim(N) : N is a connected component of Fix(M; G) . × → One measurement for the size of a transformation group G M M is the dimension of its orbit space M/G, also called the cohomogeneity of the action. This dimension is clearly ≥ constrained by the dimension of the ﬁxed point set MG of G in M. In fact, dim(M/G) dim(MG) + 1 for any non-trivial action. In light of this, the ﬁxed-point cohomogeneity of an action, denoted by cohomﬁx(M; G), is deﬁned by − − ≥ cohomﬁx(M; G) = dim(M/G) dim(MG) 1 0. A manifold with ﬁxed-point cohomogeneity 0 is also called a G-ﬁxed point homogeneous manifold. × · · · × Let G = H H = Hl act isometrically and eﬀectively on Mn. Let N denote the ⊂ 1 connected component of largest dimension in MH and let N N − denote the connected ≤ k k 1 component of largest dimension in NH− , for k l. We will call a manifold nested H-ﬁxed k 1 point homogeneous when there exists a tower of nested H-ﬁxed point sets, ⊂ ⊂ · · · ⊂ ⊂ N N − N M l l 1 1 such that H acts ﬁxed point homogeneously on Mn, and all induced actions of G/Hk on ≤ ≤ − N , for all 1 k l 1, are H-ﬁxed point homogeneous. k

text: ESCHER

text: AND

text: SEARLE

text: ARCs

text: grant

text: and

text: partial

text: support

text: from

text: a

text: Simons

text: Foundation

text: grant

text: This

text: material

text: is

text: based

text: in

text: part

text: upon

text: work

text: supported

text: by

text: the

text: National

text: Science

text: Foundation

text: under

text: Grant

text: No.

text: while

text: the

text: authors

text: were

text: in

text: residence

text: at

text: the

text: Mathematical

text: Sciences

text: Research

text: Institute

text: in

text: Berkeley,

text: California,

text: during

text: the

text: Spring

text: semester.

text: Preliminaries

text: In

text: this

text: section

text: we

text: will

text: gather

text: basic

text: results

text: and

text: facts

text: about

text: transformation

text: groups,

text: torus

text: actions,

text: torus

text: manifolds,

text: torus

text: orbifolds,

text: as

text: well

text: as

text: results

text: concerning

text: G-invariant

text: manifolds

text: of

text: non-negative

text: and

text: almost

text: non-negative

text: sectional

text: curvature.

text: Let

text: G

text: be

text: a

text: compact

text: Lie

text: group

text: acting

text: on

text: a

text: smooth

text: manifold

text: M.

text: We

text: denote

text: by

text: G

text: g

text: x

text: G

text: gx

text: x

text: the

text: isotropy

text: group

text: at

text: x

text: M

text: and

text: by

text: G(x)

text: gx

text: g

text: G

text: G/G

text: the

text: orbit

text: x

text: of

text: x.

text: Orbits

text: will

text: be

text: principal,

text: exceptional

text: or

text: singular,

text: depending

text: on

text: the

text: relative

text: size

text: of

text: their

text: isotropy

text: subgroups;

text: namely,

text: principal

text: orbits

text: correspond

text: to

text: those

text: orbits

text: with

text: the

text: smallest

text: possible

text: isotropy

text: subgroup,

text: an

text: orbit

text: is

text: called

text: exceptional

text: when

text: its

text: isotropy

text: subgroup

text: is

text: a

text: ﬁnite

text: extension

text: of

text: the

text: principal

text: isotropy

text: subgroup

text: and

text: singular

text: when

text: its

text: isotropy

text: subgroup

text: is

text: of

text: strictly

text: larger

text: dimension

text: than

text: that

text: of

text: the

text: principal

text: isotropy

text: subgroup.

text: The

text: ineﬀective

text: kernel

text: of

text: the

text: action

text: is

text: the

text: subgroup

text: K

text: G

text: We

text: say

text: that

text: G

text: acts

text: x

text: M

text: x

text: eﬀectively

text: on

text: M

text: if

text: K

text: is

text: trivial.

text: The

text: action

text: is

text: called

text: almost

text: eﬀective

text: if

text: K

text: is

text: ﬁnite.

text: The

text: action

text: is

text: free

text: if

text: every

text: isotropy

text: group

text: is

text: trivial

text: and

text: almost

text: free

text: if

text: every

text: isotropy

text: group

text: is

text: ﬁnite.

text: As

text: mentioned

text: in

text: the

text: Introduction,

text: the

text: free

text: rank

text: of

text: an

text: action

text: is

text: the

text: rank

text: of

text: the

text: maximal

text: subtorus

text: that

text: acts

text: almost

text: freely.

text: In

text: order

text: to

text: further

text: distinguish

text: between

text: the

text: case

text: when

text: the

text: free

text: rank

text: corresponds

text: to

text: a

text: free

text: action

text: and

text: the

text: case

text: when

text: it

text: corresponds

text: to

text: an

text: almost

text: free

text: action,

text: we

text: make

text: the

text: following

text: deﬁnition.

text: Deﬁnition

text: (Free

text: Dimension).

text: Suppose

text: that

text: the

text: free

text: rank

text: of

text: a

text: Tk-action

text: is

text: equal

text: to

text: r,

text: then

text: we

text: say

text: that

text: the

text: free

text: dimension

text: is

text: equal

text: to

text: the

text: dimension

text: of

text: the

text: largest

text: subtorus

text: of

text: Tr

text: that

text: acts

text: freely.

text: We

text: will

text: sometimes

text: denote

text: the

text: ﬁxed

text: point

text: set

text: MG

text: x

text: M

text: gx

text: x,

text: g

text: G

text: of

text: the

text: G-action

text: by

text: Fix(M;

text: G)

text: and

text: deﬁne

text: its

text: dimension

text: as

text: dim(Fix(M;

text: G))

text: max

text: dim(N)

text: N

text: is

text: a

text: connected

text: component

text: of

text: Fix(M;

text: G)

text: One

text: measurement

text: for

text: the

text: size

text: of

text: a

text: transformation

text: group

text: G

text: M

text: M

text: is

text: the

text: dimension

text: of

text: its

text: orbit

text: space

text: M/G,

text: also

text: called

text: the

text: cohomogeneity

text: of

text: the

text: action.

text: This

text: dimension

text: is

text: clearly

text: constrained

text: by

text: the

text: dimension

text: of

text: the

text: ﬁxed

text: point

text: set

text: MG

text: of

text: G

text: in

text: M.

text: In

text: fact,

text: dim(M/G)

text: dim(MG)

text: for

text: any

text: non-trivial

text: action.

text: In

text: light

text: of

text: this,

text: the

text: ﬁxed-point

text: cohomogeneity

text: of

text: an

text: action,

text: denoted

text: by

text: cohomﬁx(M;

text: G),

text: is

text: deﬁned

text: by

text: cohomﬁx(M;

text: G)

text: dim(M/G)

text: dim(MG)

text: A

text: manifold

text: with

text: ﬁxed-point

text: cohomogeneity

text: is

text: also

text: called

text: a

text: G-ﬁxed

text: point

text: homogeneous

text: manifold.

text: Let

text: G

text: H

text: H

text: Hl

text: act

text: isometrically

text: and

text: eﬀectively

text: on

text: Mn.

text: Let

text: N

text: denote

text: the

text: connected

text: component

text: of

text: largest

text: dimension

text: in

text: MH

text: and

text: let

text: N

text: N

text: denote

text: the

text: connected

text: k

text: k

text: component

text: of

text: largest

text: dimension

text: in

text: NH−

text: for

text: k

text: l.

text: We

text: will

text: call

text: a

text: manifold

text: nested

text: H-ﬁxed

text: k

text: point

text: homogeneous

text: when

text: there

text: exists

text: a

text: tower

text: of

text: nested

text: H-ﬁxed

text: point

text: sets,

text: N

text: N

text: N

text: M

text: l

text: l

text: such

text: that

text: H

text: acts

text: ﬁxed

text: point

text: homogeneously

text: on

text: Mn,

text: and

text: all

text: induced

text: actions

text: of

text: G/Hk

text: on

text: N

text: for

text: all

text: k

text: l

text: are

text: H-ﬁxed

text: point

text: homogeneous.

text: k


text: 13 V. Fortov, A. Ivlev, S. Khrapak, A. Khrapak, and G. Morﬁll, Phys. Rep. 421, 1 (2005). P. K. Shukla and B. Eliasson, Rev. Mod. Phys. 81, 25 (2009). M. Bonitz, C. Henning, and D. Block, Rep. Prog. Phys. 73, 066501 (2010). D. I. Zhukhovitskii, O. F. Petrov, T. W. Hyde, G. Herdrich, R. Laufer, M. Dropmann, and L. Matthews, New J. Phys. 17, 053041 (2015). G. E. Morﬁll, U. Konopka, M. Kretschmer, M. Rubin-Zuzic, H. M. Thomas, S. K. Zhdanov, and V. Tsytovich, New J. Phys. 8, 7 (2006). D. Caliebe, O. Arp, and A. Piel, Phys. Plasmas 18, 073702 (2011). A. Piel, O. Arp, M. Klindworth, and A. Melzer, Phys. Rev. E 77, 026407 (2008). K. O. Menzel, O. Arp, and A. Piel, Phys. Rev. E 83, 016402 (2011). O. Arp, D. Caliebe, and A. Piel, Phys. Rev. E 83, 066404 (2011). M. Schwabe, S. K. Zhdanov, H. M. Thomas, A. V. Ivlev, M. Rubin-Zuzic, G. E. Morﬁll, V. I. Molotkov, A. M. Lipaev, V. E. Fortov, and T. Reiter, New J. Phys. 10, 033037 (2008). G. E. Morﬁll, H. M. Thomas, U. Konopka, H. Rothermel, M. Zuzic, A. Ivlev, and J. Goree, Phys. Rev. Lett. 83, 1598 (1999). S. A. Khrapak, B. A. Klumov, P. Huber, V. I. Molotkov, A. M. Lipaev, V. N. Naumkin, H. M. Thomas, A. V. Ivlev, G. E. Morﬁll, O. F. Petrov, V. E. Fortov, Yu. Malentschenko, and S. Volkov, Phys. Rev. Lett. 106, 205001 (2011). H. M. Thomas, G. E. Morﬁll, V. E. Fortov, A. V. Ivlev, V. I. Molotkov, A. M. Lipaev, T. Hagl, H. Rothermel, S. A. Khrapak, R. K. Suetterlin, M. Rubin-Zuzic, O. F. Petrov, V. I. Tokarev, and S. K. Krikalev, New J. Phys. 10, 033036 (2008). K. Jiang, V. Nosenko, Y. F. Li, M. Schwabe, U. Konopka, A. V. Ivlev, V. E. Fortov, V. I. Molotkov, A. M. Lipaev, O. F. Petrov, M. V. Turin, H. M. Thomas, and G. E. Morﬁll, Europhys. Lett. 85, 45002 (2009). M. Schwabe, K. Jiang, S. Zhdanov, T. Hagl, P. Huber, A. V. Ivlev, A. M. Lipaev, V. I. Molotkov, V. N. Naumkin, K. R. Su¨tterlin, H. M. Thomas, V. E. Fortov, G. E. Morﬁll, A. Skvortsov, and S. Volkov, Europhys. Lett. 96, 55001 (2011). S. A. Khrapak, S. V. Ratynskaia, A. V. Zobnin, A. D. Usachev, V. V. Yaroshenko, M. H. Thoma, M. Kretschmer, H. H¨ofner, G. E. Morﬁll, O. F. Petrov, and V. E. Fortov, Phys. Rev. E 72, 016406 (2005). V. Fortov, G. Morﬁll, O. Petrov, M. Thoma, A. Usachev, H. Hoefner, A. Zobnin, M. Kretschmer, S. Ratynskaia, M. Fink, K. Tarantik, Yu. Gerasimov, and V. Esenkov, Plasma Phys. Control. Fusion 47, B537 (2005). M. A. Fink, M. H. Thoma, and G. E. Morﬁll, Microgravity Sci. Technol. 23, 169 (2011). A. V. Zobnin, A. D. Usachev, O. F. Petrov, and V. E. Fortov, Phys. Plasmas 21, 113503 (2014). N. N. Rao, P. K. Shukla, and M. Y. Yu, Planet. Space Sci. 38, 543 (1990). R. L. Merlino, J. Plasma Phys. 80, 773 (2014). O. Havnes, F. Li, F. Melandsø, T. Aslaksen, T. W. Hartquist, G. E. Morﬁll, T. Nitter, and V. Tsytovich, J. Vac. Sci. Technol. A 14, 525 (1996). D. Samsonov, J. Goree, Z. W. Ma, A. Bhattacharjee, H. M. Thomas, and G. E. Morﬁll, Phys. Rev. Lett. 83, 3649 (1999). D. Samsonov, J. Goree, H. M. Thomas, and G. E. Morﬁll, Phys. Rev. E 61, 5557 (2000). A. Melzer, S. Nunomura, D. Samsonov, Z. W. Ma, and J. Goree, Phys. Rev. E 62, 4162 (2000). V. Nosenko, J. Goree, , Z. W. Ma, and A. Piel, Phys. Rev. Lett. 88, 135001 (2002). V. Nosenko, J. Goree, Z. W. Ma, D. H. E. Dubin, and A. Piel, Phys. Rev. E 68, 056409 (2003). S. K. Zhdanov, D. Samsonov, and G. E. Morﬁll, Phys. Rev. E 66, 026411 (2002). S. K. Zhdanov, G. E. Morﬁll, D. Samsonov, M. Zuzic, and O. Havnes, Phys. Rev. E 69, 026407 (2004). D. I. Zhukhovitskii, V. E. Fortov, V. I. Molotkov, A. M. Lipaev, V. N. Naumkin, H. M. Thomas, A. V. Ivlev, M. Schwabe, and G. E. Morﬁll, Phys. Plasmas 22, 023701 (2015). D. I. Zhukhovitskii, V. I. Molotkov, and V. E. Fortov, Phys. Plasmas 21, 063701 (2014). L. D. Landau and E. M. Lifshitz, noop Statistical Physics (Elsevier, Oxford, 1980).

text: V.

text: Fortov,

text: A.

text: Ivlev,

text: S.

text: Khrapak,

text: A.

text: Khrapak,

text: and

text: G.

text: Morﬁll,

text: Phys.

text: Rep.

text: P.

text: K.

text: Shukla

text: and

text: B.

text: Eliasson,

text: Rev.

text: Mod.

text: Phys.

text: M.

text: Bonitz,

text: C.

text: Henning,

text: and

text: D.

text: Block,

text: Rep.

text: Prog.

text: Phys.

text: D.

text: I.

text: Zhukhovitskii,

text: O.

text: F.

text: Petrov,

text: T.

text: W.

text: Hyde,

text: G.

text: Herdrich,

text: R.

text: Laufer,

text: M.

text: Dropmann,

text: and

text: L.

text: Matthews,

text: New

text: J.

text: Phys.

text: G.

text: E.

text: Morﬁll,

text: U.

text: Konopka,

text: M.

text: Kretschmer,

text: M.

text: Rubin-Zuzic,

text: H.

text: M.

text: Thomas,

text: S.

text: K.

text: Zhdanov,

text: and

text: V.

text: Tsytovich,

text: New

text: J.

text: Phys.

text: D.

text: Caliebe,

text: O.

text: Arp,

text: and

text: A.

text: Piel,

text: Phys.

text: Plasmas

text: A.

text: Piel,

text: O.

text: Arp,

text: M.

text: Klindworth,

text: and

text: A.

text: Melzer,

text: Phys.

text: Rev.

text: E

text: K.

text: O.

text: Menzel,

text: O.

text: Arp,

text: and

text: A.

text: Piel,

text: Phys.

text: Rev.

text: E

text: O.

text: Arp,

text: D.

text: Caliebe,

text: and

text: A.

text: Piel,

text: Phys.

text: Rev.

text: E

text: M.

text: Schwabe,

text: S.

text: K.

text: Zhdanov,

text: H.

text: M.

text: Thomas,

text: A.

text: V.

text: Ivlev,

text: M.

text: Rubin-Zuzic,

text: G.

text: E.

text: Morﬁll,

text: V.

text: I.

text: Molotkov,

text: A.

text: M.

text: Lipaev,

text: V.

text: E.

text: Fortov,

text: and

text: T.

text: Reiter,

text: New

text: J.

text: Phys.

text: G.

text: E.

text: Morﬁll,

text: H.

text: M.

text: Thomas,

text: U.

text: Konopka,

text: H.

text: Rothermel,

text: M.

text: Zuzic,

text: A.

text: Ivlev,

text: and

text: J.

text: Goree,

text: Phys.

text: Rev.

text: Lett.

text: S.

text: A.

text: Khrapak,

text: B.

text: A.

text: Klumov,

text: P.

text: Huber,

text: V.

text: I.

text: Molotkov,

text: A.

text: M.

text: Lipaev,

text: V.

text: N.

text: Naumkin,

text: H.

text: M.

text: Thomas,

text: A.

text: V.

text: Ivlev,

text: G.

text: E.

text: Morﬁll,

text: O.

text: F.

text: Petrov,

text: V.

text: E.

text: Fortov,

text: Yu.

text: Malentschenko,

text: and

text: S.

text: Volkov,

text: Phys.

text: Rev.

text: Lett.

text: H.

text: M.

text: Thomas,

text: G.

text: E.

text: Morﬁll,

text: V.

text: E.

text: Fortov,

text: A.

text: V.

text: Ivlev,

text: V.

text: I.

text: Molotkov,

text: A.

text: M.

text: Lipaev,

text: T.

text: Hagl,

text: H.

text: Rothermel,

text: S.

text: A.

text: Khrapak,

text: R.

text: K.

text: Suetterlin,

text: M.

text: Rubin-Zuzic,

text: O.

text: F.

text: Petrov,

text: V.

text: I.

text: Tokarev,

text: and

text: S.

text: K.

text: Krikalev,

text: New

text: J.

text: Phys.

text: K.

text: Jiang,

text: V.

text: Nosenko,

text: Y.

text: F.

text: Li,

text: M.

text: Schwabe,

text: U.

text: Konopka,

text: A.

text: V.

text: Ivlev,

text: V.

text: E.

text: Fortov,

text: V.

text: I.

text: Molotkov,

text: A.

text: M.

text: Lipaev,

text: O.

text: F.

text: Petrov,

text: M.

text: V.

text: Turin,

text: H.

text: M.

text: Thomas,

text: and

text: G.

text: E.

text: Morﬁll,

text: Europhys.

text: Lett.

text: M.

text: Schwabe,

text: K.

text: Jiang,

text: S.

text: Zhdanov,

text: T.

text: Hagl,

text: P.

text: Huber,

text: A.

text: V.

text: Ivlev,

text: A.

text: M.

text: Lipaev,

text: V.

text: I.

text: Molotkov,

text: V.

text: N.

text: Naumkin,

text: K.

text: R.

text: Su¨tterlin,

text: H.

text: M.

text: Thomas,

text: V.

text: E.

text: Fortov,

text: G.

text: E.

text: Morﬁll,

text: A.

text: Skvortsov,

text: and

text: S.

text: Volkov,

text: Europhys.

text: Lett.

text: S.

text: A.

text: Khrapak,

text: S.

text: V.

text: Ratynskaia,

text: A.

text: V.

text: Zobnin,

text: A.

text: D.

text: Usachev,

text: V.

text: V.

text: Yaroshenko,

text: M.

text: H.

text: Thoma,

text: M.

text: Kretschmer,

text: H.

text: H¨ofner,

text: G.

text: E.

text: Morﬁll,

text: O.

text: F.

text: Petrov,

text: and

text: V.

text: E.

text: Fortov,

text: Phys.

text: Rev.

text: E

text: V.

text: Fortov,

text: G.

text: Morﬁll,

text: O.

text: Petrov,

text: M.

text: Thoma,

text: A.

text: Usachev,

text: H.

text: Hoefner,

text: A.

text: Zobnin,

text: M.

text: Kretschmer,

text: S.

text: Ratynskaia,

text: M.

text: Fink,

text: K.

text: Tarantik,

text: Yu.

text: Gerasimov,

text: and

text: V.

text: Esenkov,

text: Plasma

text: Phys.

text: Control.

text: Fusion

text: M.

text: A.

text: Fink,

text: M.

text: H.

text: Thoma,

text: and

text: G.

text: E.

text: Morﬁll,

text: Microgravity

text: Sci.

text: Technol.

text: A.

text: V.

text: Zobnin,

text: A.

text: D.

text: Usachev,

text: O.

text: F.

text: Petrov,

text: and

text: V.

text: E.

text: Fortov,

text: Phys.

text: Plasmas

text: N.

text: N.

text: Rao,

text: P.

text: K.

text: Shukla,

text: and

text: M.

text: Y.

text: Yu,

text: Planet.

text: Space

text: Sci.

text: R.

text: L.

text: Merlino,

text: J.

text: Plasma

text: Phys.

text: O.

text: Havnes,

text: F.

text: Li,

text: F.

text: Melandsø,

text: T.

text: Aslaksen,

text: T.

text: W.

text: Hartquist,

text: G.

text: E.

text: Morﬁll,

text: T.

text: Nitter,

text: and

text: V.

text: Tsytovich,

text: J.

text: Vac.

text: Sci.

text: Technol.

text: A

text: D.

text: Samsonov,

text: J.

text: Goree,

text: Z.

text: W.

text: Ma,

text: A.

text: Bhattacharjee,

text: H.

text: M.

text: Thomas,

text: and

text: G.

text: E.

text: Morﬁll,

text: Phys.

text: Rev.

text: Lett.

text: D.

text: Samsonov,

text: J.

text: Goree,

text: H.

text: M.

text: Thomas,

text: and

text: G.

text: E.

text: Morﬁll,

text: Phys.

text: Rev.

text: E

text: A.

text: Melzer,

text: S.

text: Nunomura,

text: D.

text: Samsonov,

text: Z.

text: W.

text: Ma,

text: and

text: J.

text: Goree,

text: Phys.

text: Rev.

text: E

text: V.

text: Nosenko,

text: J.

text: Goree,

text: Z.

text: W.

text: Ma,

text: and

text: A.

text: Piel,

text: Phys.

text: Rev.

text: Lett.

text: V.

text: Nosenko,

text: J.

text: Goree,

text: Z.

text: W.

text: Ma,

text: D.

text: H.

text: E.

text: Dubin,

text: and

text: A.

text: Piel,

text: Phys.

text: Rev.

text: E

text: S.

text: K.

text: Zhdanov,

text: D.

text: Samsonov,

text: and

text: G.

text: E.

text: Morﬁll,

text: Phys.

text: Rev.

text: E

text: S.

text: K.

text: Zhdanov,

text: G.

text: E.

text: Morﬁll,

text: D.

text: Samsonov,

text: M.

text: Zuzic,

text: and

text: O.

text: Havnes,

text: Phys.

text: Rev.

text: E

text: D.

text: I.

text: Zhukhovitskii,

text: V.

text: E.

text: Fortov,

text: V.

text: I.

text: Molotkov,

text: A.

text: M.

text: Lipaev,

text: V.

text: N.

text: Naumkin,

text: H.

text: M.

text: Thomas,

text: A.

text: V.

text: Ivlev,

text: M.

text: Schwabe,

text: and

text: G.

text: E.

text: Morﬁll,

text: Phys.

text: Plasmas

text: D.

text: I.

text: Zhukhovitskii,

text: V.

text: I.

text: Molotkov,

text: and

text: V.

text: E.

text: Fortov,

text: Phys.

text: Plasmas

text: L.

text: D.

text: Landau

text: and

text: E.

text: M.

text: Lifshitz,

text: noop

text: Statistical

text: Physics

text: (Elsevier,

text: Oxford,


text: 8 and ∇ (cid:48) ∂v − p + νv = . (28) ∂t ρ 0 ∇ We represent the velocity in the form v = ψ to deriv(cid:18)e from (28(cid:19)) (cid:48) − ∂ψ p = ρ + νψ . (29) 0 ∂t Substitution of (29) into (27) yields the wave equation ∂2ψ ∂ψ + ν = c2∆ψ, (30) ∂t2 ∂t − · where c is the sound velocity. If the sought solution has the form ψ = Aei(ωt k r), where k is the wave vector, then − we arrive at the dispersion relation c2k2 = ω(cid:112)2 iων or (cid:112) √ √ − ck 1√+ ν˜2 + 1 − 1√+ ν˜2 1 = i , (31) ω 2 2 (cid:28) where ν˜ = ν/ω. If ν˜ 1 then ck (cid:39) ν˜2 − ν˜ 1 + i . (32) ω 8 2 It is seen from (32) that the sound velocity is not much diﬀerent from c, and the damping length cν/2 is deﬁned by a half of the damping frequency ν. One can calculate c by diﬀerentiation of (5) with res(cid:18)pect t(cid:19)o ρ = Mnd, ∗ ∗ − dp ∗ dn 1 4 ∗ ∗ = 2Φn 4/3 + Φ2n 1/3. (33) dn dΦ 3 ∗ We diﬀerentiate n with due regard for (22), ∗ dn − ∗ = 8θn 2/3eΦ(1 + Φ), (34) dΦ to arrive at ∗ (cid:34) (cid:35) √aT c c = e , (cid:0) (cid:1) e 6Mλ ∗ (35) ∗ 4 ∗ − 3 n 1/3 c 2 = Φ2n 1/3 1 − ∗ , 3 2 (Φ + 1) 8 3n 1/3 ∗ − where n 1/3 = (8/3)(1 θΦeΦ). Note that the sound velocity (along with the particle compressibility) is a continuous function at the second singular point. ∗ Using (13) and (35) one can easily show that c2(Φ ) = 0, c2 > 0 at Φ > Φ , and c2 < 0 at Φ < Φ . Since n (Φ) is a c c c decreasing function, in the latter case n > n . We can conclude that the corresponding branch of solutions of Eq. (12) c relates to an unstable state of the dust cloud with a negative compressibility. Such a state would tend to collapse ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: and

label: equation

text: ∂v

text: p

text: νv

text: ∂t

text: ρ

text: We

text: represent

text: the

text: velocity

text: in

text: the

text: form

text: v

text: ψ

text: to

text: from

text: ∂ψ

text: p

text: ρ

text: νψ

text: ∂t

text: Substitution

text: of

text: into

text: yields

text: the

text: wave

text: equation

text: ∂2ψ

text: ∂ψ

text: ν

text: ∂t

text: where

text: c

text: is

text: the

text: sound

text: velocity.

text: If

text: the

text: sought

text: solution

text: has

text: the

text: form

text: ψ

text: Aei(ωt

text: k

text: r),

text: where

text: k

text: is

text: the

text: wave

text: vector,

text: then

text: we

text: arrive

text: at

text: the

text: dispersion

text: relation

text: iων

text: or

text: ck

text: ν˜2

text: ν˜2

text: i

text: ω

text: where

text: ν˜

text: ν/ω.

text: If

text: ν˜

text: then

text: ck

text: ν˜2

text: ν˜

text: i

text: ω

text: It

text: is

text: seen

text: from

text: that

text: the

text: sound

text: velocity

text: is

text: not

text: much

text: diﬀerent

text: from

text: c,

text: and

text: the

text: damping

text: length

text: is

text: deﬁned

text: by

text: a

text: half

text: of

text: the

text: damping

text: frequency

text: ν.

text: One

text: can

text: calculate

text: c

text: by

text: diﬀerentiation

text: of

text: with

text: ρ

text: Mnd,

text: dp

text: dn

text: dn

text: dΦ

text: We

text: diﬀerentiate

text: n

text: with

text: due

text: regard

text: for

text: dn

text: Φ),

text: dΦ

text: to

text: arrive

text: at

text: √aT

text: c

text: c

text: e

text: e

text: n

text: c

text: (Φ

text: where

text: n

text: θΦeΦ).

text: Note

text: that

text: the

text: sound

text: velocity

text: (along

text: with

text: the

text: particle

text: compressibility)

text: is

text: a

text: continuous

text: function

text: at

text: the

text: second

text: singular

text: point.

text: Using

text: and

text: one

text: can

text: easily

text: show

text: that

text: at

text: Φ

text: Φ

text: and

text: at

text: Φ

text: Φ

text: Since

text: n

text: (Φ)

text: is

text: a

text: c

text: c

text: c

text: decreasing

text: function,

text: in

text: the

text: latter

text: case

text: n

text: n

text: We

text: can

text: conclude

text: that

text: the

text: corresponding

text: branch

text: of

text: solutions

text: of

text: Eq.

text: c

text: relates

text: to

text: an

text: unstable

text: state

text: of

text: the

text: dust

text: cloud

text: with

text: a

text: negative

text: compressibility.

text: Such

text: a

text: state

text: would

text: tend

text: to

text: collapse

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: SEMI-SIMPLE LIE GROUPS ACTING CONFORMALLY ON COMPACT LORENTZ MANIFOLDS VINCENT PECASTAING May 27, 2020 Abstract. We give a classiﬁcation, up to local isomorphisms, of semi-simple Lie groups without compact factors that can act faithfully and conformally on a compact Lorentz manifold of dimension greater than or equal to 3. Contents 1. Introduction 1 2. Background 3 3. Proof of the main theorem 8 References 21 1. Introduction In the middle of the 1980’s, Zimmer proved a deep result on diﬀerentiable actions of Lie groups that preserve some geometric structure, called Zimmer’s embedding theorem ([Zim86], Theorem A). Let G be a real linear algebraic group and H a non-compact simple Lie group. Assuming that H acts on a manifold M so as to preserve a G-structure and a ﬁnite volume, Zimmer’s theorem gives strong algebraic constraints on H, a noticeable one being that H can be locally embedded into G. In the same paper, Zimmer gave a striking c(orolla)ry to his theorem in the special case where the G-structure is deﬁned by a Lorentz metric. Theorem ([Zim86], Theorem B). Up to local isomorphism, PSL 2,R is the o∶n=ly no(n-com)pact simple Lie group that can act faithfully and isometrically on a Lorentz manifold of ﬁnite volume. A compact Lorentz manifold admitting an isometr<ic and faithful action of H PSL 2,R= c/an be easily built. Let g( be th)e Killing metric of H. It has Lorentz signature and is invariant under K left and right translations of H on itself. Choose Γ H any uniform lattice, and set M H Γ. The action of Γ on H,g by right translations being iso=met/ric, g induces a Lorentz metric K K g on M. Moreover, the action of H on itself by left translations is also isometric for g and K centralizes the right translations. Therefore, H acts on M H Γ by isometries of g. Note that this situation contrasts with the analogous Riemannian situation: the isometry group of a compact Riemannian manifold is always a compact Lie group (it follows from Ascoli’s theorem). Thus, what we observe here is that although preserving a Lorentzian metric tensor is less restrictive than preserving a Riemannian one, it is still a rigid condition that allows us to classify groups of isometries. In fact, Zimmer’s result led Adams, Stuck and - independently - Zeghib to the full classiﬁcation up to local isomorphism of the isometry group of a compact Lorentz manifold, see [AS97], [Zeg98]. Thus, we have a good understanding of the groups of 1

text: SEMI-SIMPLE

text: LIE

text: GROUPS

text: ACTING

text: CONFORMALLY

text: ON

text: COMPACT

text: LORENTZ

text: MANIFOLDS

text: VINCENT

text: PECASTAING

text: May

text: Abstract.

text: We

label: abstract

text: give

text: a

text: classiﬁcation,

text: up

text: to

text: local

text: isomorphisms,

text: of

text: semi-simple

text: Lie

text: groups

text: without

text: compact

text: factors

text: that

text: can

text: act

text: faithfully

text: and

text: conformally

text: on

text: a

text: compact

text: Lorentz

text: manifold

text: of

text: dimension

text: greater

text: than

text: or

text: equal

text: to

text: Contents

text: Introduction

text: Background

text: Proof

text: of

text: the

text: main

text: theorem

text: References

text: Introduction

text: In

text: the

text: middle

text: of

text: the

text: Zimmer

text: proved

text: a

text: deep

text: result

text: on

text: diﬀerentiable

text: actions

text: of

text: Lie

text: groups

text: that

text: preserve

text: some

text: geometric

text: structure,

text: called

text: Zimmer’s

text: embedding

text: theorem

text: Theorem

text: A).

text: Let

text: G

text: be

text: a

text: real

text: linear

text: algebraic

text: group

text: and

text: H

text: a

text: non-compact

text: simple

text: Lie

text: group.

text: Assuming

text: that

text: H

text: acts

text: on

text: a

text: manifold

text: M

text: so

text: as

text: to

text: preserve

text: a

text: G-structure

text: and

text: a

text: ﬁnite

text: volume,

text: Zimmer’s

text: theorem

text: gives

text: strong

text: algebraic

text: constraints

text: on

text: H,

text: a

text: noticeable

text: one

text: being

text: that

text: H

text: can

text: be

text: locally

text: embedded

text: into

text: G.

text: In

text: the

text: same

text: paper,

text: Zimmer

text: gave

text: a

text: striking

text: c(orolla)ry

text: to

text: his

text: theorem

text: in

text: the

text: special

text: case

text: where

text: the

text: G-structure

text: is

text: deﬁned

text: by

text: a

text: Lorentz

text: metric.

text: Theorem

text: Theorem

text: B).

text: Up

text: to

text: local

text: isomorphism,

text: PSL

text: is

text: the

text: o∶n=ly

text: no(n-com)pact

text: simple

text: Lie

text: group

text: that

text: can

text: act

text: faithfully

text: and

text: isometrically

text: on

text: a

text: Lorentz

text: manifold

text: of

text: ﬁnite

text: volume.

text: A

text: compact

text: Lorentz

text: manifold

text: admitting

text: an

text: isometr<ic

text: and

text: faithful

text: action

text: of

text: H

text: PSL

text: c/an

text: be

text: easily

text: built.

text: Let

text: g(

text: be

text: th)e

text: Killing

text: metric

text: of

text: H.

text: It

text: has

text: Lorentz

text: signature

text: and

text: is

text: invariant

text: under

text: K

text: left

text: and

text: right

text: translations

text: of

text: H

text: on

text: itself.

text: Choose

text: Γ

text: H

text: any

text: uniform

text: lattice,

text: and

text: set

text: M

text: H

text: Γ.

text: The

text: action

text: of

text: Γ

text: on

text: H,g

text: by

text: right

text: translations

text: being

text: iso=met/ric,

text: g

text: induces

text: a

text: Lorentz

text: metric

text: K

text: K

text: g

text: on

text: M.

text: Moreover,

text: the

text: action

text: of

text: H

text: on

text: itself

text: by

text: left

text: translations

text: is

text: also

text: isometric

text: for

text: g

text: and

text: K

text: centralizes

text: the

text: right

text: translations.

text: Therefore,

text: H

text: acts

text: on

text: M

text: H

text: Γ

text: by

text: isometries

text: of

text: g.

text: Note

text: that

text: this

text: situation

text: contrasts

text: with

text: the

text: analogous

text: Riemannian

text: situation:

text: the

text: isometry

text: group

text: of

text: a

text: compact

text: Riemannian

text: manifold

text: is

text: always

text: a

text: compact

text: Lie

text: group

text: (it

text: follows

text: from

text: Ascoli’s

text: theorem).

text: Thus,

text: what

text: we

text: observe

text: here

text: is

text: that

text: although

text: preserving

text: a

text: Lorentzian

text: metric

text: tensor

text: is

text: less

text: restrictive

text: than

text: preserving

text: a

text: Riemannian

text: one,

text: it

text: is

text: still

text: a

text: rigid

text: condition

text: that

text: allows

text: us

text: to

text: classify

text: groups

text: of

text: isometries.

text: In

text: fact,

text: Zimmer’s

text: result

text: led

text: Adams,

text: Stuck

text: and

text: independently

text: Zeghib

text: to

text: the

text: full

text: classiﬁcation

text: up

text: to

text: local

text: isomorphism

text: of

text: the

text: isometry

text: group

text: of

text: a

text: compact

text: Lorentz

text: manifold,

text: see

text: Thus,

text: we

text: have

text: a

text: good

text: understanding

text: of

text: the

text: groups

text: of


text: ( ) SEMI-SIMPLE LIE GROUPS ACTING CONFORMALLY ON COMP−ACT LORENTZ MANIFOLDS 11 = ⊕ ⊕ ⊕ ⊕ ⊂ ⊕ For an(y)such H−, be it −locally isomorphic to Sp 1,k or to F 20, its root-space decomposition 4 has the form h h h h h h and there exists a subalgebra s h h isomorphic 2α α 0 α 2α α 2α to heisH 7 (Lemmas 3.2( a)nd⊂ 3.3)(. )We choose and ﬁx such an s, and note S the corresponding connected Lie subgroup of H. The linear group Ad S GL h is unipotent. Thus, it is algebraic and do not contain any h cocompact algebraic subgroup: this means that S is discompact. Moreover, this subgroup is amenable (since it is solvable) and by compactness of M, there exists a ﬁnite measure µ which is preserved by the S-action on M. Therefore, Theorem 2 ensures that ther̂e <exists Zimmer points for S. Let x be one of ∶thêm→. ( ) Proposition 2.6 then shows that there exists an algebraic subgroup Px P and an algebraic surjective m̂ orphism ρ Px Adh S . Our work consists in proving tĥa=t (such)a⋉morphism does not exist. ̂ Since Px is algebraic, it admits an algeb(ra)ic Levi decomposition Px LT U, where L, T and U are algebrai(c s)ub<group(s o)f Px, with L real semi-simple, U unipotent and T a torus (see [Mor05], Theorem 4.4.7). The group Adh S bein̂g itself a linear unipotent group, the semi- simp(le)subgroup ρ L Ad S has to be trivial. Moreover(, ρ) b=e{ing}an algebraic morphism, h it∣ se∶nds→ellipt(ic )(resp. hyperbolic) elements of Px on elliptic (resp.̂ hyperbolic) elements of Ad S (see [Mor05], Corollary 4.3.6). Thus, we must have ρ T id . Finally, the restriction h ρ U Ad S must be onto: it is not restrictive to assume that Px is unipotent. U h + + +⋯ + < ( ) Cho+ose coord+inat[es ∶x ∶,⋯...∶,x] on R2,n in such a way that t(he q)uadratic form is written 1 n 2 2x x 2x x x2 x2 and such that the parabolic subgroup P PO 2,n is the stabilizer of 1then i2sotrop2icnli1ne 13 ⎛0 n 0 . In such a ba⋯sis, any matrix of so 2,n⎞has the form ⎜ ⋯ − − ⎟ ⎜ ⎟ ⎜ − − − − − ⎟ ⎜ a b u u α 0 ⎟ ⎜⎜ c⋮ d⋮ v1 vn 2 0⋮ ⋮α ⎟⎟ ⎜ 1 n 2 ⎟ ⎜⎜⎜− z1− − w1− ⋯ −−v1− − −u1− ⎟⎟⎟ A ⎝ − ⋯ − − − ⎠ z w v u n 2 n 2 − n 2 n 2 β 0 w w d b ( − ) 1 n 2 0 β z z = c= ⋯ = a = = 1 n 2 where all the letters denote real numbers, except A which is an elem−ent of so n 2 . Moreover the Lie algebra of P corresponds to matrices verifying c z z β 0. 1 n 2 In such coordinates, let⎛umax be the⋯unipotent subalgebra o⎞f p composed with the strictly upper-triangular matrices ⎜ ⋯ − − ⎟ ⎜ ⎟ ⎜ − − − ⎟ ⎜0 t u u α 0 ⎟ ⎜⎜ 0 v1 vn 2 0⋮ ⋮α ⎟⎟ ∈ ⎜ 1 n 2 ⎟ ⎜⎜⎜ − v1− − −u1− ⎟⎟⎟ 0 p. ⎝ ⎠ v u n 2 n 2 0 t 0 ⋉ ( − ) Fact. This Lie algebra u is isomorphic to a semi-direct product R heisC 2n 3 . max ( − ) ⊕ − ∩ { = } Proof. Recall that heisC 2n 3 is Cn 2 R.Z endowed with a nilpotent Lie algebra bracket such that R.Z is the center. A basic linear algebra computation shows that u t 0 is max

text: SEMI-SIMPLE

text: LIE

text: GROUPS

text: ACTING

text: CONFORMALLY

text: ON

text: COMP−ACT

text: LORENTZ

text: MANIFOLDS

text: For

text: an(y)such

text: H−,

text: be

text: it

text: −locally

text: isomorphic

text: to

text: Sp

text: or

text: to

text: F

text: its

text: root-space

text: decomposition

text: has

text: the

text: form

text: h

text: h

text: h

text: h

text: h

text: h

text: and

text: there

text: exists

text: a

text: subalgebra

text: s

text: h

text: h

text: isomorphic

text: 2α

text: α

text: α

text: 2α

text: α

text: 2α

text: to

text: heisH

text: (Lemmas

text: a)nd⊂

text: )We

text: choose

text: and

text: ﬁx

text: such

text: an

text: s,

text: and

text: note

text: S

text: the

text: corresponding

text: connected

text: Lie

text: subgroup

text: of

text: H.

text: The

text: linear

text: group

text: Ad

text: S

text: GL

text: h

text: is

text: unipotent.

text: Thus,

text: it

text: is

text: algebraic

text: and

text: do

text: not

text: contain

text: any

text: h

text: cocompact

text: algebraic

text: subgroup:

text: this

text: means

text: that

text: S

text: is

text: discompact.

text: Moreover,

text: this

text: subgroup

text: is

text: amenable

text: (since

text: it

text: is

text: solvable)

text: and

text: by

text: compactness

text: of

text: M,

text: there

text: exists

text: a

text: ﬁnite

text: measure

text: µ

text: which

text: is

text: preserved

text: by

text: the

text: S-action

text: on

text: M.

text: Therefore,

text: Theorem

text: ensures

text: that

text: ther̂e

text: <exists

text: Zimmer

text: points

text: for

text: S.

text: Let

text: x

text: be

text: one

text: of

text: ∶thêm→.

text: Proposition

text: then

text: shows

text: that

text: there

text: exists

text: an

text: algebraic

text: subgroup

text: Px

text: P

text: and

text: an

text: algebraic

text: surjective

text: m̂

text: orphism

text: ρ

text: Px

text: Adh

text: S

text: Our

text: work

text: consists

text: in

text: proving

text: tĥa=t

text: (such)a⋉morphism

text: does

text: not

text: exist.

text: Since

text: Px

text: is

text: algebraic,

text: it

text: admits

text: an

text: algeb(ra)ic

text: Levi

text: decomposition

text: Px

text: LT

text: U,

text: where

text: L,

text: T

text: and

text: U

text: are

text: algebrai(c

text: s)ub<group(s

text: o)f

text: Px,

text: with

text: L

text: real

text: semi-simple,

text: U

text: unipotent

text: and

text: T

text: a

text: torus

text: (see

text: Theorem

text: The

text: group

text: Adh

text: S

text: bein̂g

text: itself

text: a

text: linear

text: unipotent

text: group,

text: the

text: semi-

text: simp(le)subgroup

text: ρ

text: L

text: Ad

text: S

text: has

text: to

text: be

text: trivial.

text: Moreover(,

text: ρ)

text: b=e{ing}an

text: algebraic

text: morphism,

text: h

text: it∣

text: se∶nds→ellipt(ic

text: )(resp.

text: hyperbolic)

text: elements

text: of

text: Px

text: on

text: elliptic

text: (resp.̂

text: hyperbolic)

text: elements

text: of

text: Ad

text: S

text: (see

text: Corollary

text: Thus,

text: we

text: must

text: have

text: ρ

text: T

text: id

text: Finally,

text: the

text: restriction

text: h

text: ρ

text: U

text: Ad

text: S

text: must

text: be

text: onto:

text: it

text: is

text: not

text: restrictive

text: to

text: assume

text: that

text: Px

text: is

text: unipotent.

text: U

text: h

text: Cho+ose

text: coord+inat[es

text: ∶x

text: ∶,⋯...∶,x]

text: on

text: in

text: such

text: a

text: way

text: that

text: t(he

text: q)uadratic

text: form

text: is

text: written

text: n

text: x

text: x

text: and

text: such

text: that

text: the

text: parabolic

text: subgroup

text: P

text: PO

text: is

text: the

text: stabilizer

text: of

text: ⎛0

text: n

text: In

text: such

text: a

text: ba⋯sis,

text: any

text: matrix

text: of

text: so

text: the

text: form

text: a

text: b

text: u

text: u

text: α

text: c⋮

text: d⋮

text: vn

text: 0⋮

text: ⋮α

text: n

text: −−v1−

text: −u1−

text: A

text: z

text: w

text: v

text: u

text: n

text: n

text: n

text: n

text: β

text: w

text: w

text: d

text: b

text: n

text: β

text: z

text: z

text: c=

text: a

text: n

text: where

text: all

text: the

text: letters

text: denote

text: real

text: numbers,

text: except

text: A

text: which

text: is

text: an

text: elem−ent

text: of

text: so

text: n

text: Moreover

text: the

text: Lie

text: algebra

text: of

text: P

text: corresponds

text: to

text: matrices

text: verifying

text: c

text: z

text: z

text: β

text: n

text: In

text: such

text: coordinates,

text: let⎛umax

text: be

text: the⋯unipotent

text: subalgebra

text: o⎞f

text: p

text: composed

text: with

text: the

text: strictly

text: upper-triangular

text: matrices

text: ⎜0

text: t

text: u

text: u

text: α

text: vn

text: 0⋮

text: ⋮α

text: n

text: −u1−

text: p.

text: v

text: u

text: n

text: n

text: t

text: Fact.

text: This

text: Lie

text: algebra

text: u

text: is

text: isomorphic

text: to

text: a

text: semi-direct

text: product

text: R

text: heisC

text: max

text: Proof.

text: Recall

text: that

text: heisC

text: is

text: Cn

text: R.Z

text: endowed

text: with

text: a

text: nilpotent

text: Lie

text: algebra

text: bracket

text: such

text: that

text: R.Z

text: is

text: the

text: center.

text: A

text: basic

text: linear

text: algebra

text: computation

text: shows

text: that

text: u

text: t

text: is

text: max


text: ( ) SEMI-SIMPLE LIE GROUPS ACTING CONFORMALLY ON COMPACT LORENTZ MANIFOLDS 16 ⊕ Pr(oof−. ()1) An el−ementary computation usi±ng the linear representation of so 1,k given in Section shows that the ad∶joint↦ac−tion of h on h is conjugated to the standard linear action of R 0 α so k 1 on R∈k 1′ ∩(R acting by homothet∈ies). ( ) ∈ [ ( ) ] ∈ ′ ∩ (2) We note θ X− tX a Cartan invo−lution (with respect to our root-space decomposi- t[i[on).( Le)t] Y ] =h h( no)n-z−ero an(d X ) h . Since θ( X ) h∶=, we( hav(e ))θ X ,Y h h . α α α 0 An elementary matrix computation using the linear representations given in Section≠ shows d[[eYﬁ,nθi(teX)q]u,aYd]ra∈tic′B= θf[orXm,YonY] h∈ s12i′nBcθe =Yθ[,Yis Xa −,C] wa⊂rhtea′rne iBnvθoXlu,tYion).⊂ TT′rheXreθfoYre,⊂s(ini′tceisYa ne(0gatainv)de Y,θ X ,Y h , we− get X h , proving h h . Using the matrix representation of so 1,k , α ( [ ]) < (cid:3) we verify that h h ,h and h h ,h . We then have h h , and h h . 0 α α α 0 α = ⊕ 0 α Recall that we assume that H acts faithfully and conformally on M, g . Let S H be the connected Lie subgroup whose Lie algebra is s a h . It will not be diﬃcult to see that S is α discompact and amenable, ensuring the existence of a Zimmer point for S. We ﬁrst establish the following proposition on the possible orbits of such a point. ∈ < ⩾ Propositi⊕on 3.12. Let x M be a Zimmer point for S and H H be the stabilizer of x. If x k 4, then⊕the⊕Lie algebra h is one of the following Lie subagebras of h: x (1) m h ; α (2) a= m⊕ h⊕ ; α − (3) h. When h a m h , the orbit H.x is conformal to the standard Riemannian round sphere Sk 1. x α ∈ ( ) = ∶= ( ) ∈ ( / ) [ ] Proo/f. We keep the same notations for the root-space d∈ecompo(si/tion[. L])et A a such that α A 1 and u≠ Ad etA GL h h . If q is the conformal [class ]of⊂subLorentzian metrics t x x on h h that we considered in Prop∩osition= 3.9, we have u C∶onf→h h/ , q . Assux(me )h= −h. ⩾Since x is a Zimm− er point for S,( wte )have s,hx xh (Lemma 3.7) and with Lem−max3.1−1, this implies hx h α 0. If we note−πx h h xhx thexnatu∗ral =pro−jection, dimπ h k 1 3 and the restriction of q to π h can(not⊕vani)sh⊂ide/ntically. Since u x α x x α t coincides with e t id on this latter spac(e an⊕d is)c⩽onformal for q , we must have u q e 2tq . x t x x The●refore, q must vani⊂sh identically (on )the subspace π h h h h .∩ Sin⩾ce −q i>s x x 0 α x x subLorentzian, this means that dimπ( h) h 1. We then distinguish two cases. x 0 α T[ he ﬁr∩st on]e=is h⊂ h . Since π =h ⊕is at most 1-dimensional, dimh h k 2 0 x x α x α ● 0. The adjoint action o⊄f ad h on h being ir=reduci⊕ble (Lemma 3.11), we then have 0 α h ,h h h =h . ⊕Finally, h h⩾ h ≃and(we−ar)e in situation (2). 0 x α α x x 0 α The second is when h =h . We then have h h h where h is a codimension 0 x x 0,x α 0,x 1 subalgebra o=f h⊕ ⊕a m. Since k> 4, m so k 1 does not admit a codimension 1 0 subalgebra,(prov)ing h ( m a)nd we are in sit(uatio)n (1). ( ) 0,x Now, assume that h ( a )m→ h . (Sinc)e k 2, there are only two possibilities for H: it is either x α isomorph=ic to P(SO)1,k or Spin 1,k . In fac−t, ≃Spin 1,k− is the universal cover of PSO 1,k , and∶=th[e c∶over∶in⋯g ∶Sp]in 1,k PSO 1,k is 2-sheeted. ( ) If H PSO 1,k , i/t naturally acts on Sk 1 Ein−0,k 1 (see Section ) and the stabilizer of x 1= 0 ( 0) is the only closed subgroup of P∶ SO 1(,k )w→hose L(ie al)gebra is h . The 0 x homogeneous sp∈ace−H Hx is t(hus )diﬀeomorphic to Sk 1. / → − If H Spin 1,k , the projection o>f the covering p Spin 1,k PSO 1,k sends H to the x stabilizer of x Sk 1 in PSO 1,k . Therefore, p induces a local diﬀeomorphism H H Sk 1. 0 x This must be a covering, and since k 2, a diﬀeomorphism. ##LTLine## ##LTLine##

text: SEMI-SIMPLE

text: LIE

text: GROUPS

text: ACTING

text: CONFORMALLY

text: ON

text: COMPACT

text: LORENTZ

text: MANIFOLDS

text: Pr(oof−.

text: An

text: el−ementary

text: computation

text: usi±ng

text: the

text: linear

text: representation

text: of

text: so

text: given

text: in

text: Section

text: shows

text: that

text: the

text: ad∶joint↦ac−tion

text: of

text: h

text: on

text: h

text: is

text: conjugated

text: to

text: the

text: standard

text: linear

text: action

text: of

text: R

text: α

text: so

text: k

text: on

text: R∈k

text: 1′

text: ∩(R

text: acting

text: by

text: homothet∈ies).

text: We

text: note

text: θ

text: X−

text: tX

text: a

text: Cartan

text: invo−lution

text: (with

text: respect

text: to

text: our

text: root-space

text: decomposi-

text: t[i[on).(

text: Le)t]

text: Y

text: =h

text: h(

text: no)n-z−ero

text: an(d

text: X

text: h

text: Since

text: θ(

text: X

text: h∶=,

text: we(

text: hav(e

text: ))θ

text: X

text: ,Y

text: h

text: h

text: α

text: α

text: α

text: An

text: elementary

text: matrix

text: computation

text: using

text: the

text: linear

text: representations

text: given

text: in

text: Section≠

text: shows

text: d[[eYﬁ,nθi(teX)q]u,aYd]ra∈tic′B=

text: θf[orXm,YonY]

text: h∈

text: =Yθ[,Yis

text: Xa

text: −,C]

text: wa⊂rhtea′rne

text: iBnvθoXlu,tYion).⊂

text: TT′rheXreθfoYre,⊂s(ini′tceisYa

text: Y,θ

text: X

text: ,Y

text: h

text: we−

text: get

text: X

text: h

text: proving

text: h

text: h

text: Using

text: the

text: matrix

text: representation

text: of

text: so

text: α

text: we

text: verify

text: that

text: h

text: h

text: ,h

text: and

text: h

text: h

text: ,h

text: We

text: then

text: have

text: h

text: h

text: and

text: h

text: h

text: α

text: α

text: α

text: α

text: α

text: Recall

text: that

text: we

text: assume

text: that

text: H

text: acts

text: faithfully

text: and

text: conformally

text: on

text: M,

text: g

text: Let

text: S

text: H

text: be

text: the

text: connected

text: Lie

text: subgroup

text: whose

text: Lie

text: algebra

text: is

text: s

text: a

text: h

text: It

text: will

text: not

text: be

text: diﬃcult

text: to

text: see

text: that

text: S

text: is

text: α

text: discompact

text: and

text: amenable,

text: ensuring

text: the

text: existence

text: of

text: a

text: Zimmer

text: point

text: for

text: S.

text: We

text: ﬁrst

text: establish

text: the

text: following

text: proposition

text: on

text: the

text: possible

text: orbits

text: of

text: such

text: a

text: point.

text: Propositi⊕on

text: Let

text: x

text: M

text: be

text: a

text: Zimmer

text: point

text: for

text: S

text: and

text: H

text: H

text: be

text: the

text: stabilizer

text: of

text: x.

text: If

text: x

text: k

text: then⊕the⊕Lie

text: algebra

text: h

text: is

text: one

text: of

text: the

text: following

text: Lie

text: subagebras

text: of

text: h:

text: x

text: m

text: h

text: α

text: a=

text: m⊕

text: h⊕

text: α

text: h.

text: When

text: h

text: a

text: m

text: h

text: the

text: orbit

text: H.x

text: is

text: conformal

text: to

text: the

text: standard

text: Riemannian

text: round

text: sphere

text: Sk

text: x

text: α

text: Proo/f.

text: We

text: keep

text: the

text: same

text: notations

text: for

text: the

text: root-space

text: d∈ecompo(si/tion[.

text: L])et

text: A

text: a

text: such

text: that

text: α

text: A

text: and

text: u≠

text: Ad

text: etA

text: GL

text: h

text: h

text: If

text: q

text: is

text: the

text: conformal

text: [class

text: ]of⊂subLorentzian

text: metrics

text: t

text: x

text: x

text: on

text: h

text: h

text: that

text: we

text: considered

text: in

text: Prop∩osition=

text: we

text: have

text: u

text: C∶onf→h

text: h/

text: q

text: Assux(me

text: )h=

text: −h.

text: ⩾Since

text: x

text: is

text: a

text: Zimm−

text: er

text: point

text: for

text: S,(

text: wte

text: )have

text: s,hx

text: xh

text: (Lemma

text: and

text: with

text: this

text: implies

text: hx

text: h

text: α

text: If

text: we

text: note−πx

text: h

text: h

text: xhx

text: thexnatu∗ral

text: =pro−jection,

text: dimπ

text: h

text: k

text: and

text: the

text: restriction

text: of

text: q

text: to

text: π

text: h

text: can(not⊕vani)sh⊂ide/ntically.

text: Since

text: u

text: x

text: α

text: x

text: x

text: α

text: t

text: coincides

text: with

text: e

text: t

text: id

text: on

text: this

text: latter

text: spac(e

text: an⊕d

text: is)c⩽onformal

text: for

text: q

text: we

text: must

text: have

text: u

text: q

text: e

text: x

text: t

text: x

text: x

text: The●refore,

text: q

text: must

text: vani⊂sh

text: identically

text: (on

text: )the

text: subspace

text: π

text: h

text: h

text: h

text: h

text: Sin⩾ce

text: −q

text: i>s

text: x

text: x

text: α

text: x

text: x

text: subLorentzian,

text: this

text: means

text: that

text: dimπ(

text: h)

text: h

text: We

text: then

text: distinguish

text: two

text: cases.

text: x

text: α

text: T[

text: he

text: ﬁr∩st

text: on]e=is

text: h⊂

text: h

text: Since

text: π

text: =h

text: ⊕is

text: at

text: most

text: dimh

text: h

text: k

text: x

text: x

text: α

text: x

text: α

text: The

text: adjoint

text: action

text: o⊄f

text: ad

text: h

text: on

text: h

text: being

text: ir=reduci⊕ble

text: (Lemma

text: we

text: then

text: have

text: α

text: h

text: ,h

text: h

text: h

text: =h

text: ⊕Finally,

text: h

text: h⩾

text: h

text: ≃and(we−ar)e

text: in

text: situation

text: x

text: α

text: α

text: x

text: x

text: α

text: The

text: second

text: is

text: when

text: h

text: =h

text: We

text: then

text: have

text: h

text: h

text: h

text: where

text: h

text: is

text: a

text: codimension

text: x

text: x

text: α

text: subalgebra

text: o=f

text: h⊕

text: ⊕a

text: m.

text: Since

text: k>

text: m

text: so

text: k

text: does

text: not

text: admit

text: a

text: codimension

text: subalgebra,(prov)ing

text: h

text: m

text: a)nd

text: we

text: are

text: in

text: sit(uatio)n

text: Now,

text: assume

text: that

text: h

text: a

text: )m→

text: h

text: (Sinc)e

text: k

text: there

text: are

text: only

text: two

text: possibilities

text: for

text: H:

text: it

text: is

text: either

text: x

text: α

text: isomorph=ic

text: to

text: or

text: Spin

text: In

text: fac−t,

text: ≃Spin

text: is

text: the

text: universal

text: cover

text: of

text: PSO

text: and∶=th[e

text: c∶over∶in⋯g

text: ∶Sp]in

text: PSO

text: is

text: If

text: H

text: PSO

text: i/t

text: naturally

text: acts

text: on

text: Sk

text: (see

text: Section

text: and

text: the

text: stabilizer

text: of

text: x

text: is

text: the

text: only

text: closed

text: subgroup

text: of

text: P∶

text: SO

text: )w→hose

text: L(ie

text: al)gebra

text: is

text: h

text: The

text: x

text: homogeneous

text: sp∈ace−H

text: Hx

text: is

text: t(hus

text: )diﬀeomorphic

text: to

text: Sk

text: If

text: H

text: Spin

text: the

text: projection

text: o>f

text: the

text: covering

text: p

text: Spin

text: PSO

text: sends

text: H

text: to

text: the

text: x

text: stabilizer

text: of

text: x

text: Sk

text: in

text: PSO

text: Therefore,

text: p

text: induces

text: a

text: local

text: diﬀeomorphism

text: H

text: H

text: Sk

text: x

text: This

text: must

text: be

text: a

text: covering,

text: and

text: since

text: k

text: a

text: diﬀeomorphism.

text: ##LTLine##

text: ##LTLine##


text: O ∶= SEMI-SIMPLE LIE GROUPS ACTING CONFORMALLY ON COMPACT LORENTZ MANIFOLDS 17 − Thus, in both c∶ases →the or(bitO ) H.x is a properly embedded submanif∶old o→f M, (di/ﬀeo)- x morphic to Sk 1. At last, it is a standard prop(erty) of homogeneous spaces that the is∶otrop→y repre(sen/tati[on]ρ)x Hx GL Tx′ x is conjugated to the representation Ad Hx ≃ GL( h−hx) ind′u=ced by(th−e a)djoint re(pr−ese)ntation. Since ρx Hx is confo(rm′a)l⊂, we ha(ve/ in fac)t Ad Hx Conf h hx, qx . If ′we no/te H the closed subgroup of Hx correspon(din−g )to m −so k 1 (H PSO k 1 or Spin k 1 ), by compactness we have Ad H Isom h h ,q . By Lemm[ a] x x 3.11, the action of H on h hx is conjugated tOo th≃e linea−r action of SO k 1 on Rk 1. Since this action leaves qx inva(rian)t, qx must be Euclidian. Therefore, the Lorentz conformal−class g induces a conformal Riemannian structure on x diﬀ Sk 1, which must be invariant under th(cid:3)e standard action of PSO 1,k , and has to be the standard Riemannian structure on Sk 1. ( ) ⩾ ( ) ⩾ ⩽ Corollary 3.13. When a Lie group locally isomorphic to SO 1,k , k 2, acts conformally on a compact Lorentz manifold M,g , with dimM 3, we have k dimM. Proof. In order to ensure that there exists Zimmer points for S, we prove the Lemma 3.14. The Lie subgroup S is discompact. = ⊕ ⊕ ⊕ − Proof. We compu⎧⎪⎪⎪te⎛ th−at with respe⎞ct to the root-space decompositi⎫⎪⎪⎪on h h α a m hα, we have ( ) = ⎨⎜ ⎟ ( ( )) ∈ ∈ ⎬ = ⋉ ( ( )) ⎪⎪⎪⎩⎝x 1 id ⎠ ∗ ⎪⎪⎪⎭ ∗ Zar Ad S ( (id )) exp ad( X( )); =x R(, X( h)) R exp ad h . h ∗ h α h α ( ( )) ⊂ ( ) xid ∈ ( ) = w{ her(e R ( a)c)ts on∈ ex}p ad h by x(.ex(p a)d) X exp xad X . Indeed, the linear group h α exp ad h GL h is unipotent, so it is algebraic a(nd) if A a is such that α A 1, the group α ′ ∗ exp tad A , t R normalizes exp ad h and acts on it via powers of et. α Zar Thus, an a′l=gebr∗a⋉ic coc(omp(act)S) subgroup of Adh S must contain the diagonal factor R , and since a unipotent linear group does not admit any proper algebraic cocompact subgroup, we (cid:3) must have S R exp ad h . h α Since S is discompact and amOen∶a=ble, it acts on M preserving some non-trivial ﬁnite measure and T●heorem 2 gives us the existenceOof=a Zimmer point x for S. P⩽roposition 3.12 describes three ●possibilities for the orbit OH.x. O ⩽ x In the ﬁ−rst case, wOe have dim k. −In particular,⩽we have k n. x ● In the sOeco=nd{ c}ase, the orbit being a Riemann∶ ian→subma(nifold, [we ]m)ust have dim x x dimM 1. Since ∶ ha(s di)m→ensio(n k− 1), we get k n(. ( )) = [ ( ( )) ( ( ))] ⊂ x W[ h(en − ) x (, the−iso)t]ro=py (repre−sen)tation ρ (H )Conf T M, g gives rise to a Lie x x x x algebra morphism ι s=o 1,k co 1,n 1 . Moreover, ι so 1,k ι so 1,k ,ι so 1,k co 1,n 1 ,co 1,n 1 so 1,n 1 . Since so 1,k is simple, this morphism( is tr)iv≠ial or injective. But if ι 0, ρ would be trivial by connectedness of H. In this situation, x Thurston’s stability theorem says that either H acts trivially( nea)r x, or H( 1 H−,R) 0 (se⩽e [C−G97], p. 140). The ﬁrst case contradicts the faithfulness of the action, and the second the simp⩽licity of H. Therefore, ι is an embedding of so 1,k into so 1,n 1 , and k n 1. (cid:3) In every case, we get k dimM. ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: O

text: SEMI-SIMPLE

text: LIE

text: GROUPS

text: ACTING

text: CONFORMALLY

text: ON

text: COMPACT

text: LORENTZ

text: MANIFOLDS

text: Thus,

text: in

text: both

text: c∶ases

text: →the

text: or(bitO

text: H.x

text: is

text: a

text: properly

text: embedded

text: submanif∶old

text: o→f

text: M,

text: (di/ﬀeo)-

text: x

text: morphic

text: to

text: Sk

text: At

text: last,

text: it

text: is

text: a

text: standard

text: prop(erty)

text: of

text: homogeneous

text: spaces

text: that

text: the

text: is∶otrop→y

text: repre(sen/tati[on]ρ)x

text: Hx

text: GL

text: Tx′

text: x

text: is

text: conjugated

text: to

text: the

text: representation

text: Ad

text: Hx

text: GL(

text: h−hx)

text: ind′u=ced

text: by(th−e

text: a)djoint

text: re(pr−ese)ntation.

text: Since

text: ρx

text: Hx

text: is

text: confo(rm′a)l⊂,

text: we

text: ha(ve/

text: in

text: fac)t

text: Ad

text: Hx

text: Conf

text: h

text: hx,

text: qx

text: If

text: ′we

text: no/te

text: H

text: the

text: closed

text: subgroup

text: of

text: Hx

text: correspon(din−g

text: )to

text: m

text: −so

text: k

text: (H

text: PSO

text: k

text: or

text: Spin

text: k

text: by

text: compactness

text: we

text: have

text: Ad

text: H

text: Isom

text: h

text: h

text: ,q

text: By

text: Lemm[

text: a]

text: x

text: x

text: the

text: action

text: of

text: H

text: on

text: h

text: hx

text: is

text: conjugated

text: tOo

text: th≃e

text: linea−r

text: action

text: of

text: SO

text: k

text: on

text: Rk

text: Since

text: this

text: action

text: leaves

text: qx

text: inva(rian)t,

text: qx

text: must

text: be

text: Euclidian.

text: Therefore,

text: the

text: Lorentz

text: conformal−class

text: g

text: induces

text: a

text: conformal

text: Riemannian

text: structure

text: on

text: x

text: diﬀ

text: Sk

text: which

text: must

text: be

text: invariant

text: under

text: standard

text: action

text: of

text: PSO

text: and

text: has

text: to

text: be

text: the

text: standard

text: Riemannian

text: structure

text: on

text: Sk

text: Corollary

text: When

text: a

text: Lie

text: group

text: locally

text: isomorphic

text: to

text: SO

text: k

text: acts

text: conformally

text: on

text: a

text: compact

text: Lorentz

text: manifold

text: M,g

text: with

text: dimM

text: we

text: have

text: k

text: dimM.

text: Proof.

text: In

text: order

text: to

text: ensure

text: that

text: there

text: exists

text: Zimmer

text: points

text: for

text: S,

text: we

text: prove

text: the

text: Lemma

text: The

text: Lie

text: subgroup

text: S

text: is

text: discompact.

text: Proof.

text: We

text: compu⎧⎪⎪⎪te⎛

text: th−at

text: with

text: respe⎞ct

text: to

text: the

text: root-space

text: decompositi⎫⎪⎪⎪on

text: h

text: h

text: α

text: a

text: m

text: hα,

text: we

text: have

text: ⎪⎪⎪⎩⎝x

text: id

text: Zar

text: Ad

text: S

text: (id

text: exp

text: ad(

text: X(

text: =x

text: R(,

text: X(

text: h))

text: R

text: exp

text: ad

text: h

text: h

text: h

text: α

text: h

text: α

text: xid

text: w{

text: her(e

text: R

text: a)c)ts

text: on∈

text: ex}p

text: ad

text: h

text: by

text: x(.ex(p

text: a)d)

text: X

text: exp

text: xad

text: X

text: Indeed,

text: the

text: linear

text: group

text: h

text: α

text: exp

text: ad

text: h

text: GL

text: h

text: is

text: unipotent,

text: so

text: it

text: is

text: algebraic

text: a(nd)

text: if

text: A

text: a

text: is

text: such

text: that

text: α

text: A

text: the

text: group

text: α

text: exp

text: tad

text: A

text: t

text: R

text: normalizes

text: exp

text: ad

text: h

text: and

text: acts

text: on

text: it

text: via

text: powers

text: of

text: et.

text: α

text: Zar

text: Thus,

text: an

text: a′l=gebr∗a⋉ic

text: coc(omp(act)S)

text: subgroup

text: of

text: Adh

text: S

text: must

text: contain

text: the

text: diagonal

text: factor

text: R

text: and

text: since

text: a

text: unipotent

text: linear

text: group

text: does

text: not

text: admit

text: any

text: proper

text: algebraic

text: cocompact

text: subgroup,

text: we

text: must

text: have

text: S

text: R

text: exp

text: ad

text: h

text: h

text: α

text: Since

text: S

text: is

text: discompact

text: and

text: amOen∶a=ble,

text: it

text: acts

text: on

text: M

text: preserving

text: some

text: non-trivial

text: ﬁnite

text: measure

text: and

text: T●heorem

text: gives

text: us

text: the

text: existenceOof=a

text: Zimmer

text: point

text: x

text: for

text: S.

text: P⩽roposition

text: describes

text: three

text: ●possibilities

text: for

text: the

text: orbit

text: OH.x.

text: O

text: x

text: In

text: the

text: ﬁ−rst

text: case,

text: wOe

text: have

text: dim

text: k.

text: −In

text: particular,⩽we

text: have

text: k

text: n.

text: x

text: In

text: the

text: sOeco=nd{

text: c}ase,

text: the

text: orbit

text: being

text: a

text: Riemann∶

text: ian→subma(nifold,

text: [we

text: ]m)ust

text: have

text: dim

text: x

text: x

text: dimM

text: Since

text: ha(s

text: di)m→ensio(n

text: k−

text: we

text: get

text: k

text: n(.

text: x

text: W[

text: h(en

text: x

text: the−iso)t]ro=py

text: (repre−sen)tation

text: ρ

text: (H

text: )Conf

text: T

text: M,

text: g

text: gives

text: rise

text: to

text: a

text: Lie

text: x

text: x

text: x

text: x

text: algebra

text: morphism

text: ι

text: s=o

text: co

text: Moreover,

text: ι

text: so

text: ι

text: so

text: ,ι

text: so

text: co

text: ,co

text: so

text: Since

text: so

text: is

text: simple,

text: this

text: morphism(

text: is

text: tr)iv≠ial

text: or

text: injective.

text: But

text: if

text: ι

text: ρ

text: would

text: be

text: trivial

text: by

text: connectedness

text: of

text: H.

text: In

text: this

text: situation,

text: x

text: Thurston’s

text: stability

text: theorem

text: says

text: that

text: either

text: H

text: acts

text: trivially(

text: nea)r

text: x,

text: or

text: H(

text: H−,R)

text: (se⩽e

text: p.

text: The

text: ﬁrst

text: case

text: contradicts

text: the

text: faithfulness

text: of

text: the

text: action,

text: and

text: the

text: second

text: the

text: simp⩽licity

text: of

text: H.

text: Therefore,

text: ι

text: is

text: an

text: embedding

text: of

text: so

text: into

text: so

text: and

text: k

text: n

text: In

text: every

text: case,

text: we

text: get

text: k

text: dimM.

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: × → SEMI-SIMPLE LIE GROUPS ACTING CONFORMALLY ON COMPACT LORENTZ MANIFOLDS 10 ⎡ ⎤ ⎢⎛ ⎞ ⎛ ⎞⎥ ⎛ ( )⎞ and the bracket∀hα ∈hα h⎢⎢2⎜α is give−n by⎟ ⎜ − ⎟⎥⎥ = ⎜ ⎟ ⎢ ⎥ ⎢⎝ ⎠ ⎝ ⎠⎥ ⎝ ⎠ ⎣ 0 u 0 0 v 0 ⎦ 0 0 iω u,v C u,v Ck, 0 tu , 0 tv 0 0 . ⊕ ⊂ ( − ) ( − ) 0 0 0 ( − ) Thus, the subspace h h su 1,n 1 is a Lie subalgebra isomorphic to heisC 2n 3 . α 2α triaInngsupla1r,mnatr1⎧⎪⎪⎪ic⎛etshe root sp⎞aces of posit⎫⎪⎪⎪ive roots cor⎧⎪⎪⎪re⎛spond to⎞the subspace of stri⎫⎪⎪⎪ctly upper- = ⎨⎜ − ⎟ ∈ ⎬ = ⎨⎜ ⎟ ∈ ( )⎬ ⎪⎪⎪⎝ ⎠ − ⎪⎪⎪ ⎪⎪⎪⎝ ⎠ ⎪⎪⎪ ⎩ 0 u 0 ⎭ ⎩ 0 0 q ⎭ hα 0 × tu→, u Hn 2 and h2α 0 0 , q SpanR i,j,k 0 0 ⎡ ⎤ ⎢⎛ ⎞ ⎛ ⎞⎥ ⎛ ( )⎞ and the Lie bra∀cket h∈α hα⎢⎢⎜ h2α is −given⎟ b⎜y − ⎟⎥⎥ = ⎜ ⎟ ⎢ ⎥ ⎢⎝ ⎠ ⎝ ⎠⎥ ⎝ ⎠ ⎣ 0 u 0 0 v 0 ⎦ 0 0 ω u,v H u,v Hk, 0 tu , 0 tv 0 0 . ⊕ ⊂ ( − ) ( − ) 0 0 0 (cid:3) Thus, the Lie subalgebra h h su 1,n 1 is isomorphic to heisH 4n 5 . α 2α − There exists an exceptional real rank 1 simple Lie algebra, noted f 20. Our work does not 4 need a complete description of it, we will just use the following lemma. ( ) − Lemma 3.3. The Lie algebra f 20 contains a subalgebra isomorphic to heisH 7 . 4 ( ) {± ±− } Proof. The roots system of f4 has also type BC⊕ , i.e. we have a simple root α such that 20 1 the rest=ricted roots =are α, 2α ([Kna02], p.717). We keep th≃e same notations for the root- spaces. Similarly with the previous situations, h h is a unipotent subalgebra an×d we→have α 2α dimh 8, dim(h )7.∈ Mo×reov↦er, th(ere)is−an id(ent)iﬁcation h O, algebra of octonions, and α 2α α an identiﬁcation of h with purely imaginary octotions, such that the Lie bracket h h h 2α α α 2α corresponds to x ,x O O x σ x x σ x , where σ denotes the conjugation in O. 1 2 1 2 2 1 The Cayley-Dickson construction shows that the octonions can be deﬁned as pairs of quater- nions endowe⊂d with a product⊕b[uilt wit]h≃produc(ts) and conjugations of quaternions. It is thus ′ ′ ′ ′ not diﬃcult to see (ﬁnding an appropriate copy of H in h ) that h contains a 4-dimensional α α (cid:3) subspace h h such that h h ,h heisH 7 . α α α α α ( ) ⩾ − ( ) ⩾ Let H be a Lie group locally isomorphic to some Sp 1,k , k 2 or F 20. The aim of this 4 section is to prove that there n(ever e[x]is)ts a comp⩾act Lorentz manifold M,g , with dimM 3, on which[ H] acts(faithCf)ully and conformally. − We ﬁx a compact manifold Mn, g , with n 3,=endo(wed )with a co<nformal class of Lorentz metrics g . Let M, be the corresponding norm−ali≃zed/Cartan geometry mode<led on (Ein1[,n])1 given by Cartan’s theorem (section ). We note G PO 2,n and P G the stabilize(r ofCa)n isotropic line in R2,n, so that we can identify Ein1,n 1 G P. We assume that H Conf M, g , or equivalently that H acts faithfully on M by automorphisms of the Cartan geometry M, . ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: SEMI-SIMPLE

text: LIE

text: GROUPS

text: ACTING

text: CONFORMALLY

text: ON

text: COMPACT

text: LORENTZ

text: MANIFOLDS

text: and

text: the

text: bracket∀hα

text: ∈hα

text: h⎢⎢2⎜α

text: is

text: give−n

text: by⎟

text: u

text: v

text: iω

text: u,v

text: C

text: u,v

text: Ck,

text: tu

text: tv

text: Thus,

text: the

text: subspace

text: h

text: h

text: su

text: is

text: a

text: Lie

text: subalgebra

text: isomorphic

text: to

text: heisC

text: α

text: 2α

text: root

text: sp⎞aces

text: of

text: posit⎫⎪⎪⎪ive

text: roots

text: cor⎧⎪⎪⎪re⎛spond

text: to⎞the

text: subspace

text: of

text: stri⎫⎪⎪⎪ctly

text: upper-

text: u

text: q

text: hα

text: tu→,

text: u

text: Hn

text: and

text: q

text: SpanR

text: i,j,k

text: and

text: the

text: Lie

text: bra∀cket

text: h∈α

text: hα⎢⎢⎜

text: is

text: −given⎟

text: b⎜y

text: u

text: v

text: ω

text: u,v

text: H

text: u,v

text: Hk,

text: tu

text: tv

text: Thus,

text: the

text: Lie

text: subalgebra

text: h

text: h

text: su

text: is

text: isomorphic

text: to

text: heisH

text: α

text: 2α

text: There

text: exists

text: an

text: exceptional

text: real

text: rank

text: simple

text: Lie

text: algebra,

text: noted

text: f

text: Our

text: work

text: does

text: not

text: need

text: a

text: complete

text: description

text: of

text: it,

text: we

text: will

text: just

text: use

text: the

text: following

text: lemma.

text: Lemma

text: The

text: Lie

text: algebra

text: f

text: contains

text: a

text: subalgebra

text: isomorphic

text: to

text: heisH

text: Proof.

text: The

text: roots

text: system

text: of

text: has

text: also

text: type

text: BC⊕

text: i.e.

text: we

text: have

text: a

text: simple

text: root

text: α

text: such

text: that

text: the

text: rest=ricted

text: roots

text: =are

text: α,

text: 2α

text: We

text: keep

text: th≃e

text: same

text: notations

text: for

text: the

text: root-

text: spaces.

text: Similarly

text: with

text: the

text: previous

text: situations,

text: h

text: h

text: is

text: a

text: unipotent

text: subalgebra

text: an×d

text: we→have

text: α

text: 2α

text: dimh

text: dim(h

text: Mo×reov↦er,

text: th(ere)is−an

text: id(ent)iﬁcation

text: h

text: O,

text: algebra

text: of

text: octonions,

text: and

text: α

text: 2α

text: α

text: an

text: identiﬁcation

text: of

text: h

text: with

text: purely

text: imaginary

text: octotions,

text: such

text: that

text: the

text: Lie

text: bracket

text: h

text: h

text: h

text: 2α

text: α

text: α

text: 2α

text: corresponds

text: to

text: x

text: ,x

text: O

text: O

text: x

text: σ

text: x

text: x

text: σ

text: x

text: where

text: σ

text: denotes

text: the

text: conjugation

text: in

text: O.

text: The

text: Cayley-Dickson

text: construction

text: shows

text: that

text: the

text: octonions

text: can

text: be

text: deﬁned

text: as

text: pairs

text: of

text: quater-

text: nions

text: endowe⊂d

text: with

text: a

text: product⊕b[uilt

text: wit]h≃produc(ts)

text: and

text: conjugations

text: of

text: quaternions.

text: It

text: is

text: thus

text: not

text: diﬃcult

text: to

text: see

text: (ﬁnding

text: an

text: appropriate

text: copy

text: of

text: H

text: in

text: h

text: that

text: h

text: contains

text: a

text: α

text: α

text: subspace

text: h

text: h

text: such

text: that

text: h

text: h

text: ,h

text: heisH

text: α

text: α

text: α

text: α

text: α

text: Let

text: H

text: be

text: a

text: Lie

text: group

text: locally

text: isomorphic

text: to

text: some

text: Sp

text: k

text: or

text: F

text: The

text: aim

text: of

text: this

text: section

text: is

text: to

text: prove

text: that

text: there

text: n(ever

text: e[x]is)ts

text: a

text: comp⩾act

text: Lorentz

text: manifold

text: M,g

text: with

text: dimM

text: on

text: which[

text: H]

text: acts(faithCf)ully

text: and

text: conformally.

text: We

text: ﬁx

text: a

text: compact

text: manifold

text: Mn,

text: g

text: with

text: n

text: )with

text: a

text: co<nformal

text: class

text: of

text: Lorentz

text: metrics

text: g

text: Let

text: M,

text: be

text: the

text: corresponding

text: norm−ali≃zed/Cartan

text: geometry

text: mode<led

text: on

text: given

text: by

text: Cartan’s

text: theorem

text: (section

text: We

text: note

text: G

text: PO

text: and

text: P

text: G

text: the

text: stabilize(r

text: ofCa)n

text: isotropic

text: line

text: in

text: so

text: that

text: we

text: can

text: identify

text: G

text: P.

text: We

text: assume

text: that

text: H

text: Conf

text: M,

text: g

text: or

text: equivalently

text: that

text: H

text: acts

text: faithfully

text: on

text: M

text: by

text: automorphisms

text: of

text: the

text: Cartan

text: geometry

text: M,

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: Figure 1: The P-GA establishes a race among an unbounded number of populations with expo- nentially increasing size that are evolved using a simple GA. For each 4 generations performed with population 2iN there will be one generation performed with population 2i+1N . The initial 0 0 population size N0 is chosen to be a small enough value such as N0 = 10. one generation performed with population N . Figure 1 depicts a graphical representation of a i+1 possible instance of this population evolution. The P-GA works under the assumption that solution quality grows monotonously with the population size (Lobo and Lima, 2007). In addition, the algorithm also assumes that this improve- ment in solution quality happens at a decreasing rate, i.e., when the same increase in population size happens between larger and larger populations, the gains in solution quality are expected to be fewer and fewer. For these reasons, the evolutionary race is established between populations with exponentially increasing size. Simultaneously, the P-GA allows smaller populations to use more computational resources, because if two different populations produce solutions with similar quality then the smaller population should be preferred, all other things being equal. In their paper, Harik and Lobo (1999) used a counter of base 4 to implement the population siz- ing method of the P-GA (see also, Lobo (2000); Pelikan and Lobo (1999) for detailed descriptions of this counter). However, Pelikan et al. (2007) suggested a somewhat simpler implementation for that sizing method and used it to propose a parameter-less version of the HBOA. The same sim- pler implementation was adopted to develop the work reported in this paper. Figure 2 depicts the pseudocode for the population sizing method as presented by Pelikan et al, where only the name in the title had to be changed to make it work for the simple genetic algorithm. In the P-GA, the performance of a population at any point in time is measured by the current average ﬁtness of all its individuals and by how many resources it has consumed, i.e., by how many ﬁtness function evaluations it has already performed. Naturally, a population that has consumed more resources is expected to have a better ﬁtness than any other population that has performed 3 ##LTFigure## ##LTLine## ##LTLine##

text: Figure

text: The

label: caption

text: P-GA

text: establishes

text: a

text: race

text: among

text: an

text: unbounded

text: number

text: of

text: populations

text: with

text: expo-

text: nentially

text: increasing

text: size

text: that

text: are

text: evolved

text: using

text: a

text: simple

text: GA.

text: For

text: each

text: generations

text: performed

text: with

text: population

text: there

text: will

text: be

text: one

text: generation

text: performed

text: with

text: population

text: The

text: initial

text: population

text: size

text: is

text: chosen

text: to

text: be

text: a

text: small

text: enough

text: value

text: such

text: as

text: one

text: generation

text: performed

text: with

text: population

text: N

text: Figure

text: depicts

text: a

text: graphical

text: representation

text: of

text: a

text: possible

text: instance

text: of

text: this

text: population

text: evolution.

text: The

text: P-GA

text: works

text: under

text: the

text: assumption

text: that

text: solution

text: quality

text: grows

text: monotonously

text: with

text: the

text: population

text: size

text: (Lobo

text: and

text: Lima,

text: In

text: addition,

text: the

text: algorithm

text: also

text: assumes

text: that

text: this

text: improve-

text: ment

text: in

text: solution

text: quality

text: happens

text: at

text: a

text: decreasing

text: rate,

text: i.e.,

text: when

text: the

text: same

text: increase

text: in

text: population

text: size

text: happens

text: between

text: larger

text: and

text: larger

text: populations,

text: the

text: gains

text: in

text: solution

text: quality

text: are

text: expected

text: to

text: be

text: fewer

text: and

text: fewer.

text: For

text: these

text: reasons,

text: the

text: evolutionary

text: race

text: is

text: established

text: between

text: populations

text: with

text: exponentially

text: increasing

text: size.

text: Simultaneously,

text: the

text: P-GA

text: allows

text: smaller

text: populations

text: to

text: use

text: more

text: computational

text: resources,

text: because

text: if

text: two

text: different

text: populations

text: produce

text: solutions

text: with

text: similar

text: quality

text: then

text: the

text: smaller

text: population

text: should

text: be

text: preferred,

text: all

text: other

text: things

text: being

text: equal.

text: In

text: their

text: paper,

text: Harik

text: and

text: Lobo

text: used

text: a

text: counter

text: of

text: base

text: to

text: implement

text: the

text: population

text: siz-

text: ing

text: method

text: of

text: the

text: P-GA

text: (see

text: also,

text: Lobo

text: Pelikan

text: and

text: Lobo

text: for

text: detailed

text: descriptions

text: of

text: this

text: counter).

text: However,

text: Pelikan

text: et

text: al.

text: suggested

text: a

text: somewhat

text: simpler

text: implementation

text: for

text: that

text: sizing

text: method

text: and

text: used

text: it

text: to

text: propose

text: a

text: parameter-less

text: version

text: of

text: the

text: HBOA.

text: The

text: same

text: sim-

text: pler

text: implementation

text: was

text: adopted

text: to

text: develop

text: the

text: work

text: reported

text: in

text: this

text: paper.

text: Figure

text: depicts

text: the

text: pseudocode

text: for

text: the

text: population

text: sizing

text: method

text: as

text: presented

text: by

text: Pelikan

text: et

text: al,

text: where

text: only

text: the

text: name

text: in

text: the

text: title

text: had

text: to

text: be

text: changed

text: to

text: make

text: it

text: work

text: for

text: the

text: simple

text: genetic

text: algorithm.

text: In

text: the

text: P-GA,

text: the

text: performance

text: of

text: a

text: population

text: at

text: any

text: point

text: in

text: time

text: is

text: measured

text: by

text: the

text: current

text: average

text: ﬁtness

text: of

text: all

text: its

text: individuals

text: and

text: by

text: how

text: many

text: resources

text: it

text: has

text: consumed,

text: i.e.,

text: by

text: how

text: many

text: ﬁtness

text: function

text: evaluations

text: it

text: has

text: already

text: performed.

text: Naturally,

text: a

text: population

text: that

text: has

text: consumed

text: more

text: resources

text: is

text: expected

text: to

text: have

text: a

text: better

text: ﬁtness

text: than

text: any

text: other

text: population

text: that

text: has

text: performed

text: ##LTFigure##

text: ##LTLine##

text: ##LTLine##