TITLE: Is a rhombus rigid on a sphere or torus? And generalizations
QUESTION [19 upvotes]: If a rectangle is formed from rigid bars for edges and joints
at vertices, then it is flexible in the plane: it can flex
to a parallelogram.
On any smooth surface with a metric, one can define a linkage
(e.g., a rectangle) whose edges are geodesics of fixed length,
and whose vertices are joints, and again ask if it is rigid
or flexible on the surface.
This leads to my first, specific question:

Q1.
  Is a rhombus, or a rectangle,
  always flexible on a sphere?


It seems the answer should be Yes but I am a bit uncertain
if there must be a restriction on the edge lengths.
(In the above figure, the four arcs are each $49^\circ$ in length, comfortably short.)

Q2. 
  The same question for other surfaces: Arbitrary convex surfaces? A torus?

I am especially interested to learn if there are situations where a linkage that is flexible
in the plane is rendered rigid when embedded on some surface.
It seems this should be possible...?

Q3.
  More generally, 
  Laman's theorem
  provides a combinatorial characterization of the rigid linkages in the plane.
  The $n{=}4$ rectangle is not rigid because it has fewer than $2n-3 = 5$ bars:
  it needs a 5th diagonal bar to rigidify.
  Has Laman's theorem been extended to arbitary (closed, smooth) surfaces
  embedded in $\mathbb{R}^3$?
  Perhaps at least to spheres, or to all convex surfaces?

Thanks for any ideas or pointers to relevant literature!
Addendum.
I found one paper related to my question:
"Rigidity of Frameworks Supported on Surfaces"
by A. Nixon, J.C. Owen, S.C. Power.
arXiv:1009.3772v1 math.CO
In it they prove an analog of Laman's theorem for
the circular cylinder in $\mathbb{R}^3$.
If one phrases Laman's theorem as requiring for
rigidity that the number of edges $E \ge 2 V - 3$
in both the graph and in all its subgraphs,
then their result (Thm. 5.3) is that, on the cylinder, rigidity requires
$E \ge 2 V -2$ in the graph and in all its subgraphs.
This is not the precise statement of their theorem.
They must also insist that the graph be regular
in a sense that depends on the rigidity matrix achieving maximal rank
(Def. 3.3).
They give as examples of irregular linkages on a sphere
one that
contains an edge with antipodal endpoints, or one that includes
a triangle all three of whose vertices lie on a great circle.
But modulo excluding irregular graphs and other minor technical
details, they essentially replace the constant 3 in Laman's
theorem for the plane with 2 for the cylinder.
Theirs is a very recent paper but contains few citations to related
work on surfaces, suggesting that perhaps the area
of linkages embedded on surfaces is
not yet well explored.
In light of this apparent paucity of information, it seems appropriate that I 'accept'
one of the excellent answers received.  Thanks!
Addendum [31Jan11].
I just learned of a 2010 paper by Justin Malestein and Louis Theran,
"Generic combinatorial rigidity of periodic frameworks"
arXiv:1008.1837v2 (math.CO),
which pretty much completely solves the problem of linkages on a flat 2-torus,
generalizing to flat orbifolds.  They obtain a combinatorial characterization for generic minimal
rigidity for "planar periodic frameworks," which encompass these surfaces.

REPLY [3 votes]: The first paragraph of Some notes on the equivalence of first-order rigidity in various geometries by Franco V. Saliola and Walter Whiteley states:

In this paper, we explore the connections among the theories of first-order rigidity of bar and joint frameworks (and associated structures) in various metric geometries extracted from the underlying projective space of dimension n, or $\mathbb{R}^{n+1}$. The standard examples include Euclidean space, elliptical (or spherical) space, hyperbolic space, and a metric on the exterior of hyperbolic space.

Section 4 of the notes proves that a framework is first-order rigid in the upper hemisphere $S^n_+$ iff a corresponding framework in $\mathbb{R}^n$ is first-order rigid.  
Section 5 extends this to geometries on the space $X^n_{c,k}=\{x\in\mathbb{R}^{n+1}|\langle x,x\rangle_k=c,x_{n+1}>0\}$ where $\langle x,x\rangle_k=x_1y_1+\cdots+x_{n-k+1}y_{n-k+1}-x_{n-k+2}y_{n-k+2}-\cdots-x_{n+1}y_{n+1}$.  Note that $k=1,c=-1$ is hyperbolic space $\mathbb{H}^n$, the case $k=0,c=1$ is $S^n_+$.