TITLE: Torsion of submanifolds
QUESTION [5 upvotes]: Studying curves in the Euclidean three dimensional space, one usually defines the curvature and the torsion of a curve. If I am not missunderstanding the thing, I guess that a curve has zero torision if and only if it is contained in a plane.
The curvature is generalised by the second fundamental form of a submanifold.
What it is the generalisation of the torsion for a sub manifold $N$ of a Riemannian manifold $M$? I would like to preserve the property that the torsion vanishes if and only if the submanifold is contaned in a totally geodesic hypersurface.
My guess, if $N$ is a curve, would be something like $\nabla_vII(v,v)-\nabla_{II(v,v)}v-[v,II(v,v)]$, where $\nabla$ is the Levi-Civita connection, $II$ is the seocond fundamental form, and $v$ is a tangent vector to $N$. This guess is insipired by the other use of the word torsion in Riemanninan geometry, and this is likely to be missleading, in view of the question Relating curvature and torsion of a connection to those of a curve
Probably this material is standard, but the references I found are either about torision of a curve in the Euclidean space (and usually they are very computational), or torsion of a connection.

REPLY [3 votes]: CLARIFICATION: The connection $\nabla$ used below is the Levi-Civita connection for the Riemannian metric on $M$ and not $N$.
First, assume that $N$ is a curve
and recall how to construct the Frenet frame of a curve.
Let $e_1 \in T_*N$ be a unit tangent vector along the
curve. If $\nabla_{e_1}e_1 \ne 0$, then, since $e_1\cdot\nabla_{e_1}e_1
= 0$, there isa a unique unit vector $e_2$ and function
$\kappa_1 > 0$ such that $e_2\cdot e_1 = 0$
and $\nabla_{e_1}e_1 = \kappa_{1} e_2$. 
If, in turn, $\nabla_{e_1}e_2 \ne 0$, then, $e_2\cdot \nabla_{e_1}e_2 =
0$ and $e_1\cdot \nabla_{e_1}e_2 = -e_2\cdot\nabla_{e_1}e_1 =
-\kappa_{1}$. Therefore, there exists a unique unit vector $e_3$
and function $\kappa_2 > 0$ such
that $e_3\cdot e_1 = e_3\cdot e_2 = 0$ and
$$
\nabla_{e_1}e_2 = -\kappa_{1} e_1 + \kappa_{2} e_3.
$$
If, at each stage, $\nabla_{e_1}e_k \ne 0$, then this leads to a
uniquely defined orthonormal frame of vectors $e_1, \dots, e_n \in
T_*M$ along $N$, known as the Frenet frame and corresponding functions
$\kappa_1, \dots, \kappa_{n-1}$ such that
$$
\nabla_{e_1}e_k = -\kappa_{k-1}e_{k-1} + \kappa_{k}e_{k+1}.
$$
If $M$ is flat, then the vanishing of $\kappa_{j},
\dots, \kappa_{n-1}$ implies that $N$ lies in an affine subspace of
dimension $j$. The function $\kappa_1$ is usually called the curvature
of $N$, and the functions $\kappa_2, \dots, \kappa_{n-1}$ considered as
torsion functions.
This can be generalized to higher dimensional submanifolds
as described by Liviu. Here is a sketch of one way to do it:
If $n_1 = \dim N$, let $e_1, \dots, e_{n_1}$ be an
orthonormal frame of vectors tangent to $N$. Suppose at each point
in $N$, the vectors $e_1, \dots, e_{n_1},
\nabla_{e_j}e_i$, where $1 \le i, j \le n$, span an $n_2$-dimensional
subspace, where $n < n_2$. Extend the orthonormal frame to a basis $e_1, \dots,
e_{n_2}$ of this larger subspace. The generalization of $\kappa_1$ for a
curve to a higher dimensional submanifold is the second fundamental
form
$H_{ij} = H^\mu_{ij}e_\mu$, where $H^\mu_{ij} =
e_\mu\cdot\nabla_{e_i}e_j$ and $n+1 \le \mu \le n_2$. Now suppose that $e_i, \nabla_{e_j}e_i$,
$1 \le i,j \le n_2$ span an $n_3$-dimensional subspace of
$T_*M$. Again, extend the orthonormal frame to one that spans this
subspace.
Then one can define a torsion tensor $T_{\mu,\nu} = T_{\mu\nu}^\eta
e_\eta$, where $n+1 \le \mu,\nu \le n_2$ and $n_2+1\le \eta \le n_3$.
This process can be continued. If the resulting frame does not span $T_*M$, then
one might call $N$ degenerate. If $M$ is flat, this implies that $N$
lies in an affine subspace. Otherwise, one gets a higher dimensional
version of a Frenet frame. The frame is not unique, but the
the nested sequence of osculating subspaces obtained in this way are.