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](https://mathoverflow.net/questions/122182/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.



Torsion of submanifolds
The torsion of a curve $C$ *in* $\newcommand{\bR}{\mathbb{R}}$ $\bR^3$ is a a manifestation of two ``freak'' low dimensional accidents: the curve is $1$-dimensional and it lives in a $3$-dimensional Euclidean space.


Being $1$-dimensional allows us, after fixing an orientation, to choose an orthonormal basis of the tangent space $T\_pC$ at each point $p\in C$. Just fix the arclength parametrization compatible with the orientation. This allows us to defined the first vector $T(p)$ of the Frenet frame.


If there is a bit of curvature, then we can define the 2nd vector $N(p)$ of the Frenet frame. Since the curve is in $\bR^3$ we can now define the third vector of the Frenet frame $B(p)=T(p)\times N(p)$. (A [result of Massey](https://www.jstor.org/stable/2323537?origin=crossref) based on a deep theorem of J. Adams shows that cross-like products exist only in dimensions $3$ and $7$.)


Note that if the curve lived in a higher dimensional space $\bR^n$, $n>3$, the above process would stop after we've defined the normal vector $N(p)$.


For a submanifold $M$ of dimension $m$ living in an Euclidean space $\bR^N$, $N>m$, the 2nd fundamental form $\DeclareMathOperator{\Gr}{\mathbf{Gr}}$ has an alternate description as the *shape operator*, i.e., the differential of the Gauss map $\Gamma: M\to\Gr\_m(\bR^N)$ that associates to a point $p\in M$ the tangent space $T\_pM$ viewed as an element of the Grassmannian $\Gr\_m(\bR^N)$ of $m$-dimensional subspaces of $\bR^m$. In this sense the shape operator describes the rate of change along $M$ of an orthonormal moving tangent frame.


The Frenet equations of a curve describe a bit more that the shape operator. They describe the motion along $C$ of an orthonormal frame of $\bR^3$ which for some fortunate accident can be chosen more or less canonically. Thus, I do not believe there is a natural simple notion of torsion beyond curves in $\bR^3$.


On the other hand, given $M\subset \bR^N$ we can still ask ask if $M$ is contained in an affine subspace of dimension $n<N$. This corresponds to asking when the image of the Gauss map be contained in a sub-Grassmannian $\Gr\_m(\bR^n)$ but, be careful, there are many ways of canonically embedding $\Gr\_m(\bR^n)$ in $\Gr\_m(\bR^N)$.


In any case, this phenomenon can be expressed in terms of certain constraints on the 2nd fundamental form that you could call torsions. I don't believe they are too useful, though I could be proven wrong.


