EDIT: Let me modify the question then: for what submanifolds $N$ does the torsion $T$ preserve tangent vectors to $N$?


If $\nabla$ is a connection on a manifold $M$, then torsion is defined to be the map
$$
T(X,Y)=\nabla\_XY-\nabla\_YX-[X,Y]
$$


where $X$ and $Y$ are vector fields on $M$. It can be shown that $T$ is a $2 \choose 1$ tensor on $M$; that is, for all $p\in M$,
$$
T:T\_pM\times T\_pM\longrightarrow T\_pM
$$


where $T\_pM$ is the tangent space to $M$ at $p$.


Suppose $N\subset M$ is a submanifold of $M$. Is it the case that $T$ preserves tangent vectors to $N$? That is, does
$$
T:T\_pN\times T\_pN\longrightarrow T\_pN
$$


for $p\in N$?



Torsion and submanifolds
Edit: Answering Robert, moved to 3-space to give a general example:


A simple example in $M=\mathbb R^3$: Let $N=0\times \mathbb R^2$ and put 
$$
\nabla\_XY = dY(X) + \begin{pmatrix}X^T\,A^1\,Y \\ X^T\,A^2\,Y \\ X^T\,A^3\,Y\end{pmatrix},
\quad
A=\begin{pmatrix} a^i\_{11} & a^i\_{12} & a^i\_{13}\\ \dots \\a^i\_{31} & a^i\_{32} & a^i\_{33} \end{pmatrix},
\quad
a^2\_{kl} = a^3\_{kl} = 0 \text{ for } 2\le k,l\le 3.
$$
Then $Tor^i\_{kl} = A^i\_{kl}-A^i\_{lk}$ maps $T\_{(0,x,y)}(0\times \mathbb R^2)\times T\_{(0,x,y)}(0\times \mathbb R^2)$ skew linearly into $T\_{(0,x,y)}\mathbb R\times 0$.


