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 $p\in N$?

allsubspaces of $T_pM$! $\endgroup$3more comments