The tangential connection on an embedded submanifold $M ⊂ R^n$ is symmetric.
the hint is let $X,Y$ be vector fields that are tangent of M at points of M, so is $[X,Y]$
I start with $$T(X,Y)=\nabla_x Y-\nabla_yX-[X,Y]$$ and thinking prove first part of equation would equal to the second part of equation ..
not sure if it's right or not
OK. The tangential connection is nothing but the Levi-Civita connection for the induced metric. So it is symmetric. Besides this fact we can calculate it directly: Let D be the tangential connection, $\bar{D}$ be the connection of the ambient Euclidean space, and P be the projection operator then $D=P(\bar{D})$, and $$ D_X(Y)-D_Y(X)=P(\bar{D}_X(Y))-P(\bar{D}_Y(X))=P(\bar{D}_X(Y)-\bar{D}_Y(X))=P([X,Y])=[X,Y]$$