On a Riemannain manifold $M$ with a submanifold $S$, we can define the second fundamental form $H$ to be $-H(X, Y) = (\nabla_{X} Y)^{N}$ the component of $(\nabla_{X} Y)^{N}$ normal to $S$. Then mean curvature vector is defined to be the trace of the second fundamental form. I am confused as to what does it mean for a vector to be the trace of the second fundamental form.
Usually contractions/trace will yield a real-valued function on the manifold, but here we want a vector.
The definition of the trace is essentially the same. You simply define the trace of to be $$ \operatorname{tr} = \sum_{k=1}^n H(e_k,e_k) $$ where $(_1,…,_)$ is an orthonormal basis of $_$. Or, if you have coordinate vector fields $\partial_1,\dots,\partial_$ and dual 1-forms $^1,\dots,^$ on and the metric is $g_{ij}dx^i\,dx^j$, then $$ \operatorname{tr} = g^{ij} H(\partial_i,\partial_j) $$ If a basis of the normal bundle is $\nu_1,\dots,\nu_{n-m}$, then you can write $$=\nu_\mu H^\mu_{ij}dx^idx^j$$ and its trace as $$^{ij}^\mu_{ij}\nu_\mu$$