Second Fundamental Form of the Graph of a Function of Higher Codimension

Question

Let $f:\mathbb{R}^n\to\mathbb{R}^m$, $m\geq 2$, be a smooth function. I would like to find an explicit description for the second fundamental form of the graph of $f$ in terms of the Hessians and gradients of $f_i$, where $f=(f_1, \ldots,f_m)$ . I could find such a description for $m=1$, but except the definition, I could not find an explicit formula for $m>1$. Any reference or remarks would be highly appreciated.

Answer 1

$\newcommand\R{\mathbb{R}}$ It's easiest to explain this for an arbitrary embedding $\Phi: \R^n \rightarrow \R^{n+m}$ and then apply it to a graph.

The metric induced by $\Phi$ is given by $$ g_{ij}\,dx^i\,dx^j = \partial_i\Phi\cdot\partial_j\Phi\,dx^i\,dx^j. $$ The second fundamental form is a normal vector valued symmetric $2$-tensor field given by $$ N_{ij}\,dx^i\,dx^j, $$ where $N_{ij}$ is the normal component of $\partial^2_{ij}\Phi$ at the point $\Phi(x)$ and is given in terms of $\Phi$ by $$ N_{ij} = \partial^2_{ij}\Phi - \partial_k\Phi (g^{kl}\partial_l\Phi\cdot\partial^2_{ij}\Phi). $$

You can get the formula for the graph by setting $$ \Phi(x) = (x^1, \dots, x^n, f_1(x), \dots, f_m(x)). $$

As an aside, the tangential component of $\partial^2_{ij}\Phi$ is comprised of the Christoffel symbols, $$ \partial_k\Phi\cdot\partial^2_{ij}\Phi = g_{kl}\Gamma^l_{ij}. $$