I'm reading a paper and I came across a computation that is giving me trouble.
Here is the setup. Let $(M,g)$ be a Riemannian manifold and $u:M\to\mathbb{R}$ a smooth function with $\nabla u \neq 0$. Then the level set $\Sigma = \{x \in M | u(x)=c\}$ is a closed embedded submanifold, with unit normal $\nu = \frac{\nabla u}{|\nabla u|}$. Let $\nabla^2_{\Sigma} u$ denote the Hessian of $u$ restricted to $T\Sigma \otimes T\Sigma$ and $\nabla^2 u$ denote the full Hessian on $TM \otimes TM$ . I know the second fundamental form of $\Sigma$ is given by $\Pi=\frac{\nabla^2_{\Sigma} u}{|\nabla u|}$. Then the claim is that $$ |\Pi|^2=|\nabla u|^{-2}\big(|\nabla^2 u)|^2-2|\nabla|\nabla u||^2+[\nabla^2 u(\nu,\nu)]^2\big).$$ To verify this I tried computing $\nabla_{\Sigma}^2u$ in terms of $\nabla^2 u$, but couldn't find a nice expression. Is there a nice formula for the induced Hessian? And can anyone help in verifying the claim?