Normal vector field to the hypersurface

Question

Consider a hypersurface $\Sigma$ in a manifold $M$ specified by setting a single function to a constant: $$f(x)=f_{*}$$

Define the vector field $$\zeta^{\nu}=g^{\mu\nu}\nabla_{\mu}f$$

How to show that $\zeta^{\mu}$ is orthogonal to all vectors $V^{\nu} \in T_p\Sigma$?

That is:

$$g_{\mu\nu}\zeta^{\mu}V^{\nu}=0$$

Answer 1

$V^\nu \nabla_\mu$ is the derivative along the integral curve of $V$. This curve lies on the hypersurface. If $f(x)$ is constant on the hypersurface, this means that

$V^\nu \nabla_\mu f= 0$

that is what you wanted to prove.