Statement: Let $f$ be a real-valued smooth function on a compact n-manifold $M$.Suppose it have finitely many critical points $\left\{p_1,...,p_k\right\}$ with associated critical values $\left\{c_1,...,c_k\right\}$.Choose a Riemannian metric $g$ on $M$ , let $X$ be the vector field $X=\text{grad}f/|\text{grad}f|_{g}^{2}$ on $M\backslash\{p_{1},...,p_{k}\}$ , and let $\theta$ denote the flow of $X$ . Show that $f(\theta_{t}(p))=f(p)+t$ whenever $\theta_{t}(p)$ is defined.
Question: Can someone give me some hints as to how to start this problem. Thanks.