Let $(M,g)$ be a Riemannian manifold, and let $f$ be a smooth function whose Hessian is conformal to the metric, i.e. $\text{Hess} f=\lambda g$ for some smooth $\lambda:M\to \mathbb{R}$. Suppose that $p\in M$ satisfies $df(p)=0$, $\lambda(p)\neq 0$. For the sake of simplicity, I'll assume that $f(p)=0$ and $\lambda(p)=1$.
I was wondering if there always exists a local coordinate system $(y^1,\ldots,y^n)$ in a neighborhood $U$ of $p$ satisfying
- $f(y^1,...,y^n)=\frac{1}{2}((y^1)^2+\ldots (y^n)^2)$
- $\nabla f$ points directly in the "outward" direction, i.e. $(\nabla f)_x$ is a multiple of \begin{equation} y^1(x)\frac{\partial}{\partial y^1}+\ldots + y^n(x)\frac{\partial}{\partial y^1}. \end{equation}
The Morse lemma guarantees the existence of a coordinate system satisfying (1), but I'm not quite sure how to find coordinates that simultaneously satisfy (2).
It seems that the statement I'm trying to prove is an analog of Gauss's lemma. A consequence of Gauss's lemma is that for all $p\in M$, there exists a coordinate system $x^1,\ldots, x^n$ such that $\nabla r$ points in the "outward" direction, where $r$ is the distance function from $p$. Its proof relies on how $r$ behaves under its parametrization in exponential coordinates, and I'm not sure how I could apply it to the statement I'm trying to prove about $\nabla f$. I would appreciate any hints!
My question comes from reading the proof of Theorem 4.3.3 of Petersen's Riemannian geometry book. It seems to be something that the author implicitly assumes.