$\newcommand{\bR}{\mathbb{R}}\newcommand{\pa}{\partial}$This questions has some nebulous roots in Morse theory. The most general version goes as follows. Fix an integer $n\geq 2$. Suppose that we have a smooth function $f$ defined on an open neighborhood $U$ of $0$ in $\bR^n$. To a smooth Riemannian metric $g$ defined on $U$ we associated the gradient $\nabla^g f$.

For which smooth functions functions $f$ is the correspondence

$$g\mapsto \nabla^g f $$

injective, i.e., from the knowledge that $\nabla^{g_0} f=\nabla^{g_1} f$ on $U$ we can conclude that $g_0=g_1$ on some neighborhood $V$ of $0$ in $\bR^n$, $V\subset U$.

Clearly, the above correspondence is not injective if $f\equiv const$, or $f(x)$ is linear in $x$.

The next interesting case is when $q$ is quadratic. Suppose that

$$ f(x)=q(x):=\frac{1}{2}(x_1^2+\cdots +x_n^2). $$

Denote by $g_0$ the canonical Euclidean metric on $\bR^n$ so that

$$\nabla^{g_0}q =\sum_i x_i\pa_{x_i}. $$

The special case of the question goes as follows.

If $g$ is a real analytic Riemann metric defined on a neighborhood $U$ of $0$ and $\nabla^g q(x)= \nabla^{g_0} q(x)$, $\forall x\in U$, can we conclude that $g=g_0$ in a neighborhood of $0$?

I was not able to track results of this kind in the literature, and I would appreciate any pointers.

$\newcommand{\bR}{\mathbb{R}}\newcommand{\pa}{\partial}$This questions has some nebulous roots in Morse theory. The most general version goes as follows. Fix an integer $n\geq 2$. Suppose that we have a smooth function $f$ defined on an open neighborhood $U$ of $0$ in $\bR^n$. To a smooth Riemannian metric $g$ defined on $U$ we associated the gradient $\nabla^g f$.

For which smooth functions   $f$ is the correspondence

$$g\mapsto \nabla^g f $$

injective, i.e., from the knowledge that $\nabla^{g_0} f=\nabla^{g_1} f$ on $U$ we can conclude that $g_0=g_1$ on some neighborhood $V$ of $0$ in $\bR^n$, $V\subset U$.

Clearly, the above correspondence is not injective if $f\equiv const$, or $f(x)$ is linear in $x$.

The next interesting case is when $q$ is quadratic. Suppose that

$$ f(x)=q(x):=\frac{1}{2}(x_1^2+\cdots +x_n^2). $$

Denote by $g_0$ the canonical Euclidean metric on $\bR^n$ so that

$$\nabla^{g_0}q =\sum_i x_i\pa_{x_i}. $$

The special case of the question goes as follows.

If $g$ is a real analytic Riemann metric defined on a neighborhood $U$ of $0$ and $\nabla^g q(x)= \nabla^{g_0} q(x)$, $\forall x\in U$, can we conclude that $g=g_0$ in a neighborhood of $0$?

I was not able to track results of this kind in the literature, and I would appreciate any pointers.

$\newcommand{\bR}{\mathbb{R}}$ $\newcommand{\pa}{\partial}$

This questions has some nebulous roots in Morse theory. The most general version goes as follows. Fix an integer $n\geq 2$. Suppose that we have a smooth function $f$ defined on an open neighborhood $U$ of $0$ in $\bR^n$. To a smooth Riemannian metric $g$ defined on $U$ we associated the gradient $\nabla^g f$.

For which smooth functions functions $f$ is the correspondence

$$g\mapsto \nabla^g f $$

injective, i.e., from the knowledge that $\nabla^{g_0} f=\nabla^{g_1} f$ on $U$ we can conclude that $g_0=g_1$ on some neighborhood $V$ of $0$ in $\bR^n$, $V\subset U$.

Clearly, the above correspondence is not injective if $f\equiv const$, or $f(x)$ is linear in $x$.

The next interesting case is when $q$ is quadratic. Suppose that

$$ f(x)=q(x):=\frac{1}{2}(x_1^2+\cdots +x_n^2). $$

Denote by $g_0$ the canonical Euclidean metric on $\bR^n$ so that

$$\nabla^{g_0}q =\sum_i x_i\pa_{x_i}. $$

The special case of the question goes as follows.

If $g$ is a real analytic Riemann metric defined on a neighborhood $U$ of $0$ and $\nabla^g q(x)= \nabla^{g_0} q(x)$, $\forall x\in U$, can we conclude that $g=g_0$ in a neighborhood of $0$?

I was not able to track results of this kind in the literature, and I would appreciate any pointers.

$\newcommand{\bR}{\mathbb{R}}\newcommand{\pa}{\partial}$This questions has some nebulous roots in Morse theory. The most general version goes as follows. Fix an integer $n\geq 2$. Suppose that we have a smooth function $f$ defined on an open neighborhood $U$ of $0$ in $\bR^n$. To a smooth Riemannian metric $g$ defined on $U$ we associated the gradient $\nabla^g f$.

For which smooth functions functions $f$ is the correspondence

$$g\mapsto \nabla^g f $$

injective, i.e., from the knowledge that $\nabla^{g_0} f=\nabla^{g_1} f$ on $U$ we can conclude that $g_0=g_1$ on some neighborhood $V$ of $0$ in $\bR^n$, $V\subset U$.

Clearly, the above correspondence is not injective if $f\equiv const$, or $f(x)$ is linear in $x$.

The next interesting case is when $q$ is quadratic. Suppose that

$$ f(x)=q(x):=\frac{1}{2}(x_1^2+\cdots +x_n^2). $$

Denote by $g_0$ the canonical Euclidean metric on $\bR^n$ so that

$$\nabla^{g_0}q =\sum_i x_i\pa_{x_i}. $$

The special case of the question goes as follows.

If $g$ is a real analytic Riemann metric defined on a neighborhood $U$ of $0$ and $\nabla^g q(x)= \nabla^{g_0} q(x)$, $\forall x\in U$, can we conclude that $g=g_0$ in a neighborhood of $0$?

I was not able to track results of this kind in the literature, and I would appreciate any pointers.