# Finding a metric to write a vector field as a gradient of a given function

Let $M$ be a smooth manifold, and $f$ a smooth function with an isolated local minimum at $p$. Furthermore, let $X$ be a vector field vanishing at $p$ such that for some neighborhood $U$ of $p$, $df_q(X_q)<0$ for all $q\in U\setminus\{p\}$. Is it possible to find a Riemannian metric g on $U$ such that $X|_U=-\text{grad}_gf$? If not, what is a counterexample?

• I think the answer should be no, in the same way not every 1-form is the differential of a function. However, since you allow to change the metric for each case... the answer is more difficult. Jul 4, 2018 at 13:49
• Do you know something about the order of vanishing at $p$? For example, if $f$ is Morse while $X$ vanishes at $p$ up to some high order, I believe your desired metric does not exist. However, if both $df$ and $X$ have a simple zero at $p$, there may be something you could do. Jul 4, 2018 at 15:19
• No, I would like a statement in full generality. So if there is one, any explicit counterexample of the kind you mention would be helpful. Jul 4, 2018 at 16:01
• In the one-dimensional case this reduces to smooth divisibility of functions: $p=0,f=t^4, X=-t^2 \partial_t$ is a simple counterexample on $M=\mathbb R.$ Jul 5, 2018 at 9:37
• Yeah, you're right, I forgot about the derivative when choosing the signs. $-t \partial_t$ is the right choice. Jul 5, 2018 at 11:50

The following is a general fact: Let $$M$$ be a smooth manifold of dimension $$n$$, let $$E\to M$$ be a vector bundle of rank $$n$$, and let $$s\in\Gamma(M,E)$$ be a smooth section vanishing at $$p\in M$$. Let $$\nabla$$ be a linear connection on $$E$$. Then $$s$$ is transverse to the zero section at $$p$$ if and only if we have $$(\nabla_Xs)_p\neq0$$ for every $$X\in T_pM$$, $$X\neq 0$$, in other words, iff the endomorphism $$X\mapsto(\nabla_Xs)_p$$ is invertible.
Now, a Riemannian metric induces an isomorphism of vector bundles between the tangent and cotangent bundles. Transversality of sections is preserved under a bundle isomorphism. Suppose $$f$$ is Morse. In particular, the Hessian $$\nabla^2f$$ is non-degenerate at $$p$$, where $$\nabla$$ can be any linear connection on $$TM$$, or equivalently, the section $$df$$ of the cotangent bundle is transverse to the zero section at $$p$$. Take any vector field $$X$$ which vanishes at $$p$$ and which is not transverse to the zero section. Then no bundle isomorphism carries $$X$$ to $$df$$.