Riemannian metric making a given function harmonic
Solution 1:
The pullback under $f$ of the canonical 1-form $d\theta$ on $S^1$ is a closed form $\omega$ on $M$. Such a form is called intrinsically harmonic if there exists a Riemannian metric on $M$ making it harmonic. This term was introduced by Calabi in 1969, who proved the following:
Suppose $\omega$ has only Morse-type zeros (this is interpreted in terms of a locally-defined real-valued function $\phi$ such that $d\phi=\omega$). Then the following are equivalent:
- $\omega$ is intrinsically harmonic
- For any point $p\in M$ which is not a zero of $\omega$ there exists a smooth path $\gamma\colon [0,1]\to M$ such that $\gamma(0)=\gamma(1)=p$ and $\omega(\dot \gamma(t))>0$ for all $t$. (Such $\gamma$ is called an $\omega$-positive loop.)
If $f\colon M\to S^1$ is a submersion then $\omega = f^*(d\theta)$ has no zeroes, so we are in good shape on the Morse side. To get an $\omega$-positive loop at $p\in M$, we must lift some loop $\sigma_n(t)=f(p)e^{i t}$, $0\le t\le 2\pi n$ to a loop based at $p$. It's not clear to me how to do this for an arbitrary fibration over $S^1$, but in the paper the fibration is assumed to be locally trivial, in which case $\sigma_1$ should have such a lift.
Sources:
- Topology of closed one-forms by M. Farber, Chapter 9.
- MathOverflow answer by Dan Fox.