Non existence of harmonic maps

Question

Is there an example of two compact Riemannian manifolds $M,N$, such that the only harmonic mappings between them are constants?

Answer 1

Yep. Take $M = S^2$ with any metric and $N = S^1$ (thought of as the unit circle in the plane) with the induced metric. Let $\pi \colon \mathbb{R} \rightarrow S^1$ be the covering map $\pi(t) = e^{it}$ which is a local isometry with respect to the Euclidean metric on $\mathbb{R}$.

Let $\phi \colon S^2 \rightarrow S^1$ be a harmonic map. Since $S^2$ is simply-connected, we can lift $\phi$ to $\tilde{\phi} \colon S^2 \rightarrow \mathbb{R}$ which is also harmonic (because locally, $\tilde{\phi} = s \circ \phi$ where $s$ is a local section of $\pi$ which is an isometry). But since $S^2$ is compact, $\tilde{\phi}$ must be constant and so is $\phi = \pi \circ \tilde{\phi}$.