Assume that $(M,g)$ is a Riemannian manifold. A vector field $X$ on $M$ is called a harmonic vector field if the corresponding $1$-form $\alpha$ with $\alpha(Y)= \langle X,Y \rangle\_g$ is a harmonic $1$-form.


Motivated by [this conversations](https://mathoverflow.net/questions/269940/vector-field-with-holomorphic-flow/269971#269971) we ask the following question:



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> Assume that $X$ is a vector field on $M$. Assume that for every $t$ the flow $\phi\_t$ of the vector field is a harmonic map. Does this imply that the vector field is a harmonic vector field?
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Vector field with Harmonic flow
Well, right away, you can see that the answer is 'no', in general. Consider the round $n$-sphere $S^n$ with its standard metric. When $n>1$, it has no nonzero harmonic $1$-forms, but it has nontrivial Killing vector fields since it is a homogeneous Riemannian manifold. Since the flow $\phi\_t$ of such a Killing field $X$ is a harmonic map of $S^n$ to itself, this example shows that the flow of $X$ can be harmonic even though $X$ is not.


