In this short note we present a result in a paper due to Edward M. Fan [here]. To be precise, we prove
Given a sphere with standard metric, if and , then must be a constant.
Proof. To prove the result, we shall use the following well-known formula
Therefore, the fact that is harmonic implies that
Since , multiplying both sides by , we get
Integrating both sides on using standard volume form, we get
Now integration by parts shows that
This in turn shows that
We conclude that must be a constant function.