Let $(M,g)$ be a riemannian manifold and $f$ a function. $f$ is called a Killing function if:
$$\nabla_X (df^*)=\mu f .X$$
where $X$ is a vector field, $\nabla$ is the Levi-Civita connection and $\mu$ is a scalar number.
What is the shape of a manifold with a Killing function?