The Killing functions

Question

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?

Comments

Is that your definition or is it taken from some source?

What is the meaning of the star on the l.h.s.?

What is the meaning of the low dot on the r.h.s.?