# What is a minimal fiber of a Riemannian submersion

I am reading "Spectral Geometry, Riemannian Submersions and the Gromov-Lawson conjecture" by Gilkey, Leahy and Park, and I'm having some trouble with some of the terms they introduce without explanation. I am somewhat familiar with differential/Riemannian geometry, but not an expert.

The setting: let $\pi: Z\rightarrow Y$ be a surjection, with the pushforward $\pi_*: T_z Z \rightarrow T_{\pi z}Y$ also a surjection (i.e. $\pi$ is a submersion). If the pushforward is an isometry, it is called a Riemannian submersion. Then $X:=\pi^{-1}(y_0), y_0\in Y$ is called the fiber of the submersion (and in fact defines a fiber bundle, i.e. locally trivial). Now they state (without proof, the claim is it's obvious from the definition) that the fibers of a Riemannian submersion are minimal if and only if the submersion is harmonic, or if and only if the mean normalized curvature vanishes. My question is: what is the definition of minimal fibers in this setting? I've tried to search the internet, but I couldn't find one.

(By the way, not $\pi_*$ is an isometry but just its restriction to the orthogonal complement of the fiber.)