# 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.

## 1 Answer

It is meant that the fiber is a minimal submanifold in the sense of http://www.impa.br/opencms/pt/biblioteca/mono/Mon_14.pdf

In the codimension 1 case, minimal submanifolds are characterised by vanishing of the mean curvature. I don't know what is the corresponding condition is in higher codimensions, but presumably that should help for proving the claimed equivalence.

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

• Ah thanks, I knew it was something not too hard, but just couldn't find the right term (minimal is a bit overloaded, and so is fiber). Also, good catch with the isometry, I forgot the restriction to the horizontal distribution (also a nice overloaded term...). – MathJJ Dec 3 '14 at 17:36