Hypersurfaces d'un fibr\'e vectoriel Riemannien \`a courbures moyennes verticale et horizontale prescrites

TL;DR

This paper investigates the problem of embedding the sphere subbundle of a Riemannian vector bundle into the bundle itself, with the goal of achieving prescribed mean curvatures in both vertical and horizontal directions.

Contribution

It introduces a method to find embeddings of sphere subbundles with specified mean curvatures, extending previous work on curvature prescriptions in Riemannian geometry.

Findings

Established existence results for embeddings with prescribed mean curvatures

Developed new techniques for controlling vertical and horizontal curvature components

Provided examples illustrating the prescribed curvature embeddings

Abstract

Let M be a compact Riemannian manifold without boundary and let E be a Riemannian vector bundle over M. If $\Sigma$ denotes the sphere subbundle of E, we look for embeddings of $\Sigma$ into E admitting a prescribed mean curvatures of various type.