Manifolds with many hyperbolic planes

TL;DR

This paper constructs complete Riemannian manifolds where every geodesic is contained in a totally geodesic hyperbolic plane, highlighting the existence of such structures beyond locally homogeneous spaces.

Contribution

It introduces new examples of manifolds with pervasive hyperbolic planes that are not locally homogeneous, expanding understanding of geometric structures.

Findings

Existence of non-homogeneous manifolds with all geodesics in hyperbolic planes

Construction of explicit examples of such manifolds

Demonstration that abundance of hyperbolic planes does not imply local homogeneity

Abstract

We construct examples of complete Riemannian manifolds having the property that every geodesic lies in a totally geodesic hyperbolic plane. Despite the abundance of totally geodesic hyperbolic planes, these examples are not locally homogenous.