Extrinsic homogeneity of parallel submanifolds

TL;DR

This paper investigates conditions under which parallel submanifolds in Riemannian symmetric spaces are extrinsically homogeneous, establishing homogeneity for certain complete, irreducible cases of dimension at least 3.

Contribution

It provides necessary and sufficient conditions for the existence of extrinsically homogeneous parallel submanifolds based on their 2-jet data and proves homogeneity for a broad class of complete, irreducible submanifolds.

Findings

Necessary and sufficient conditions for extrinsic homogeneity based on 2-jet data.

Complete, irreducible parallel submanifolds of dimension ≥ 3 are extrinsically homogeneous.

Homogeneity results apply to submanifolds in symmetric spaces of compact or non-compact type.

Abstract

We consider parallel submanifolds $M$ of a Riemannian symmetric space $N$ and study the question whether $M$ is extrinsically homogeneous in $N$\,, i.e.\ whether there exists a subgroup of the isometry group of $N$ which acts transitively on $M$\,. First, given a "2-jet" $(W,b)$ at some point $p\in N$ (i.e. $W\subset T_pN$ is a linear space and $b:W\times W\to W^\bot$ is a symmetric bilinear form)\,, we derive necessary and sufficient conditions for the existence of a parallel submanifold with extrinsically homogeneous tangent holonomy bundle which passes through $p$ and whose 2-jet at $p$ is given by $(W,b)$\,. Second, we focus our attention on complete, (intrinsically) {\em irreducible} parallel submanifolds of $N$\,. Provided that $N$ is of compact or non-compact type, we establish the extrinsic homogeneity of every complete, irreducible parallel submanifold of $N$ whose dimension is at least 3 and which is not contained in any flat of $N$\,.