Extrinsic homogeneity of parallel submanifolds

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 $N$\,.