Parallel submanifolds with an intrinsic product structure

TL;DR

This paper characterizes parallel isometric immersions of product symmetric spaces into other symmetric spaces, showing under certain conditions that the image is a homogeneous submanifold, with a focus on the intrinsic and extrinsic geometric structures involved.

Contribution

It introduces a new framework linking the intrinsic product structure of the domain with the extrinsic geometry of the immersion, leading to conditions for homogeneity of the submanifold.

Findings

Intrinsic product structure reflected in a fiberwise orthogonal decomposition

Existence of commuting, parallel involutions on the osculating bundle

Under specified conditions, the immersed submanifold is homogeneous

Abstract

Let $M$ and $N$ be Riemannian symmetric spaces and $f:M\to N$ be a parallel isometric immersion. We additionally assume that there exist simply connected, irreducible Riemannian symmetric spaces $M_i$ with $\dim(M_i)\geq 2$ for $i=1,...,r$ such that $M\cong M_1\times...\times M_r$ . As a starting point, we describe how the intrinsic product structure of $M$ is reflected by a distinguished, fiberwise orthogonal direct sum decomposition of the corresponding first normal bundle. Then we consider the (second) osculating bundle $\osc f$, which is a $\nabla^N$-parallel vector subbundle of the pullback bundle $f^*TN$, and establish the existence of $r$ distinguished, pairwise commuting, $\nabla^N$-parallel vector bundle involutions on $\osc f$ . Consequently, the "extrinsic holonomy Lie algebra" of $\osc f$ bears naturally the structure of a graded Lie algebra over the Abelian group which is given by the direct sum of $r$ copies of $\Z/2 \Z$ . Our main result is the following: Provided that $N$ is of compact or non-compact type, that $\dim(M_i)\geq 3$ for $i=1,...,r$ and that none of the product slices through one point of $M$ gets mapped into any flat of $N$, we can show that $f(M)$ is a homogeneous submanifold of $N$ .