Parallel submanifolds with an intrinsic product structure

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 $\dim (M_i)\ge 2$ 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 $f^{\ast }TN$, 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 $\mathbb{Z}/2\mathbb{Z}$ . Our main result is the following: Provided that $N$ is of compact or non-compact type, that $\dim (M_i)\ge 3$ 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 $N$ .

25 pages, Appendix A added, a few corrections, new numbering of the theorems

References in corpus (2)

• Extrinsic homogeneity of parallel submanifolds
• The extrinsic holonomy Lie algebra of a parallel submanifold