Mathematics – Differential Geometry
Scientific paper
2009-11-19
Mathematics
Differential Geometry
25 pages, Appendix A added, a few corrections, new numbering of the theorems
Scientific paper
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 $N$ .
No associations
LandOfFree
Parallel submanifolds with an intrinsic product structure does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.
If you have personal experience with Parallel submanifolds with an intrinsic product structure, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Parallel submanifolds with an intrinsic product structure will most certainly appreciate the feedback.
Profile ID: LFWR-SCP-O-453793