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Parallel submanifolds with an intrinsic product structure
Let and be Riemannian symmetric spaces and be a parallel
isometric immersion. We additionally assume that there exist simply connected,
irreducible Riemannian symmetric spaces with for
such that . As a starting point, we
describe how the intrinsic product structure of 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 -parallel vector subbundle of the
pullback bundle , and establish the existence of distinguished,
pairwise commuting, -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 copies of . Our main result is the following:
Provided that is of compact or non-compact type, that for
and that none of the product slices through one point of gets
mapped into any flat of , we can show that is a homogeneous
submanifold of .Comment: 25 pages, Appendix A added, a few corrections, new numbering of the
theorem
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