Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Parallel submanifolds with an intrinsic product structure   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 $N$ .

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The categories of this paper are: comp-gas
9
The categories of this paper are: nlin
10
The categories of this paper are: q-bio
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The categories of this paper are: bayes-an
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The categories of this paper are: eess
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The categories of this paper are: math-ph
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The categories of this paper are: q-fin
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The categories of this paper are: cmp-lg
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The categories of this paper are: q-alg
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The categories of this paper are: atom-ph
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The categories of this paper are: physics
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The categories of this paper are: nucl-ex
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The categories of this paper are: dg-ga
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The categories of this paper are: mtrl-th
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The categories of this paper are: gr-qc
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The categories of this paper are: acc-phys