Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Normal holonomy of orbits and Veronese submanifolds   It was conjectured, twenty years ago, the following result that would
generalize the so-called rank rigidity theorem for homogeneous Euclidean
submanifolds: let M^n, n>=2, be a full and irreducible homogeneous submanifold
of the sphere $S^{N-1}\subset R^N$ and such that the normal holonomy group is
not transitive (on the unit sphere of the normal space to the sphere). Then M^n
must be an orbit of an irreducible s-representation (i.e. the isotropy
representation of a semisimple Riemannian symmetric space).
  If n=2, then the normal holonomy is always transitive, unless M is a
homogeneous isoparametric hypersurface of the sphere (and so the conjecture is
true in this case). We prove the conjecture when n=3. In this case M^3 must be
either isoparametric or a Veronese submanifold. The proof combines geometric
arguments with (delicate) topological arguments that uses information from two
different fibrations with the same total space (the holonomy tube and the
caustic fibrations).
  We also prove the conjecture for n>= 3 when the normal holonomy acts
irreducibly and the codimension is the maximal possible n(n+1)/2. This gives a
characterization of Veronese submanifolds in terms of normal holonomy. We also
extend this last result by replacing the homogeneity assumption by the
assumption of minimality (in the sphere).
  Another result of the paper, used for the case n=3, is that the number of
irreducible factors of the local normal holonomy group, for any Euclidean
submanifold M^n, is less or equal than [n/2] (which is the rank of the
orthogonal group SO(n)). This bound is sharp and improves the known bound
n(n-1)/2.

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The categories of this paper are: acc-phys
5
The categories of this paper are: gr-qc
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The categories of this paper are: q-fin
3
The categories of this paper are: physics
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The categories of this paper are: dg-ga
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The categories of this paper are: cmp-lg
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The categories of this paper are: patt-sol
10
The categories of this paper are: q-bio
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The categories of this paper are: nlin
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The categories of this paper are: adap-org
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The categories of this paper are: cond-mat
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The categories of this paper are: alg-geom
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The categories of this paper are: ao-sci
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The categories of this paper are: astro-ph
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The categories of this paper are: plasm-ph
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The categories of this paper are: nucl-th