Normal holonomy of orbits and Veronese submanifolds

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It was conjectured, twenty years ago, the following result that would generalize the so-called rank rigidity theorem for homogeneous Euclidean submanifolds: let Mn, n ≥ 2, be a full and irreducible homogeneous submanifold of the sphere SN−1 ⊂ RN such that the normal holonomy group is not transitive...

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Autores principales: Olmos, Carlos Enrique, Riaño Riaño, Richar Fernando
Formato: article
Lenguaje:Inglés
Publicado: 2022
Materias:
Normal holonomy
Orbits of s-representations
Veronese submanifolds
Acceso en línea:http://hdl.handle.net/11086/23638
https://doi.org/10.2969/jmsj/06730903
Aporte de:
Repositorio Digital Universitario (UNC) de Universidad Nacional de Córdoba
id I10-R141-11086-23638
record_format dspace
institution Universidad Nacional de Córdoba
institution_str I-10
repository_str R-141
collection Repositorio Digital Universitario (UNC)
language Inglés
topic Normal holonomy
Orbits of s-representations
Veronese submanifolds
spellingShingle Normal holonomy
Orbits of s-representations
Veronese submanifolds
Olmos, Carlos Enrique
Riaño Riaño, Richar Fernando
Normal holonomy of orbits and Veronese submanifolds
topic_facet Normal holonomy
Orbits of s-representations
Veronese submanifolds
description It was conjectured, twenty years ago, the following result that would generalize the so-called rank rigidity theorem for homogeneous Euclidean submanifolds: let Mn, n ≥ 2, be a full and irreducible homogeneous submanifold of the sphere SN−1 ⊂ RN such that the normal holonomy group is not transitive (on the unit sphere of the normal space to the sphere). Then Mn 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 M3 must be either isoparametric or a Veronese submanifold. The proof combines geometric arguments with (delicate) topological arguments that use 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 Mn, 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.
format article
author Olmos, Carlos Enrique
Riaño Riaño, Richar Fernando
author_facet Olmos, Carlos Enrique
Riaño Riaño, Richar Fernando
author_sort Olmos, Carlos Enrique
title Normal holonomy of orbits and Veronese submanifolds
title_short Normal holonomy of orbits and Veronese submanifolds
title_full Normal holonomy of orbits and Veronese submanifolds
title_fullStr Normal holonomy of orbits and Veronese submanifolds
title_full_unstemmed Normal holonomy of orbits and Veronese submanifolds
title_sort normal holonomy of orbits and veronese submanifolds
publishDate 2022
url http://hdl.handle.net/11086/23638
https://doi.org/10.2969/jmsj/06730903
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AT rianorianoricharfernando normalholonomyoforbitsandveronesesubmanifolds
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