Carlos Enrique Olmos - Academia.edu (original) (raw)

Papers by Carlos Enrique Olmos

Transformation Groups

This is a substantially improved version of an earlier preprint of the authors with a similar tit... more This is a substantially improved version of an earlier preprint of the authors with a similar title. We develop a general theory for irreducible homogeneous spaces M = G/H, in relation to the nullity distribution ν of their curvature tensor. We construct natural invariant (different and increasing) distributions associated with the nullity, that give a deep insight of such spaces. In particular, there must exist an order-two transvection, not in the nullity, with null Jacobi operator. This fact was very important for finding out the first homogeneous examples with non-trivial nullity, i.e. where the nullity distribution is not parallel. Moreover, we construct irreducible examples of conullity k = 3, the smallest possible, in any dimension. None of our examples admit a quotient of finite volume. We also proved that H is trivial and G is solvable if k = 3. Another of our main results is that the leaves, i.e. the integral manifolds, of the nullity are closed (we used a rather delicate argument). This implies that M is a Euclidean affine bundle over the quotient by the leaves of ν. Moreover, we prove that ν ⊥ defines a metric connection on this bundle with transitive holonomy or, equivalently, ν ⊥ is completely non-integrable (this is not in general true for an arbitrary autoparallel and flat invariant distribution). We also found some general obstruction for the existence of non-trivial nullity: e.g., if G is reductive (in particular, if M is compact), or if G is two-step nilpotent.

Journal für die reine und angewandte Mathematik (Crelles Journal)

In 1980, Oniščik [A. L. Oniščik, Totally geodesic submanifolds of symmetric spaces, Geometric met... more In 1980, Oniščik [A. L. Oniščik, Totally geodesic submanifolds of symmetric spaces, Geometric methods in problems of algebra and analysis. Vol. 2, Yaroslav. Gos. Univ., Yaroslavl’ 1980, 64–85, 161] introduced the index of a Riemannian symmetric space as the minimal codimension of a (proper) totally geodesic submanifold. He calculated the index for symmetric spaces of rank {\leq 2}, but for higher rank it was unclear how to tackle the problem. In [J. Berndt, S. Console and C. E. Olmos, Submanifolds and holonomy, 2nd ed., Monogr. Res. Notes Math., CRC Press, Boca Raton 2016], [J. Berndt and C. Olmos, Maximal totally geodesic submanifolds and index of symmetric spaces, J. Differential Geom. 104 2016, 2, 187–217], [J. Berndt and C. Olmos, The index of compact simple Lie groups, Bull. Lond. Math. Soc. 49 2017, 5, 903–907], [J. Berndt and C. Olmos, On the index of symmetric spaces, J. reine angew. Math. 737 2018, 33–48], [J. Berndt, C. Olmos and J. S. Rodríguez, The index of exceptional s...

manuscripta mathematica

Let M be a most singular orbit of the isotropy representation of a simple symmetric space. Let (ν... more Let M be a most singular orbit of the isotropy representation of a simple symmetric space. Let (ν i , Φ i) be an irreducible factor of the normal holonomy representation (νpM, Φ(p)). We prove that there exists a basis of a section Σ i ⊂ ν i of Φ i such that the corresponding shape operators have rational eigenvalues (this is not in general true for other isotropy orbits). Conversely, this property, if referred to some non-transitive irreducible normal holonomy factor, characterizes the isotropy orbits. We also prove that the definition of a submanifold with constant principal curvatures can be given by using only the traceless shape operator, instead of the shape operator, restricted to a non-transitive (non necessarily irreducible) normal holonomy factor. This article generalizes previous results of the authors that characterized Veronese submanifolds in terms of normal holonomy.

São Paulo Journal of Mathematical Sciences

Journal of Differential Geometry, 2016

Let M be an irreducible Riemannian symmetric space. The index i(M) of M is the minimal codimensio... more Let M be an irreducible Riemannian symmetric space. The index i(M) of M is the minimal codimension of a totally geodesic submanifold of M. In [1] we proved that i(M) is bounded from below by the rank rk(M) of M , that is, rk(M) ≤ i(M). In this paper we classify all irreducible Riemannian symmetric spaces M for which the equality holds, that is, rk(M) = i(M). In this context we also obtain an explicit classification of all non-semisimple maximal totally geodesic submanifolds in irreducible Riemannian symmetric spaces of noncompact type and show that they are closely related to irreducible symmetric R-spaces. We also determine the index of some symmetric spaces and classify the irreducible Riemannian symmetric spaces of noncompact type with i(M) ∈ {4, 5, 6}.

Journal of Differential Geometry, 1994

Journal of Differential Geometry, 1997

Journal of Differential Geometry, 1993

Proceedings of the American Mathematical Society, 2001

A recent result of C. Gorodski and G. Thorbergsson, involving classification, asserts that a vari... more A recent result of C. Gorodski and G. Thorbergsson, involving classification, asserts that a variationally complete representation is polar. The aim of this paper is to give a conceptual and very short proof of this fact, which is the converse of a result of Conlon.

In this paper we give a geometric proof of the Karpelevich's theorem that asserts that a semisimp... more In this paper we give a geometric proof of the Karpelevich's theorem that asserts that a semisimple Lie subgroup of isometries, of a symmetric space of non compact type, has a totally geodesic orbit. In fact, this is equivalent to a well-known result of Mostow about existence of compatible Cartan decompositions.

Revista De La Union Matematica Argentina, Jun 1, 2006

Page 1. Rev. Un. Mat. Argentina, Vol. 47-1 II Encuentro de Geometría Diferencial 6 al 11 de junio... more Page 1. Rev. Un. Mat. Argentina, Vol. 47-1 II Encuentro de Geometría Diferencial 6 al 11 de junio de 2005. La Falda, Sierras de Córdoba, Argentina. Las motivaciones para la organización de este II Encuentro de Geometría ...

Revista De La Union Matematica Argentina, Apr 7, 2011

Proceedings of the American Mathematical Society, 2015

Journal für die reine und angewandte Mathematik (Crelles Journal), 2015

... Since the sum of the 2 -th powers of the eigenvalues up to order $ = dim M generate all symme... more ... Since the sum of the 2 -th powers of the eigenvalues up to order $ = dim M generate all symmetric polynomials on © 1 , WiWiW , ©m , the character-... In this case, if we diagonalize Qξ, (letting © 1 , WiWiW , © be the different eigenvalues) ...