Submanifold geometry in symmetric spaces (original) (raw)

Submanifolds Of Maximal Nullity In Symmetric Spaces

Results in Mathematics, 1999

A non-totally-geodesic submanifold of relative nullity n − 1 in a symmetric space M is a cylinder over one of the following submanifolds: a surface F 2 of nullity 1 in a totally geodesic submanifold N 3 ⊂ M locally isometric to S 2 (c) × R or H 2 (c) × R; a submanifold F k+1 spanned by a totally geodesic submanifold F k (c) of constant curvature moving by a special curve in the isometry group of M ; a submanifold F k+1 of nullity k in a flat totally geodesic submanifold of M ; a curve.

Closed geodesics and flat tori in spectral theory on symmetric spaces

Séminaire de théorie spectrale et géométrie, 1993

L'accès aux archives de la revue « Séminaire de Théorie spectrale et géométrie » implique l'accord avec les conditions générales d'utilisation (http://www.numdam.org/conditions). Toute utilisation commerciale ou impression systématique est constitutive d'une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright. Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques http://www.numdam.org/ Séminaire de théorie spectrale et géométrie GRENOBLE 1992-1993 (85-103) CLOSED GEODESICS AND FLAT TORI IN SPECTRAL THEORY ON SYMMETRIC SPACES 7 David GURARIE Closed geodesics are known to play an important rôle in spectral theory of Laplacians on Riemannian manifolds Jk. They also contribute to spectral theory of Schrôdinger operators A + V, typically in the form of higher order correction to the principal (Laplace) eigenvalue. We give a brief survey of the classical "Shape of metric* and "Shape of potentiaP problems of spectral theory, and explore the rôle of "length spectrum" (the length of ail closed path/geodesics), and the related "Radon transform of V. Then we outline some récent progress in spécial cases: the n-sphere theory, and Schrôdinger operators on higher rank symmetric spaces. The latter case brings in new players: the flat tori. They naturally appear in higher rank compact symmetric spaces and play the rôle of closed geodesics hère. We conclude by a list of open problems. David GURARIE CLEVELAND STATE UNIVERSITY Department of Math. Euclid Ave at 24 th St Cleveland OHIO 44115 U.S.A.

A differential geometric characterization of symmetric spaces of higher rank

Publications Mathématiques de l'IHÉS, 1990

In this paper we consider simply connected Riemannian manifolds l~I of non-positive sectional curvature whose isometry group I(~) is large in an appropriate sense. We do not assume that M has a lower bound for the sectional curvature, and we do not assume that I(~I) contains ...

Totally geodesic submanifolds of symmetric spaces, II

Duke Mathematical Journal, 1978

... 746 BANG-YEN CHEN AND TADASHI NAGANO The basic facts on symmetric spaces we need in this paper may be found in [4] and [5]. 2. The space Qm and its totally geodesic submanifolds. We write Q Qm for the Grassman manifold G2(E2 + m) and P for the orient-ed 2 ...

On Riemannian symmetric spaces of rank one

Advances in Mathematics, 1970

In this paper, we present an elementary (i.e., without the structure theory of compact Lie groups) exposition of the theory of Riemannian symmetric spaces of rank 1. In particular, we are able to derive directly, using the theory of curvature and geodesics in Riemannian spaces: (i) If M is a Riemannian symmetric space of rank 1 which is not simply connected, then it is a real projective space of constant sectional curvature. (ii) If M is Riemannian symmetric of rank 1 with odd dimension, then it has constant sectional curvature.

Curvature on reductive homogeneous spaces

2007

Here we consider the general flag manifold F Θ as a naturally reductive homogeneous space endowed with an U -invariant metric Λ Θ and an invariant almost-complex structure J Θ . The main objective of this work is to explore the riemannian connection associated with the metric Λ Θ in order to calculate some classes of curvatures which should allow us to confirm, in a simple way, that flag manifolds are either not biholomorfically equivalent nor holomorphically isometric to any complex projective space.

Maximal totally geodesic submanifolds and index of symmetric spaces

Journal of Differential Geometry, 2016

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}.