Hubert Goldschmidt

Complexes of Differential Operators and Symmetric Spaces

We present here an introduction to the theory of overdetermined linear partial differential equations and of the resulting complexes of differential operators. We then illustrate the theory by describing explicitly the various sequences which are associated to the Killing operator on a symmetric space. This paper may be considered as a complement to Chapter II of our expository essay [8].

Linear Differential Systems

The goal of this chapter is to develop the formalism of linear Pfaffian differential systems in a form that will facilitate the computation of examples.

Infinitesimal Isospectral Deformations of Symmetric Spaces, II: Quotients of the Special Unitary Group of Rank Two

We study the space I(X) of infinitesimal isospectral deformations of an irreducible and reduced symmetric space X of compact type when X is a quotient of the special unitary group G = SU(n), with n ≥ 3. If X is the reduced space of the special unitary group SU(n) or of the special Lagrangian Grassmannian SU(n)/SO(n), the non-zero G-invariant symmetric 3-form on X gives rise to a linear mapping Φ0 : C∞ R (X) → I(X), where C∞ R (X) is the space of real-valued functions on X. Previously, we constructed a subspace FX of C∞ R (X) of finite-codimension and showed that the restriction Φ : FX → I(X) of Φ0 is a monomorphism. Here we prove that, when n = 3, the mapping Φ is an isomorphism and thus obtain in this case an explicit description of the deformation space I(X).

Infinitesimal Isospectral Deformations of Symmetric Spaces: Quotients of The Special Unitary Group

Abstract: We show that the reduced spaces of the special unitary group SU(n) and the symmetric space SU(2n)/Sp(n), with n ≥ 3, possesses nontrivial infinitesimal isospectral deformations. For the reduced space X of the unitary group SU(n), we also prove a related result: in all degrees p ≥ 2, there exist symmetric p-forms on X which satisfy the Guillemin condition and are not symmetrized covariant derivatives of symmetric (p − 1)-forms, unless n = p = 3.

Radon Transforms and the Rigidity of the Grassmannians (AM-156)

University Press. All rights reserved. No part of this book may be reproduced in any form by any electronic or mechanical means (including photocopying, recording, or information storage and retrieval) without permission in writing from the publisher, except for reading and browsing via the World Wide Web. Users are not permitted to mount this file on any network servers. INTRODUCTION This monograph is motivated by a fundamental rigidity problem in Riemannian geometry: determine whether the metric of a given Rieman-nian symmetric space of compact type can be characterized by means of the spectrum of its Laplacian. An infinitesimal isospectral deformation of the metric of such a symmetric space belongs to the kernel of a certain Radon transform defined in terms of integration over the flat totally geodesic tori of dimension equal to the rank of the space. Here we study an infinitesi-mal version of this spectral rigidity problem: determine all the symmetric spaces of compact type for which this Radon transform is injective in an appropriate sense. We shall both give examples of spaces which are not infinitesimally rigid in this sense and prove that this Radon transform is injective in the case of most Grassmannians. At present, it is only in the case of spaces of rank one that infinitesimal rigidity in this sense gives rise to a characterization of the metric by means of its spectrum. In the case of spaces of higher rank, there are no analogues of this phenomenon and the relationship between the two rigidity problems is not yet elucidated. However, the existence of infinitesimal deformations belonging to the kernel of the Radon transform might lead to non-trivial isospectral deformations of the metric. Here we also study another closely related rigidity question which arises from the Blaschke problem: determine all the symmetric spaces for which the X-ray transform for symmetric 2-forms, which consists in integrating over all closed geodesics, is injective in an appropriate sense. In the case of spaces of rank one, this problem coincides with the previous Radon transform question. The methods used here for the study of these two problems are similar in nature. Let (X, g) be a Riemannian symmetric space of compact type. Consider a family of Riemannian metrics {g t } on X, for |t| < ε, with g 0 = g. The family {g t } is said to be an isospectral deformation of g if the spectrum …

Injectivité de la transformation de Radon sur les grassmanniennes

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Linear Differential Operators

In this chapter, we consider only linear systems of partial differential equations, and use the notation and terminology introduced in Chapter IX. In general, if D: e → ℱ is a linear differential operator, where E, F are vector bundles over the manifold X, and if f is a section of F, the inhomogeneous equation

Cartan-Kähler Theory

Du = f