Atsushi Katsuda

Closed orbits in homology classes

© Publications mathématiques de l’I.H.É.S., 1990, tous droits réservés. L’accès aux archives de la revue « Publications mathématiques de l’I.H.É.S. » (http:// www.ihes.fr/IHES/Publications/Publications.html) implique l’accord avec les conditions générales d’utilisation (http://www.numdam.org/legal.php). 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.

Isospectral graphs and isoperimetric constants

Abstract We give two methods for constructing isospectral graphs. As an application, families of infinite pairs of regular isospectral graphs with different magnification are presented.

Homology of closed geodesics in certain Riemannian manifolds

It is shown, by using the trace formula of Selberg type, that every primitive, one-dimensional homology class of a negatively curved compact locally symmetric space contains infinitely many prime closed geodesics. 0. Let M be a compact space form of a symmetric space of rank one. In this note, we prove that each homology class in H1 (M, Z) contains infinitely many free homotopy classes of closed curves, that is, the mapping induced from the Hurewicz homomorphism [XI (M)] -? H1 (M, Z) is an oo-to-one correspondence. One of the geometric consequences is that any primitive homology class contains infinitely many prime closed geodesics, since, as was shown by Hadamard, every nonnull homotopy class contains a closed geodesic which is automatically prime if the homology class is primitive. Here a homology class a is called primitive if Ol is not a nontrivial integral multiple of another homology class. If dimM = 2, then one can prove the much stronger assertion that every homology class contains infinitely many prime closed geodesics (see ?2). The following theorem, which can be shown by means of a number-theoretic argument applied to the L-functions associated to length spectrum of closed geodesics (see [1, 4] for proof), is somewhat related to the result. THEOREM. Let H be a subgroup of H1(M, Z) of finite index, and let al be a coset in Hi/H. If M is negatively curved, then there exist infinitely many prime closed geodesics whose homology classes are in ca. 1. The proof relies heavily on the trace formula for the heat kernel function. We shall start with a general setting. Let 7r: M -? M be the universal covering of a compact Riemannian manifold M. The fundamental group ri (M) acts on M in the usual way. For brevity we write r for 7ri(M). For an element -y in r, we denote by rF the centralizer of -y, and by [-y] the conjugacy class of -y. The set of all conjugacy classes is denoted by [r]. Let p: r -, U(N) be a unitary representation, and let Ep be the flat vector bundle associated to p. We denote by AP the Laplacian acting on the sections of Ep. The fundamental solution of the heat equation on M will be denoted by k(t; x, y). The following lemma is proved in the same way as the proof of the Selberg trace formula (see [6]). LEMMA 1. tr(e-tAP) E tr p(y)J k(t; x, ay) dx. [I]E[r] Mr Received by the editors January 22, 1985. 1980 Mathematics Subject Classification. Primary 53C35, 53C22. tSupported by the Ishida Foundation. ?1986 American Mathematical Society 0002-9939/86

Stability and reconstruction in Gel'fand inverse boundary spectral problem

1.00 +

The isometry groups of compact manifolds with negative Ricci curvature

.25 per page

Correction to: "Gromov's convergence theorem and its application''

In this paper we study stability and approximate reconstruction in the inverse boundary spectral problem (the generalized Gelfand inverse problem [12]) for Riemannian manifolds. We denote by (M, g) an unknown, m-dimensional, compact connected Riemannian manifold with a (smooth) metric g and non-empty boundary ∂M. The boundary ∂M is itself an (m − 1)-dimensional compact differentiable manifold. We do not assume the knowledge of i* (g) on ∂M, where i : ∂M → M is an embedding or the corresponding area element dS g . Because the boundary ∂M is known, we will consider a class M = M ∂M of compact, connected Riemannian manifolds which have the same boundary, ∂M.

Boundary regularity for the Ricci equation, geometric convergence, and Gel’fand’s inverse boundary problem

We estimate the order of the isometry groups of compact manifolds with negative Ricci curvature in terms of geometric quantities: the sectional curvature, the Ricci curvature, the diameter, and the injectivity radius.

Gromov's convergence theorem and its application

In the proof of Lemma 12.2, the following inequality (p. 43, line 11) is incorrect. This should be corrected as follows.