By Isaac Chavel

The fundamental objectives of the booklet are: (i) to introduce the topic to these attracted to studying it, (ii) to coherently current a few uncomplicated thoughts and effects, presently utilized in the topic, to these operating in it, and (iii) to give a few of the effects which are appealing of their personal correct, and which lend themselves to a presentation now not overburdened with technical equipment.

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Ce cours de topologie a été dispensé en licence à l'Université de Rennes 1 de 1999 à 2002. Toutes les constructions permettant de parler de limite et de continuité sont d'abord dégagées, puis l'utilité de los angeles compacité pour ramener des problèmes de complexité infinie à l'étude d'un nombre fini de cas est explicitée.

This publication is the 6th version of the vintage areas of continuing Curvature, first released in 1967, with the former (fifth) variation released in 1984. It illustrates the excessive measure of interaction among team concept and geometry. The reader will enjoy the very concise remedies of riemannian and pseudo-riemannian manifolds and their curvatures, of the illustration thought of finite teams, and of symptoms of contemporary development in discrete subgroups of Lie teams.

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Then P(u) is afundarnental domain of T, that is, no two distinct points of P(u) __ represent the same point of T, and the image of the closed parallelopiped P(u), under the covering, is all of T. Note that v01 T = V O P(u) ~ for every u E r. We simply refer to P(u) as a copy of T. Also note that if A: R" + R" is a linear transformation for which r = A(Z'), then r*= (A*)-'(E"),where A* is the adjoint of A. Therefore, if T* is the torus determined by r*then we have vol T = ldet A1 = (vol T*)-'; 3.

R, be pairwise disjoint normal domains in M , whose boundaries, when intersecting d M , do so transversally. Given an eigenvalue problem on M , consider, for each r = 1, . , m, the eigenvalue problem on R, obtained by requiring vanishing Dirichlet data on dR, n M and by leaving the 18 1. The Laplacian original data on aR, n aM unchanged. Arange all the eigenvalues of in an increasing sequence a,, . ,R, 0I v1 I I v2 * * a with each eigenvalue repeated according to its multiplicity, and let the eigenvalues of M be given as in (79).

From (44) we would conclude that t , E (0,n/2&). But 1(6) I A(n/2fi) = nK implies that { ( V c , - v's,)s"-'}(t) = {n(6) - n K } 1: Vs: is nonnegative on (0, 6). Therefore at t = t , we have V ( t , ) 2 0-a tradiction. Thus for K > 0, 6 > n/2<~, we have p(6) > 1(6). con- THEOREM4. If K = 0, that is, if M K= R", then there exist positive constants c D ,cNsuch that A(6) = Ci/62, p(6) = Ci/62 for all 6 > 0. PROOF:For K = 0 we have S,(t) = t, C,(t) = 1, so the dzerential equation under study is y" + ( n -t 1) ~ 1(1 + n - 2) y = 0.