By Roger Godement

Ce 4?me quantity de l'ouvrage *Analyse math?matique* initiera le lecteur ? l'analyse fonctionnelle (int?gration, espaces de Hilbert, examine harmonique en th?orie des groupes) et aux m?thodes de l. a. th?orie des fonctions modulaires (s?ries L et theta, fonctions elliptiques, utilization de l'alg?bre de Lie de SL2). Tout comme pour les volumes 1 ? three, on reconna?tra ici encore, le sort inimitable de l'auteur et pas seulement par son refus de l'ecriture condens?e en utilization dans de nombreux manuels. Mariant judicieusement les math?matiques dites 'modernes' et' classiques', l. a. premi?re partie (Int?gration) est d'utilit? universelle tandis que l. a. seconde oriente le lecteur vers un domaine de recherche sp?cialis? et tr?s actif, avec de vastes g?n?ralisations possibles.

**Read or Download Analyse mathématique IV: Intégration et théorie spectrale, analyse harmonique, le jardin des délices modulaires PDF**

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R = Σnj=1 Ric(ej , ej ). The plane E ⊂ Tp (M) spanned by X, Y is called a section of the tangent space. The sectional curvature with respect to E is sec(E) = sec(X, Y ) = R(X, Y, Y, X) . 3), R = gij Rij = gij gkl Rkijl = −gij gkl Rikjl . Rij = ∂k Γkij − ∂i Γkkj + Γkkp Γpij − Γkip Γpkj . 16) 52 Chapter 3. 6 The curvature tensor has a rich structure. For example, it can be decomposed into the sum of three parts. The first part is determined solely by the scalar curvature, the second part by the Ricci curvature.

E. {dxi } is the canonical basis for the cotangent space. We define w= U φ(U ) f ◦ φ−1 dx1 . . dxn . Let {(Ui , φi )} be a family of local charts for M such that {(Ui , hi )} is a partition of unity for M, then we define w = Σi M hi w = Σi hi w. 4 One needs to prove that the above integration is independent of the choice of local charts, or the partition of unity. This is where we are using antisymmetry of forms. Let (U, ψ), ψ = (y 1 , , , , y n ) be another local chart. These y i are smooth functions on M.

E. (∇T )(X, X1 , . . Xp , η1 , . . , ηq ) ≡ (∇X T )(X1 , . . Xp , η1 , . . , ηq ). 1. 5 The motivation behind the above definition is the Leibnitz rule for differentiation. The term X(T (X1 , . . Xp , η1 , . . , ηq )) is nothing but the directional derivative of the scalar function T (X1 , . . Xp , η1 , . . , ηq ) in the direction of X. 14 (Riemann manifold) A Riemann manifold is a smooth manifold with a Riemann metric, a smooth, positive definite, symmetric (2, 0) tensor field. The Riemann metric is also called an inner product for vector fields.