By Michael Reed, Barry Simon

BESTSELLER of the XXth Century in Mathematical Physics voted on by means of contributors of the XIIIth overseas Congress on Mathematical Physics

This revision will make this ebook extra beautiful as a textbook in useful research. additional refinement of insurance of actual issues also will toughen its well-established use as a direction publication in mathematical physics.

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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.