By Arthur L. Besse

*From the reviews:*

"[...] an effective reference e-book for plenty of basic ideas of Riemannian geometry. [...] regardless of its size, the reader could have no trouble in getting the texture of its contents and learning first-class examples of all interplay of geometry with partial differential equations, topology, and Lie teams. primarily, the ebook offers a transparent perception into the scope and variety of difficulties posed by way of its title."*S.M. Salamon* in *MathSciNet 1988*

*"*It appeared more likely to a person who learn the former ebook via an identical writer, specifically *"Manifolds all of whose geodesic are closed",* that the current ebook will be some of the most vital ever released on Riemannian geometry. This prophecy is certainly fulfilled."*T.J. Wilmore* in *Bulletin of the London Mathematical Society 1987*

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**Additional info for Einstein Manifolds**

**Example text**

D F. Einstein Manifolds We first collect various properties ofthe Riemann curvature tensor R that we have met before. 85 Proposition. , R(X, Y) = -R(Y,X); 42 1. 85f) (differential Bianchi identity) (DR)(X, Y,Z) + (DR)(Y,Z,X) + (DR)(Z, X, Y) = O. ). 86. Using the metrie g, we mayaiso eonsider the eurvature as a (4, O)-tensor, namely (X, Y, Z, U) --+ g(R(X, Y)Z, U). We will also use the (2, 2)-tensor dedueed from R, that we denote by Bt. Due to the symmetries, Bt may be eonsidered as a linear map from /\2 M to /\2 M, satisfying g(Pll(X A Y), Z A U) = g(R(X, Y)Z, U) for any veetors X, Y, Z, U.

C) The assumption that the connection D be torsion-free enables one to express brackets of vector fields in terms of D: (X, Y) H Dx Y - [X, Y] = DxY - DyX Dually, the exterior differential of differential forms mayaiso be expressed in terms of D. For cxEQPM, and vectors X o, ... 20. We now come back to the relations between the iterated covariant derivatives. The simplest one is the following so-called Rieci formula. 21) where T is the torsion field of D and RV the curvature of V. e. symmetrie), then the right hand side does not involve Vs.

16 Definition. A linear connection D on a manifold M is a linear connection on the tangent bundle TM of M. 17. Let V be a linear connection on a vector bundle E over M. 8a)). Now let D be a linear connection on M. Then D and V induce a linear connection (that we still denote by V) on T*M ® E, so we may define V(Vs), and we denote it by V2 s. It is the section of T(2,0)M ® E defined by (V 2 sh,y = Vx(Vys) - V(DxY)S. Instead of(V 2 sh,y we may write (VVs)(X, Y), or V;,yS, or even (V 2 s)(X, Y). Now, using an obvious induction, we may define the iterated covariant derivative VP s as the section of T(p, 0) M ® E such that 26 1.