Riemannian Geometry and Geometric Analysis (6th Edition) (Universitext)
Jürgen Jost
Language: English
Pages: 616
ISBN: B00FC983BM
Format: PDF / Kindle (mobi) / ePub
This established reference work continues to lead its readers to some of the hottest topics of contemporary mathematical research. The previous edition already introduced and explained the ideas of the parabolic methods that had found a spectacular success in the work of Perelman at the examples of closed geodesics and harmonic forms. It also discussed further examples of geometric variational problems from quantum field theory, another source of profound new ideas and methods in geometry.
The 6th edition includes a systematic treatment of eigenvalues of Riemannian manifolds and several other additions. Also, the entire material has been reorganized in order to improve the coherence of the book.
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⊗C ClC (R2 ) we put l(vj ) := ivj ⊗ e1 e2 , for j = 1, . . . , n, l(eα ) := 1 ⊗ eα , for α = 1, 2. Since for example l(vj vk + vk vj ) = (−vj vk − vk vj ) ⊗ e1 e2 e1 e2 = vj vk + vk vj ⊗ 1 l(vj eα + eα vj ) = ivj ⊗ (e1 e2 eα + eα e1 e2 ) = 0 for α = 1, 2 we have an extension of l as an algebra homomorphism l : Cl(V ⊕ R2 ) → ClC (V ) ⊗C ClC (R2 ). Extending scalars from R to C, we obtain an algebra homomorphism l : ClC (V ⊕ R2 ) → ClC (V ) ⊗C ClC (R2 ). Now l has become a homomorphism between
4.1 Connections in Vector Bundles Let X be a vector field on Rd , V a vector at x0 ∈ Rd . We want to analyze how one takes the derivative of X at x0 in the direction V. For this derivative, one forms X(x0 + tV ) − X(x0 ) . t→0 t lim Thus, one first adds the vector tV to the point x0 . Next, one compares the vector X(x0 + tV ) at the point x0 + tV and the vector X(x0 ) at x0 ; more precisely, one subtracts the second vector from the first one. Division by t and taking the limit then are obvious
(M ). Thus D : Ωp (E) → Ωp+1 (E), 0 ≤ p ≤ d. We want to compare this with the exterior derivative d : Ωp → Ωp+1 . Here, we have d ◦ d = 0. Such a relation, however, in general does not hold anymore for D. 4.1 Connections in Vector Bundles 139 Definition 4.1.5. The curvature of a connection D is the operator F := D ◦ D : Ω0 (E) → Ω2 (E). The connection is called flat, if its curvature satisfies F = 0. The exterior derivative d thus yields a flat connection on the trivial bundle M × R. We
Chapter 4 Connections and Curvature that M then has the same intersection form as this union of CP2 ’s. As will be demonstrated in §6.1, H 2 (CP2 , R) = R, and the intersection form of CP2 is 1. These facts then imply Donaldson’s theorem. The main work in the proof goes into deriving the stated properties of the moduli space M. In particular, one uses a theorem of Taubes on the existence of selfdual connections over four-manifolds with definite intersection form. Donaldson then went on to use
one then assumes that the curvature is such that the curvature term also is nonnegative, both terms have to vanish identically, because the integral of the left-hand side vanishes. The vanishing of the square term then implies that the object is parallel. If the curvature is even positive, the vanishing of the curvature term implies that the object itself vanishes. We shall see another instance of the Bochner method in §8.2. When combining the preceding reasoning with the Weitzenb¨ock formula of
