By Jürgen Jost

ISBN-10: 3540571132

ISBN-13: 9783540571131

ISBN-10: 3662031183

ISBN-13: 9783662031186

This textbook introduces suggestions from nonlinear research at an early degree. Such concepts have lately turn into an imperative software in learn in geometry, and they're handled the following for the 1st time in a textbook. issues handled comprise: Differentiable and Riemannian manifolds, metric homes, tensor calculus, vector bundles; the Hodge Theorem for de Rham cohomology; connections and curvature, the Yang-Mills practical; geodesics and Jacobi fields, Rauch comparability theorem and functions; Morse idea (including an creation to algebraic topology), functions to the life of closed geodesics; symmetric areas and K?hler manifolds; the Palais-Smale and closed geodesics; Harmonic maps, minimum surfaces.

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6. 7 Let M carry a Riemannian metric g = gijdxi ® dxi. A vector field X on M is cailed Killing field or infinitesimal isometry if Lx(g) = 0. 6. 7 A vector field X on a Riemannian manifold M is a Killing field if and only if the local1-parameter group generated by X consists of local isometrics. 21) :t(1/J;g)Jt=0 = 0. 22) Since this holds for every point of M, we obtain 1/J; g =g for all t E I. Therefore, the diffeomorphisms 1/Jt are isometries. 21). 8 The Killing fields of a Riemannian manifold constitute a Lie algebra.

6. 4. 7 Let M be a compact Riemannian manifold. Then for any p E M, the exponential map expp is defined on all of TpM, and any geodesic may be extended indefinitely in each direction. Proof. For v E TpM, let A:= {t E JR+ : Cv is defined on [-t, t]}, where Cv is, as usual, the geodesic with Cv(O) = p, cv(O) = v. It follows from cv(-t) = c_v(t) that ev may also be defined for negative t, at the moment at least for those with sufficiently small absolute value. 2 implies A ::j:. 0. The compactness of M implies the closedness of A.

The Jacobi identity follows by direct computation. 5 A Lie algebra (over IR) is a real vector space V equipped with a bilinear map[·,·] : V x V ~ V, the Lie bracket, satisfying: (i) [X, X]= 0 for all X E V. (ii) [X, [Y, Z]] + [Y, [Z, X]]+ [Z, [X, Y]] = 0 for all X, Y, Z E V. 2 The space of vector fields on M, equipped with the Lie bracket, is a Lie algebra. 6 Let '1/J : M M. 12) Thus, '1/J* induces a Lie algebra lsomorphism. Proof. 3. 0 We now want to investigate how one might differentiate tensor fields.

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