"Affine Manifolds with Parallel Torsion 001.ps.gz" - читать интересную книгу автораAFFINE MANIFOLDS WITH PARALLEL TORSION AND CURVATURE WERNER BALLMANN In these lecture notes I discuss the theorem of Ambrose and Hicks on parallel translation of torsion and curvature [Amb56], [Hic59] and the Lie theoretic description of affine manifolds with parallel torsion and curvature by Nomizu [Nom54]. These results are based on earlier work of E. Cartan on the corresponding problems for locally symmetric Riemannian spaces. Some immediate applications concern results of Kostant [Kos60] and Ambrose and Singer [AS58]. A related discussion and more references can be found in [KN63], [KN69] and [TV83]. This is not the final version of the notes, comments and criticism are welcome. I would like to thank Gregor Weingart for his valuable comments which led to an improvement of an earlier version. Contents 1. Preliminaries about Affine Manifolds 2 2. Parallel Translation of Torsion and Curvature 5 3. Homogeneous Structures: Uniqueness 8 4. Homogeneous Structures: Existence 10 5. Riemannian Homogeneous Spaces 16 References 18 Date: November 16, 1998. 1 2 WERNER BALLMANN For the convenience of the reader I recall some definitions and general facts. Let M be an affine manifold with connection D, torsion T and curvature R. One of the formulas we need is as follows, SfR(X; Y )Zg = \Gamma SfT (X; T (Y; Z))g + SfDX T (Y; Z)g :(1.1) where S denotes the sum over all cyclic permutations of X, Y and Z. Another formula we need is the Jacobi equation, J 00 + R(J; .c) .c = \Gamma T (J 0; .c) + T 0(J; .c) ;(1.2) where c is a geodesic and the prime denotes covariant differentiation along c. A continous curve c : I ! M is called a geodesic polygon if there is a subdivision : : : ti\Gamma 1 ! ti ! ti+1 : : : of I such that cj[ti\Gamma 1; ti] is a geodesic for all i. For a point p 2 M , we denote by \Pi p the space of all geodesic polygons c : [0; 1] ! M with c(0) = p, endowed with the compact-open topology. Let p 2 M and c 2 \Pi p. Let 0 = t0 ! t1 ! \Delta \Delta \Delta ! tk = 1 be a subdivision of [0; 1] such that cj[ti\Gamma 1; ti] is a geodesic, 1 ^ i ^ k. Let vi 2 TpM be the parallel translate of .c(ti+0) along cj[t0; ti], 0 ^ i ^ k\Gamma 1. Then c is completely determined by the data (v0; t1); : : : ; (vk\Gamma 1; tk) 2 TpM \Theta [0; 1] : If M is geodesically complete, any such family of pairs in TpM \Theta [0; 1] determines a geodesic polygon in \Pi p. Since subdivision points may also occur in smooth points of c, the correspondence between c and the data is not one to one. The data correspond to geodesic polygons together with a subdivision of [0; 1]. |
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