Spaces of Constant Curvature
American Mathematical Soc., 2011 - 420 pages
This book is the sixth edition of the classic Spaces of Constant Curvature, first published in 1967, with the previous (fifth) edition published in 1984. It illustrates the high degree of interplay between group theory and geometry. The reader will benefit from the very concise treatments of riemannian and pseudo-riemannian manifolds and their curvatures, of the representation theory of finite groups, and of indications of recent progress in discrete subgroups of Lie groups. Part I is a brief introduction to differentiable manifolds, covering spaces, and riemannian and pseudo-riemannian geometry. It also contains a certain amount of introductory material on symmetry groups and space forms, indicating the direction of the later chapters. Part II is an updated treatment of euclidean space form. Part III is Wolf's classic solution to the Clifford-Klein Spherical Space Form Problem. It starts with an exposition of the representation theory of finite groups. Part IV introduces riemannian symmetric spaces and extends considerations of spherical space forms to space forms of riemannian symmetric spaces. Finally, Part V examines space form problems on pseudo-riemannian symmetric spaces. At the end of Chapter 12 there is a new appendix describing some of the recent work on discrete subgroups of Lie groups with application to space forms of pseudo-riemannian symmetric spaces. Additional references have been added to this sixth edition as well.
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abelian acts freely assume automorphism bilinear Cartan centralizer classiﬁcation commutes compact complete connected conjugate connected riemannian constant curvature coordinate Corollary deﬁne deﬁnite denote diffeomorphism differentiable dimension eigenvalue element of order euclidean space euclidean space forms ﬁnite group ﬁrst ﬁxed point free ﬂat follows frame bundle fundamental group g e G geodesic given Grassmann manifold group of isometries groups of type homogeneous implies integers involutive Lie algebra isomorphism isotropic Lemma Let F Let G Let H Lie group matrix metric neighborhood nondegenerate nonzero normal subgroup open set orthogonal involutive Lie point free representation prime Proof properly discontinuously proves pseudo-riemannian manifold quotient representation of G resp riemannian manifold riemannian symmetric space satisﬁes says sectional curvature semisimple simply connected space form problem spherical space forms subalgebra subgroup of G subspace Suppose Sylow 2-subgroup tangent space Theorem torus translation vector ﬁeld vector space