A Comprehensive Introduction to Differential Geometry, Vol. by Michael Spivak

By Michael Spivak

E-book by means of Michael Spivak, Spivak, Michael

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Extra resources for A Comprehensive Introduction to Differential Geometry, Vol. 3, 3rd Edition

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II . ) A consequence of Margulis' superrigidity theorem is his arithmeticity theorem that we now describe. 2: A lattice IF in a connected semisimple Lie group G with trivial center and without compact factors is arithmetic if there exist a semisimple algebraic Q-group H and a surjective homomorphism c : HR - C with compact kernel for which

Let c : [0, 1] -' N be the (unique) geodesic from 71(1) to 72(1). Then for each t E [0. 1], there exists s E [0, 1] with vature < -A < 0. Let p E N. and let 71, 72 d(c(t),7;(s)) < c:(A) for i = 1 or 2. 26) p This property has been used by Gromov as an axiomatic characterization of metric spaces of negative curvature, with important applications to group theory (see [G31). In this context, see also the discussion of word hyperbolic groups on p. 10. 6 does not imply negative, not even nonpositive sectional curvature.

Step 1) will not use the special geometry of symmetric spaces, apart from the nonpositive sectional curvature. Step 2), by contrast, will depend on certain particular identities and inequalities that only hold for symmetric spaces. rEG/K. When compared with the previous situation of maps between compact quotients, we are now looking at equivariant lifts of such maps to universal covers. Thus, the topological context has not changed, but we have gained a more convenient formulation. iemannian manifolds with isometry groups I(X) and I(Y), resp.

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