![]() ![]() We'll look at how humans learned to ask questions about the universe, and even before the invention of modern instruments like the telescope, learned some amazing things about their place in nature. Note that Lemma 2.1 is not required in the above proofs.This is an introductory level course about the history and philosophy of astronomy, the oldest science. By Corollary 1.9, we then conclude that \(\pi _1(N_0)\) is cyclic. If \(\dim (N_0)=11\), then it is easy to see, by analyzing the isotropy representation of \(T^2\) on the normal space of \(N_0\), that there is \(T^1\subset T^2\) with a fixed point set of dimension 13. Since the case \(n=17\) has been proved, we may assume that \(\dim (N_i)\le 15\). So we may assume both \(H_1\) and \(H_2\) are circle subgroups. Without loss of generality, we may assume that \(\dim (N_1)\le 17\) (Corollary 1.9). Let \(H_1\) and \(H_2\) be subgroups of \(T^5\) of rank at most 1 with fixed point sets \(N_1\) and \(N_2\) of dimension \(\ge 11\) (Lemma 1.14) and such that \(T^5/H_i\) acts effectively on \(N_i\). We then complete the proof by applying Lemma 3.2. So we can assume that the \(T^5\)-action has no fixed point. If the \(T^5\)-fixed point set is not empty, then by the last part of (1.12.1), \(\pi _1(M)\) is cyclic. So we may assume that \(H_i\) are circles. In : McGraw-Hill Series in Higher Mathematics. Wolf, J.A.: The spaces of constant curvature. Wilking, B.: Group Actions on Manifolds of Positive Sectional Curvature. Wilking, B.: Torus actions on manifolds of positive sectional curvature. Wang, Y.: On Fundamental Groups of Closed Positively Curved Manifolds with Symmetry. Sugahara, K.: The isometry group of and the diameter of a Riemannian manifold with positive curvature. Smale, S.: Generalized Poincaré conjecture in dimension \(>4\). Rong, X., Wang, Y.: Fundamental groups of \((4k 1)\)-manifolds with positive curvature and isometric \(T^k\)-actions are cyclic (Preprint) Rong, X., Wang, Y.: Fundamental group of manifolds with positive curvature and torus actions. Rong, X.: Fundamental group of positively curved manifolds admitting compatible local torus actions. Rong, X.: Positively curved manifolds with almost maximal symmetry rank. Rong, X.: On the fundamental group of manifolds of positive sectional curvature. Kobayashi, S.: Transformation Groups in Differential Geometry. Hsiang, W., Kleiner, B.: On the topology of positively curved \(4\)-manifolds with symmetry. Hamilton, R.: Three-manifolds with positive Ricci curvature. Grove, K., Searle, C.: Positively curved manifolds with maximal symmetry-rank. ![]() Grove, K.: Geometry of, and via symmetries. Thesis (2005)įreedman, M.: Topology of four manifolds. ![]() 221, 830–860 (2009)įrank, P.: The Fundamental Groups of Positively Curved Manifolds with Symmetry. 332(1), 81–101 (2005)įang, F., Rong, X.: Collapsed \(5\)-manifolds with pinched positive sectional curvature. Math 126, 227–245 (2004)įang, F., Rong, X.: Homeomorphic classification of positively curved manifolds with almost maximal symmetry rank. 13(2), 479–501 (2005)įang, F., Rong, X.: Positively curved manifolds with maximal discrete symmetry rank. Springer, New York (1982)įang, F., Mendonca, S., Rong, X.: A connectedness principle in the geometry of positive curvature. Academic Press, Dublin (1972)īrown, K.S.: Cohomology of Groups. Bredon, G.: Introduction to Compact Transformation Groups, vol. ![]()
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