![]() ![]() 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. Schmitz.Bredon, G.: Introduction to Compact Transformation Groups, vol. The total absolute curvature of closed curves in Riemannian manifolds. I would really prefer something metric independent if possible. However, this would depend strongly on the metrics/contraction maps involved. It seems that one could derive some inequality from the usual one using contraction maps to and from Euclidean space and estimating the resulting effect on the total curvature along $\gamma$. Is there a version of the Fáry-Milnor theorem for positively curved metrics on $S^3$? I'm interested in the opposite situation. the usual hyperbolic metric), maybe satisfying some extra hypotheses (for example, see ). (Fáry-Milnor) If a simple closed curve $\gamma \subset \mathbb^3$ is equipped with a non-positively curved metric (e.g. I'm interested in generalizations the following well-known theorem of Fáry and Milnor. ![]()
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