![]() 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. I think this would make it harder to solve the Friedmann equations compared to Standard cosmology, since you would have boundary conditions for $a(t)$ at the beginning and everytime the time component loops, which for example forbids exponential growth as in a dark energy dominated universe.Bredon, G.: Introduction to Compact Transformation Groups, vol. You could think of a universe where time loops, being a circle, I think this would correspond to your idea of time having $k=1$ (which, again, it has not, since mathematically a 1D circle still has no curvature). Mathematically you cannot assign a curvature to ONLY the time component, because a one-dimensional space cannot have curvature (there are no angles between different points that you could measure). The idea of assigning ONE curvature to all four space-time coordinates, including time, does not make a lot of sense in a universe that we expect to be different at different points in time, since including the time dimension, the universe is NOT homogenous and isotropic (it DOES matter at which time you look, and it DOES matter if you look in the direction of the past or towards the future). This leaves the three possibilities of $k=1,0,-1$, if all spatial coordinates are normalised correspondingly, because these three spatial metrics fulfill the conditions. ![]() observation, that the spatial part of the universe at a certain instance in time (!) should be homogenous and isotropic. The idea of constant spatial curvature comes from the idea, i.e.
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