Victor Patrangenaru

Department of Mathematics and Statistics, Texas Tech University, Lubbock, TX 79409


Metric classification of geometries of positive Ricci curvature in  3D


Using Cartan's approach, we give an explicit formula of the metric tensor of a simply connected  3D  homogeneous space of positive scalar curvature in terms of the principal Ricci curvatures at one point. The principal Ricci curvatures at a single point fully determine such an isometry class of a  3D  Riemannian homogeneous space, and each isometry class can be represented by a Lie group with a left invariant metric. The problem whether a  3D  compact connected simply connected manifold has a Lie group structure, raised in my paper "On the  3D  Riemannian homogeneous spaces of positive sectional curvature" [Algebra Geom. Appl. Semin. Proc. 2(2002), 5-13], is an open oalternative to the Ricci flow approach to Poincaré’s conjecture.



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