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.