Local and Global Homogeneity for Manifolds that admit a Positive Curvature Metric

2019/06/15 by Joseph A. Wolf, Wolf, Joseph A.
Physics and Astronomy · Mathematics · #Advanced Differential Geometry Research #Geometric Analysis and Curvature Flows #Geometry and complex manifolds

paper · pdf · doi:10.48550/arxiv.1906.06596

Abstract

In this note we study globally homogeneous Riemannian quotients Γ\backslash (M,ds²) of homogeneous Riemannian manifolds (M,ds²). The Homogeneity Conjecture is that Γ\backslash (M,ds²) is (globally) homogeneous if and only if (M,ds²) is homogeneous and every γ∈ Γ is of constant displacement on (M,ds²). We provide further evidence for that conjecture by (i) verifying it for normal homogeneous Riemannian manifolds that also admit an invariant Riemannian metric of strictly positive sectional curvature and (ii) showing that in most (three or less) cases the normality condition can be dropped.

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