A generalization of Gallot's Theorem

Details A generalization of Gallot's Theorem A generalization of Gallot's Theorem

2009-07-18

arxiv.org/abs/0907.3182v3

Mathematics

Differential Geometry

28 pages, 5 figures

Scientific paper

A classical theorem of Gallot states that a Riemannian cone over a compact manifold is either irreducible or flat. Such a cone has compact quotients by radial homotheties (which form a 1-parameter group). More generally, we define cone-like manifolds to be those non-compact manifolds that admit compact quotients by discrete subgroups of homotheties and show that, under some tameness assumption (concerning the life-time of incomplete geodesics), all cone-like manifolds are either irreducible or flat. This assumption holds, in particular, for any small cone-like deformation of Riemannian cones. Using the natural correspondence between cone-like manifolds and compact conformal manifolds with a closed Weyl structure, our result can be restated as follows: Every closed, non-exact, tame Weyl structure on a compact conformal manifold is either flat, or has irreducible holonomy. As an application, we describe the compact conformal manifolds carrying a tame closed Weyl structure with non-generic holonomy.

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