Manifolds of Nonpositive Curvature pp 120-124 | Cite as

# Analytic manifolds of nonpositive curvature

## Abstract

Is it possible to generalize the theorems proved for manifolds of negative curvature to manifolds of nonpositive curvature? Let us first look to the Margulis-Heintze theorem. There is an obvious counterexample: Let V’ be a compact manifold with curvature -1 ≤ K ≤ 0, then define V: = V’ × S<Stack><Subscript>€</Subscript><Superscript>1</Superscript></Stack>, where S<Stack><Subscript>€</Subscript><Superscript>1</Superscript></Stack> is the circle of length *2π€*. Then also V satisfies -1 ≤ K ≤ 0, but the injectivity radius is everywhere smaller than €π. Thus to prove a generalization one has to exclude these products. For example we can assume that the universal cover X of V has no Euclidean de Rham factor. But also in this case, there is a counterexample: Take a Riemann surface of genus g ≥ 1 with one cusp, cut it off, then there is a Riemannian metric on this surface with curvature -1 ≤ K ≤ 0 which is isometric to S<Stack><Subscript>€</Subscript><Superscript>1</Superscript></Stack> × [0,1] at the boundary. Cross this surface with S<Stack><Subscript>€</Subscript><Superscript>1</Superscript></Stack> and glue it together with a second copy of itself by interchanging the factors. Thus globally the resulting manifold is not a product and the universal covering has no Euclidean de Rham factor, but the injectivity radius is small everywhere.

## Keywords

Riemannian Manifold Riemann Surface Fundamental Group Universal Covering Nonpositive Curvature## Preview

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