# Analytic manifolds of nonpositive curvature

• Werner Ballmann
• Mikhael Gromov
• Viktor Schroeder
Part of the Progress in Mathematics book series (PM, volume 61)

## 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
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

## Authors and Affiliations

• Werner Ballmann
• 1
• 2
• Mikhael Gromov
• 3
• Viktor Schroeder
• 4
• 5
1. 1.Dept. of MathematicsUniversity of MarylandCollege ParkUSA
2. 2.Math. Institut der UniversitätBonnWest Germany
3. 3.Inst. des Hautes Etudes ScientifiquesBures-sur-YvetteFrance
4. 4.Math. Institut der UniversitätMünsterGermany
5. 5.Math. Institut der UniversitätBaselSwitzerland