Mathematische Annalen

, Volume 321, Issue 4, pp 775–788 | Cite as

Asymptotical flatness and cone structure at infinity

  • Anton Petrunin
  • Wilderich Tuschmann
Original article


We investigate asymptotically flat manifolds with cone structure at infinity. We show that any such manifold M has a finite number of ends, and we classify (except for the case dim M=4, where it remains open if one of the theoretically possible cones can actually arise) for simply connected ends all possible cones at infinity. This result yields in particular a complete classification of asymptotically flat manifolds with nonnegative curvature: The universal covering of an asymptotically flat m-manifold with nonnegative sectional curvature is isometric to \({\mathbb R}^{m-2}\times S\), whereS is an asymptotically flat surface.


Manifold Finite Number Flat Surface Sectional Curvature Universal Covering 
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Copyright information

© Springer-Verlag Berlin Heidelberg 2001

Authors and Affiliations

  • Anton Petrunin
    • 1
  • Wilderich Tuschmann
    • 2
  1. 1.Math. Dept. Pennsylvania State University, University Park, PA 16802, USA (e-mail:
  2. 2.Max-Planck-Institute for Mathematics in the Sciences, Inselstrasse, D-04103 Leipzig, Germany (e-mail:

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