Volume 18, issue 2 (2014)
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Sweeping out sectional curvature

Dmitri Panov and Anton Petrunin
Geometry & Topology 18 (2014) 617–631
DOI: 10.2140/gt.2014.18.617
Abstract

We observe that the maximal open set of constant curvature κ in a Riemannian manifold of curvature ≥ κ or ≤ κ has a convexity-type property, which we call two-convexity. This statement is used to prove a number of rigidity statements in comparison geometry.

Keywords
rigidity, comparison geometry
Mathematical Subject Classification 2010
Primary: 53C24
Secondary: 53C20
References
Publication
Received: 29 March 2013
Revised: 7 September 2013
Accepted: 28 September 2013
Published: 30 January 2014
Proposed: Dmitri Burago
Seconded: John Lott, Simon Donaldson
Authors
Dmitri Panov
Department of Mathematics
King’s College London
London WC2R 2LS
UK
http://www.kcl.ac.uk/nms/depts/mathematics/people/atoz/panovd.aspx
Anton Petrunin
Department of Mathematics
Pennsylvania State University
University Park, PA 16802
USA
http://www.math.psu.edu/petrunin/