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Proceedings of the American Mathematical Society

Published by the American Mathematical Society since 1950, Proceedings of the American Mathematical Society is devoted to shorter research articles in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2024 MCQ for Proceedings of the American Mathematical Society is 0.85.

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Almost positive curvature on the Gromoll-Meyer 7-sphere
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by J.-H. Eschenburg
Proc. Amer. Math. Soc. 130 (2002), 1165-1167
DOI: https://doi.org/10.1090/S0002-9939-01-06151-2
Published electronically: September 19, 2001
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Abstract:

D. Gromoll and W. Meyer have represented a certain exotic 7-sphere $M$ as a biquotient of the compact Lie group $Sp(2)$. Thus any invariant normal homogeneous metric on $Sp(2)$ induces a metric of nonnegative sectional curvature on $M$. We show that the simplest such metrics (except the bi-invariant one) induce metrics which have in fact strictly positive curvature outside a subset of $M$ with measure zero.
References
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  • J.-H. Eschenburg, Inhomogeneous spaces of positive curvature, Differential Geom. Appl. 2 (1992), no. 2, 123–132. MR 1245552, DOI 10.1016/0926-2245(92)90029-M
  • Detlef Gromoll and Wolfgang Meyer, An exotic sphere with nonnegative sectional curvature, Ann. of Math. (2) 100 (1974), 401–406. MR 375151, DOI 10.2307/1971078
  • Peter Petersen and Frederick Wilhelm, Examples of Riemannian manifolds with positive curvature almost everywhere, Geom. Topol. 3 (1999), 331–367. MR 1714915, DOI 10.2140/gt.1999.3.331
  • F. Wilhelm: An exotic sphere with positive curvature almost everywhere, Preprint Riverside 1999
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Bibliographic Information
  • J.-H. Eschenburg
  • Affiliation: Institut für Mathematik, Universität Augsburg, D-86135 Augsburg, Germany
  • Email: eschenburg@math.uni-augsburg.de
  • Received by editor(s): September 28, 2000
  • Received by editor(s) in revised form: October 23, 2000
  • Published electronically: September 19, 2001
  • Communicated by: Wolfgang Ziller
  • © Copyright 2001 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 130 (2002), 1165-1167
  • MSC (2000): Primary 53C20, 53C30; Secondary 57S25, 57R60
  • DOI: https://doi.org/10.1090/S0002-9939-01-06151-2
  • MathSciNet review: 1873792