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Almost positive curvature on the Gromoll-Meyer 7-sphere

Author(s): J.-H. Eschenburg
Journal: Proc. Amer. Math. Soc. 130 (2002), 1165-1167.
MSC (2000): Primary 53C20, 53C30; Secondary 57S25, 57R60
Posted: 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:

[E1]
J.-H. Eschenburg: Freie isometrische Aktionen auf kompakten Lie-Gruppen mit positiv gekrümmten Orbiträumen, Schriftenr. Math. Inst. Univ. Münster (2) 32 (1984) MR 86a:53045
[E2]
J.-H. Eschenburg: Inhomogeneous spaces of positive curvature, Diff. Geom. Appl. 2 (1992), 123-132 MR 94j:53044
[GM]
D. Gromoll and W.T. Meyer: An exotic sphere with nonnegative sectional curvature, Ann. of Math. 100 (1974), 401 - 406 MR 51:11347
[PW]
P. Petersen and F. Wilhelm: Examples of Riemannian manifolds with positive curvature almost everywhere, Geom. and Top. 3 (1999), 331 - 367 MR 2000g:53030
[W]
F. Wilhelm: An exotic sphere with positive curvature almost everywhere, Preprint Riverside 1999
[Wk]
B. Wilking: Manifolds with positive sectional curvature almost everywhere, preprint

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Additional Information:

J.-H. Eschenburg
Affiliation: Institut für Mathematik, Universität Augsburg, D-86135 Augsburg, Germany
Email: eschenburg@math.uni-augsburg.de

DOI: 10.1090/S0002-9939-01-06151-2
PII: S 0002-9939(01)06151-2
Keywords: Biquotients, exotic 7-sphere, quaternions, zero curvature set
Received by editor(s): September 28, 2000
Received by editor(s) in revised form: October 23, 2000
Posted: September 19, 2001
Communicated by: Wolfgang Ziller
Copyright of article: Copyright 2001, American Mathematical Society

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