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15 January 2009 E7, Wirtinger inequalities, Cayley 4-form, and homotopy
Victor Bangert, Mikhail G. Katz, Steven Shnider, Shmuel Weinberger
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Duke Math. J. 146(1): 35-70 (15 January 2009). DOI: 10.1215/00127094-2008-061

Abstract

We study optimal curvature-free inequalities of the type discovered by C. Loewner and M. Gromov, using a generalization of the Wirtinger inequality for the comass. Using a model for the classifying space BS3 built inductively out of BS1, we prove that the symmetric metrics of certain two-point homogeneous manifolds turn out not to be the systolically optimal metrics on those manifolds. We point out the unexpected role played by the exceptional Lie algebra E7 in systolic geometry, via the calculation of Wirtinger constants. Using a technique of pullback with controlled systolic ratio, we calculate the optimal systolic ratio of the quaternionic projective plane, modulo the existence of a Joyce manifold with Spin(7)-holonomy and unit middle-dimensional Betti number

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Victor Bangert. Mikhail G. Katz. Steven Shnider. Shmuel Weinberger. "E7, Wirtinger inequalities, Cayley 4-form, and homotopy." Duke Math. J. 146 (1) 35 - 70, 15 January 2009. https://doi.org/10.1215/00127094-2008-061

Information

Published: 15 January 2009
First available in Project Euclid: 17 December 2008

zbMATH: 1160.53023
MathSciNet: MR2475399
Digital Object Identifier: 10.1215/00127094-2008-061

Subjects:
Primary: 53C23
Secondary: 17B25 , 55R37

Rights: Copyright © 2009 Duke University Press

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Vol.146 • No. 1 • 15 January 2009
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