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A min-max characterization of Zoll Riemannian metrics

Part of: Variational problems in infinite-dimensional spaces Global differential geometry

Published online by Cambridge University Press:  29 April 2021

MARCO MAZZUCCHELLI  and
STEFAN SUHR
MARCO MAZZUCCHELLI
Affiliation:
Unité de Mathématiques Pures et Appliquées École Normale Supérieure de Lyon, 46 allée d’Italie, 69364Lyon, France. e-mail: [email protected]
STEFAN SUHR
Affiliation:
Fakultät für Mathematik, Ruhr–Universität Bochum, IB 3/81, Universitätsstr. 150, 44780Bochum, Germany e-mail: [email protected]
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Abstract

We characterise the Zoll Riemannian metrics on a given simply connected spin closed manifold as those Riemannian metrics for which two suitable min-max values in a finite dimensional loop space coincide. We also show that on odd dimensional Riemannian spheres, when certain pairs of min-max values in the loop space coincide, every point lies on a closed geodesic.

MSC classification

Primary: 53C22: Geodesics 58E10: Applications to the theory of geodesics (problems in one independent variable)
Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 172 , Issue 3 , May 2022 , pp. 591 - 615
Copyright
© The Author(s), 2021. Published by Cambridge University Press on behalf of Cambridge Philosophical Society

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