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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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References

Besse, A. L.. Manifolds all of whose geodesics are closed. Ergeb. Math. Grenzgeb., vol. 93 (Springer-Verlag, Berlin-New York, 1978).CrossRefGoogle Scholar
Bott, R.. On manifolds all of whose geodesics are closed. Ann. of Math. 60 (1954), no. 2, 375–382.CrossRefGoogle Scholar
Bott, R.. On the iteration of closed geodesics and the Sturm intersection theory. Comm. Pure Appl. Math. 9 (1956), 171–206.CrossRefGoogle Scholar
Ballmann, W., Thorbergsson, G., & Ziller, W.. Existence of closed geodesics on positively curved manifolds. J. Differential Geom. 18 (1983), no. 2, 221–252.CrossRefGoogle Scholar
Gromoll, D. & Grove, K.. On metrics on S 2 all of whose geodesics are closed. Invent. Math. 65 (1981), 175–177.CrossRefGoogle Scholar
Ginzburg, V. L., Gürel, B. Z. & Mazzucchelli, M.. On the spectral characterization of Besse and Zoll Reeb flows. Ann. Inst. H. Poincaré Anal. Non Linéaire 38 (2021), no. 3, 549–576.CrossRefGoogle Scholar
Goresky, M. & Hingston, N.. Loop products and closed geodesics. Duke Math. J. 150 (2009), no. 1, 117–209.CrossRefGoogle Scholar
Ljusternik, L. A.. The topology of the calculus of variations in the large. Trans. Math. Monogr. vol. 16, (Amer. Mathematical Society, Providence, R.I., 1966).Google Scholar
Milnor, J.. Morse theory. Ann. Math. Stud., no. 51 (Princeton University Press, Princeton, N.J., 1963).Google Scholar
Mazzucchelli, M. & Suhr, S.. A characterisation of Zoll Riemannian metrics on the 2-sphere. Bull. Lond. Math. Soc. 50 (2018), 997–1006.CrossRefGoogle Scholar
Radeschi, M. & Wilking, B.. On the Berger conjecture for manifolds all of whose geodesics are closed. Invent. Math. 210 (2017), 911–962.CrossRefGoogle Scholar
Samelson, H.. On manifolds with many closed geodesics. Portugal. Math. 22 (1963), 193–196.Google Scholar
Viterbo, C.. Some remarks on Massey products, tied cohomology classes, and the Lusternik-Shnirelman category. Duke Math. J. 86 (1997), no. 3, 547–564.CrossRefGoogle Scholar
Wadsley, A. W.. Geodesic foliations by circles. J. Differential Geom. 10 (1975), no. 4, 541–549.CrossRefGoogle Scholar
Wilking, B.. Index parity of closed geodesics and rigidity of Hopf fibrations. Invent. Math. 144 (2001), 281–295.CrossRefGoogle Scholar
Ziller, W.. The free loop space of globally symmetric spaces. Invent. Math. 41 (1977), no. 1, 1–22.CrossRefGoogle Scholar