On metrics on 2-orbifolds all of whose geodesics are closed

Christian Lange

doi:10.1515/crelle-2017-0050

Published by De Gruyter January 13, 2018

On metrics on 2-orbifolds all of whose geodesics are closed

Christian Lange

From the journal Journal für die reine und angewandte Mathematik (Crelles Journal)

https://doi.org/10.1515/crelle-2017-0050

Showing a limited preview of this publication:

Abstract

We show that the periods and the topology of the space of closed geodesics on a Riemannian 2-orbifold all of whose geodesics are closed depend, up to scaling, only on the orbifold topology and compute it. In the manifold case we recover the fact proved by Gromoll, Grove and Pries that all prime geodesics have the same length, without referring to the existence of simple geodesics. We partly strengthen our result in terms of conjugacy of contact forms and explain how to deduce rigidity on the projective plane based on a systolic inequality due to Pu.

Funding statement: The author is partially supported by the DFG funded project SFB/TRR 191 “Symplectic Structures in Geometry, Algebra and Dynamics”.

A Appendix

A.1 Rigidity on the real projective plane

For a Riemannian metric g on 𝒪=ℝ⁢ℙ2 we denote its corresponding area measure by νg, its total area by Ag and the length of a shortest non-contractible loop by ag. The following inequality is due to Pu [25]:

Ag≥2π⁢ag2.

We recall its proof and show how it implies rigidity for Besse metrics on ℝ⁢ℙ2. This argument was explained to us by A. Abbondandolo.

Table 1

Orbifolds of geodesics and (labeled) geodesic periods of Besse 2-orbifolds. For definitions and notations see Sections 2.1 and 2.7. Expressions that only hold in the good orbifold case, i.e. when p=q, are stated in parentheses. For the good orbifolds 𝒪=S2/G, G<O⁢(3), appearing in the table the second column specifies G in terms of the Schönflies notation, see for example [19]). Recall that G+=G∩SO⁢(3), G×={g∈G:det⁢(g)⁢g} and G*=〈G,-1〉. A detailed discussion of the finite subgroups of O⁢(3) and their relations can for instance be found in [19].

	(G<O⁢(3))	𝒪(≅S2/G)	𝒪g(≅S2/G×)	𝒪g/i(≅S2/G*)	geod. periods
	(G<SO⁢(3), G=G+)		(≅S2/G)	(≅S2/G*)
	(Cp)	S2⁢(p,q)
01		2∣(p+q), 2∣pq	S2/Cp+q2	S2/Cp+q2⁢h	(1∞,p+q2¯)
		2∣(p+q), 2∤p⁢q	S2/Cp+q2	S2/Sp+q	(p+q2¯)
01’		2∤(p+q), 2∣pq	S2/Cp+q	S2/Cp+q⁢h	(1∞,p+q¯)
02	D2⁢n	S2⁢(2,2,2⁢n)	S2/D2⁢n	S2/D2⁢n⁢h	(1∞,2,2,2⁢n)
03	D2⁢n+1	S2⁢(2,2,2⁢n+1)	S2/D2⁢n+1	S2/D2⁢n+1⁢d	(1∞,2¯,2⁢n+1)
04	T	S2⁢(2,3,3)	S2/T	S2/Th	(1∞,2,3¯)
05	O	S2⁢(2,3,4)	S2/O	S2/Oh	(1∞,2,3,4)
06	I	S2⁢(2,3,5)	S2/I	S2/Ih	(1∞,2,3,5)
	(-1∉G≮SO⁢(3), G≅G×)		(≅S2/G×)	(≅S2/G*)
	(Cp⁢v)	D2(;p,q)
07		2∣(p+q), 2∣pq	S2/Dp+q2	S2/Dp+q2⁢h	(1∞,p+q2)
		2∣(p+q), 2∤p⁢q	S2/Dp+q2	S2/Dp+q2⁢d	(1∞,p+q2)
07’		2∤(p+q), 2∣pq	S2/Dp+q	S2/Dp+q⁢h	(1∞,2,p+q)
08	S4⁢n	ℝ⁢ℙ2⁢(2⁢n)	S2/C4⁢n	S2/C4⁢n⁢h	(1∞,4⁢n¯)
09	C2⁢n+1⁢h	D2(2n+1;)	S2/C4⁢n+2	S2/C4⁢n+2⁢h	(1∞,2⁢n+1¯)
10	D2⁢n⁢d	D2⁢(2;2⁢n)	S2/D4⁢n	S2/D4⁢n⁢h	(1∞,2,4⁢n)
11	D2⁢n+1⁢h	D2(;2,2,2n+1)	S2/D4⁢n+2	S2/D4⁢n+2⁢h	(1∞,2,2⁢n+1)
12	Td	D2(;2,3,3)	S2/O	S2/Oh	(1∞,3,4)
	(-1∈G=G*≅G+×ℤ2)		(≅S2/G+)	(≅S2/G)
13	S4⁢n+2	ℝ⁢ℙ2⁢(2⁢n+1)	S2/C2⁢n+1	S2/S4⁢n+2	(2⁢n+1¯)
14	C2⁢n⁢h	D2(2n;)	S2/C2⁢n	S2/C2⁢n⁢h	(1∞,n¯)
15	D2⁢n+1⁢d	D2⁢(2;2⁢n+1)	S2/D2⁢n+1	S2/D2⁢n+1⁢d	(1∞,2⁢n+1)
16	D2⁢n⁢h	D2(;2,2,2n)	S2/D2⁢n	S2/D2⁢n⁢h	(1∞,n)
17	Th	D2⁢(3;2)	S2/T	S2/Th	(1∞,3¯)
18	Oh	D2(;2,3,4)	S2/O	S2/Oh	(1∞,2,3)
19	Ih	D2(;2,3,5)	S2/I	S2/Ih	(1∞,3,5)

Suppose that g is some Riemannian metric on ℝ⁢ℙ2 and let g0 be the standard Riemannian metric on ℝ⁢ℙ2 of constant curvature 1. The group G=SO⁢(3) acts on ℝ⁢ℙ2 in its standard way. By the uniformization theorem there is some positive smooth function φ on ℝ⁢ℙ2 such that g=φ⋅g0. We endow G with its Haar measure μ and define

φ¯=(∫G(g*⁢φ)12⁢𝑑μ)2

and g¯=φ¯⋅g0. By construction g¯ is a G-invariant Riemannian metric on M=ℝ⁢ℙ2 and hence has constant curvature. We claim that Ag≥Ag¯ and ag¯≥ag . Indeed, we have

Ag¯=∫Mφ¯⁢𝑑νg0=∫M(∫G(h*⁢φ)12⁢𝑑μ)2⁢𝑑νg0≤∫M(∫Gh*⁢φ⁢𝑑μ)⁢𝑑νg0=∫G(∫Mh*⁢φ⁢𝑑νg0)⁢𝑑μ=∫GAg⁢𝑑μ=Ag,

where we have applied the Cauchy–Schwarz inequality. Moreover, with a shortest non-contractible loop (and hence geodesic) γ on ℝ⁢ℙ2 with respect to g¯ we have

ag¯=∫01φ¯⁢(γ⁢(s))12⁢∥γ˙⁢(s)∥g0⁢𝑑s=∫01(∫G((h*⁢φ)⁢(γ⁢(s)))12⁢∥γ˙⁢(s)∥g0⁢𝑑μ)⁢𝑑s=∫G(∫01((h*⁢φ)⁢(γ⁢(s)))12⁢∥γ˙⁢(s)∥g0⁢𝑑s)⁢𝑑μ≥∫Gag⁢𝑑μ=ag.

In particular, this proves Pu’s inequality, since we have Ag¯=2π⁢ag¯2 for the metric of constant curvature g¯ (see [25]). Now suppose that g is Besse. Theorem 4.1 implies that the same equality also holds for the Besse metric g. In fact, after normalizing g such that ag=π, Theorem 4.1 implies vol⁢(T1⁢ℝ⁢ℙ2,α0∧d⁢α0)=vol⁢(T1⁢ℝ⁢ℙ2,α∧d⁢α) which in turn implies Ag=2⁢π by fiberwise integration with respect to T1⁢ℝ⁢ℙ2→ℝ⁢ℙ2. (Alternatively, this follows from a theorem of Weinstein: The two-fold covering (S2,g^) of (ℝ⁢ℙ2,g) has area 2⁢Ag and the minimal geodesic period is 2⁢ag due to the fact that 𝒪g≅S2. Now the theorem by Weinstein says that for a Besse metric g^ on S2 we have area⁢(S2,g^)=l2π, where l is the minimal geodesic period [33], cf. [3, Proposition 2.24].) It follows that Ag¯=Ag, i.e. we have equality in the Cauchy–Schwarz inequality implying that φ is constant. Hence, g is proportional to g0 and has constant curvature.

Acknowledgements

The author would like to thank Alexander Lytchak for drawing his attention to the subject. He is grateful to Alberto Abbondandolo for explaining to him how this paper’s result combined with work by Pu implies rigidity on the real projective plane (see Section A.1). He thanks the referee for critical comments and suggestions that helped to improve the exposition. The research in this paper is part of a project in the SFB/TRR 191 “Symplectic Structures in Geometry, Algebra and Dynamics”. A partial support is gratefully acknowledged.

References

[1] A. Abbondandolo, B. Bramham, U. L. Hryniewicz and P. A. S. Salomão, A systolic inequality for geodesic flows on the two-sphere, Math. Ann. 367 (2017), no. 1–2, 701–753. 10.1007/s00208-016-1385-2Search in Google Scholar

[2] A. Adem, J. Leida and Y. Ruan, Orbifolds and stringy topology, Cambridge Tracts in Math. 171, Cambridge University Press, Cambridge 2007. 10.1017/CBO9780511543081Search in Google Scholar

[3] A. L. Besse, Manifolds all of whose geodesics are closed, Ergeb. Math. Grenzgeb. (3) 93, Springer, Berlin 1978. 10.1007/978-3-642-61876-5Search in Google Scholar

[4] M. Boileau, S. Maillot and J. Porti, Three-dimensional orbifolds and their geometric structures, Panor. Synthèses 15, Société Mathématique de France, Paris 2003. Search in Google Scholar

[5] W. M. Boothby and H. C. Wang, On contact manifolds, Ann. of Math. (2) 68 (1958), 721–734. 10.2307/1970165Search in Google Scholar

[6] M. R. Bridson and A. Haefliger, Metric spaces of non-positive curvature, Grundlehren Math. Wiss. 319, Springer, Berlin 1999. 10.1007/978-3-662-12494-9Search in Google Scholar

[7] M. G. Brin, Seifert fibered spaces, Notes for a course given in the Spring of 1993. Search in Google Scholar

[8] D. Burago, Y. Burago and S. Ivanov, A course in metric geometry, Grad. Stud. Math. 33, American Mathematical Society, Providence 2001. 10.1090/gsm/033Search in Google Scholar

[9] M. W. Davis, Lectures on orbifolds and reflection groups, Transformation groups and moduli spaces of curves, Adv. Lect. Math. (ALM) 16, International Press, Somerville (2011), 63–93. Search in Google Scholar

[10] U. Frauenfelder, C. Lange and S. Suhr, A Hamiltonian version of a result of Gromoll and Grove, preprint (2016), https://arxiv.org/abs/1603.05107v2. 10.5802/aif.3247Search in Google Scholar

[11] H. Geiges, An introduction to contact topology, Cambridge Stud. Adv. Math. 109, Cambridge University Press, Cambridge 2008. 10.1017/CBO9780511611438Search in Google Scholar

[12] H. Geiges and C. Lange, Seifert fibrations of lens spaces, Abh. Math. Semin. Univ. Hambg. (2017), 10.1007/s12188-017-0188-z. 10.1007/s12188-017-0188-zSearch in Google Scholar

[13] L. W. Green, Auf Wiedersehensflächen, Ann. of Math. (2) 78 (1963), 289–299. 10.2307/1970344Search in Google Scholar

[14] D. Gromoll and K. Grove, On metrics on S2 all of whose geodesics are closed, Invent. Math. 65 (1981/82), no. 1, 175–177. 10.1007/BF01389300Search in Google Scholar

[15] V. Guillemin, The Radon transform on Zoll surfaces, Adv. Math. 22 (1976), no. 1, 85–119. 10.1016/0001-8708(76)90139-0Search in Google Scholar

[16] V. Guillemin, A. Uribe and Z. Wang, Geodesics on weighted projective spaces, Ann. Global Anal. Geom. 36 (2009), no. 2, 205–220. 10.1007/s10455-009-9159-7Search in Google Scholar

[17] M. Jankins and W. D. Neumann, Lectures on Seifert manifolds, Brandeis Lect. Notes 2, Brandeis University, Waltham 1983. Search in Google Scholar

[18] T. Konno, Unit tangent bundle over two-dimensional real projective space, Nihonkai Math. J. 13 (2002), no. 1, 57–66. Search in Google Scholar

[19] L. D. Landau and E. M. Lifshitz, Mechanics, Course Theor. Phys. 1, Pergamon Press, Oxford 1960. Search in Google Scholar

[20] C. Lange, Characterization of finite groups generated by reflections and rotations, J. Topol. 9 (2016), no. 4, 1109–1129. 10.1112/jtopol/jtw020Search in Google Scholar

[21] C. Lange, Orbifolds from a metric viewpoint, preprint (2018), https://arxiv.org/abs/1801.03472. 10.1007/s10711-020-00521-xSearch in Google Scholar

[22] C. Lange, Equivariant smoothing of piecewise linear manifolds, Math. Proc. Camb. Philos. Soc. (2017), 10.1017/S0305004117000275. 10.1017/S0305004117000275Search in Google Scholar

[23] A. Lytchak and G. Thorbergsson, Curvature explosion in quotients and applications, J. Differential Geom. 85 (2010), no. 1, 117–139. 10.4310/jdg/1284557927Search in Google Scholar

[24] C. Pries, Geodesics closed on the projective plane, Geom. Funct. Anal. 18 (2009), no. 5, 1774–1785. 10.1007/s00039-008-0682-7Search in Google Scholar

[25] P. M. Pu, Some inequalities in certain nonorientable Riemannian manifolds, Pacific J. Math. 2 (1952), 55–71. 10.2140/pjm.1952.2.55Search in Google Scholar

[26] M. Radeschi and B. Wilking, On the Berger conjecture for manifolds all of whose geodesics are closed, Invent. Math. 210 (2017), no. 3, 911–962. 10.1007/s00222-017-0742-4Search in Google Scholar

[27] F. Raymond, Classification of the actions of the circle on 3-manifolds, Trans. Amer. Math. Soc. 131 (1968), 51–78. 10.1090/S0002-9947-1968-0219086-9Search in Google Scholar

[28] P. Scott, The geometries of 3-manifolds, Bull. Lond. Math. Soc. 15 (1983), no. 5, 401–487. 10.1112/blms/15.5.401Search in Google Scholar

[29] H. Seifert, Topologie dreidimensionaler gefaserter Räume, Acta Math. 60 (1933), no. 1, 147–238. 10.1007/BF02398271Search in Google Scholar

[30] E. Swartz, Matroids and quotients of spheres, Math. Z. 241 (2002), no. 2, 247–269. 10.1007/s002090200414Search in Google Scholar

[31] W. P. Thurston, The geometry and topology of three-manifolds, Princeton University Press, Princeton 1979. Search in Google Scholar

[32] A. W. Wadsley, Geodesic foliations by circles, J. Differential Geom. 10 (1975), no. 4, 541–549. 10.4310/jdg/1214433160Search in Google Scholar

[33] A. Weinstein, On the volume of manifolds all of whose geodesics are closed, J. Differential Geom. 9 (1974), 513–517. 10.1090/pspum/027.1/9930Search in Google Scholar

[34] B. P. Zimmermann, On finite groups acting on spheres and finite subgroups of orthogonal groups, Sib. Èlektron. Mat. Izv. 9 (2012), 1–12. Search in Google Scholar

[35] O. Zoll, Über Flächen mit Scharen geschlossener geodätischer Linien, Math. Ann. 57 (1903), no. 1, 108–133. 10.1007/BF01449019Search in Google Scholar

Received: 2016-05-26

Revised: 2017-09-22

Published Online: 2018-01-13

Published in Print: 2020-01-01

On metrics on 2-orbifolds all of whose geodesics are closed

Abstract

A Appendix

A.1 Rigidity on the real projective plane

Acknowledgements

References

Journal and Issue

Articles in the same Issue