Abstract
A conjecture of Burns and Knieper (1991) asks whether a 2-plane with a metric without conjugate points, and with a geodesic foliation whose lines are at bounded Hausdorff distance, is necessarily flat. We prove this conjecture in two cases: under the hypothesis that the plane admits total curvature, and under the hypothesis of visibility at some point. Along the way, we show that all geodesic line foliations on a Riemannian 2-plane must be homeomorphic to the standard one.
Funding statement: The first author was partially supported by National Key R&D Program of China grant 2020YFA0712800 and the Fundamental Research Funds for the Central Universities. The second author was supported in part by research grants MTM2017-85934-C3-2-P from the Ministerio de Economía y Competitividad de España (MINECO), PID2021-124195NB-C32 from the Ministerio de Ciencia e Innovación (MICINN), and by ICMAT Severo Ochoa project CEX2019-000904-S (MINECO).
Acknowledgements
We are very grateful to the anonymous referee for the detailed suggestions which greatly improved the clarity of the paper.
References
[1] V. Bangert and P. Emmerich, Area growth and rigidity of surfaces without conjugate points, J. Differential Geom. 94 (2013), no. 3, 367–385. 10.4310/jdg/1370979332Search in Google Scholar
[2] K. Burns, The flat strip theorem fails for surfaces with no conjugate points, Proc. Amer. Math. Soc. 115 (1992), no. 1, 199–206. 10.1090/S0002-9939-1992-1093593-0Search in Google Scholar
[3] K. Burns and G. Knieper, Rigidity of surfaces with no conjugate points, J. Differential Geom. 34 (1991), no. 3, 623–650. 10.4310/jdg/1214447537Search in Google Scholar
[4] S. Cohn-Vossen, Kürzeste Wege und Totalkrümmung auf Flächen, Compositio Math. 2 (1935), 69–133. Search in Google Scholar
[5] P. Eberlein, Geodesic flow in certain manifolds without conjugate points, Trans. Amer. Math. Soc. 167 (1972), 151–170. 10.1090/S0002-9947-1972-0295387-4Search in Google Scholar
[6] P. Eberlein and B. O’Neill, Visibility manifolds, Pacific J. Math. 46 (1973), 45–109. 10.2140/pjm.1973.46.45Search in Google Scholar
[7] J. Ge, L. Guijarro and P. Solórzano, Riemannian rigidity of the parallel postulate in total curvature, Math. Ann. 376 (2020), no. 1–2, 177–185. 10.1007/s00208-019-01883-8Search in Google Scholar
[8] L. W. Green, A theorem of E. Hopf, Michigan Math. J. 5 (1958), 31–34. 10.1307/mmj/1028998009Search in Google Scholar
[9] F. F. Guimarães, The integral of the scalar curvature of complete manifolds without conjugate points, J. Differential Geom. 36 (1992), no. 3, 651–662. 10.4310/jdg/1214453184Search in Google Scholar
[10] A. Haefliger and G. Reeb, Variétés (non séparées) à une dimension et structures feuilletées du plan, Enseign. Math. (2) 3 (1957), 107–125. Search in Google Scholar
[11] E. Hopf, Closed surfaces without conjugate points, Proc. Nat. Acad. Sci. USA 34 (1948), 47–51. 10.1073/pnas.34.2.47Search in Google Scholar PubMed PubMed Central
[12] H. Hopf, Differential geometry in the large. Notes taken by Peter Lax and John W. Gray, With a preface by S. S. Chern, With a preface by K. Voss, 2nd ed., Lecture Notes in Math. 1000, Springer, Berlin 1989. 10.1007/3-540-39482-6Search in Google Scholar
[13] A. Huber, Über die Darstellung vollständiger offener Flächen durch konforme Metriken, Festband 70. Geburtstag R. Nevanlinna, Springer, Berlin (1966), 35–41. 10.1007/978-3-642-86699-9_6Search in Google Scholar
[14] W. Kaplan, Regular curve-families filling the plane, I, Duke Math. J. 7 (1940), 154–185. 10.1215/S0012-7094-40-00710-4Search in Google Scholar
[15] W. Kaplan, Regular curve-families filling the plane, II, Duke Math. J. 8 (1941), 11–46. 10.1215/S0012-7094-41-00802-5Search in Google Scholar
[16] K. Shiohama, T. Shioya and M. Tanaka, The geometry of total curvature on complete open surfaces, Cambridge Tracts in Math. 159, Cambridge University Press, Cambridge 2003. 10.1017/CBO9780511543159Search in Google Scholar
[17] T. Shioya, The ideal boundaries of complete open surfaces, Tohoku Math. J. (2) 43 (1991), no. 1, 37–59. 10.2748/tmj/1178227534Search in Google Scholar
[18] T. Shioya, Geometry of total curvature, Actes de la Table Ronde de Géométrie Différentielle (Luminy 1992), Sémin. Congr. 1, Société Mathématique de France, Paris (1996), 561–600. Search in Google Scholar
© 2023 Walter de Gruyter GmbH, Berlin/Boston
