Geodesic families characterizing flat metrics on a cylinder and a plane. (English) Zbl 1523.53043
Summary: We prove that a complete non-compact surface contains a domain which is isometric to a pipe cylinder if all prime closed geodesics in it have the same length. As an application, we show that a flat cylinder is conjugacy rigid in the class of surfaces whose universal covering planes satisfy the divergence property. We study the divergence property from the view point of geodesic conjugacy for the Euclidean plane.
Keywords:
complete non-compact surfaces; pipe cylinder; conjugacy rigid manifolds; divergence propertyReferences:
| [1] | Aravinda, C. S., and Gururaja, H. A., On certain geodesic conjugacies of flat cylinders, Proc. Indian Acad. Sci. Math. Sci., 127 (2017), no. 3, 517-523. https://doi.org/10.1007/s12044-017-0343-6 Bangert, V., Closed geodesics on complete surfaces, Math. Ann. 251 (1980), no. 1, 83-96. https://doi.org/10.1007/BF01420283 Bangert, V., and Emmerich, P., Area growth and the rigidity of surfaces without conjugate points, J. Differential Geom. 94 (2013), no. 3, 367-385. http://projecteuclid.org/euclid.jdg/1370979332 Besse, A., Manifolds all of whose geodesics are closed, Ergebnisse der Mathematik und ihrer Grenzgebiete 93, Springer-Verlag, Berlin-New York, 1978. Burns, K., and Kneiper, G., Rigidity of surfaces with no conjugate points, J. Differential Geom. 34 (1991), no. 3, 623-650. http://projecteuclid.org/euclid.jdg/1214447537 Busemann, H., The geometry of geodesics, Academic Press, Inc., New York, N.Y., 1955. Busemann, H., and Pedersen, F. P., Tori with one-parameter groups of motions, Math. Scand. 3 (1955), 209-220. https://doi.org/10.7146/math.scand.a-10438 Croke, C., and Kleiner, B., Conjugacy and rigidity for manifolds with a parallel vector field, J. Differential Geom. 39 (1994), no. 3, 659-680. http://projecteuclid.org/euclid.jdg/1214455076 Hopf, E., Closed surfaces without conjugate points, Proc. Nat. Acad. Sci. U.S.A, 34 (1948), 47-51. https://doi.org/10.1073/pnas.34.2.47 Innami, N., Differentiability of Busemann functions and total excess, Math. Z. 180 (1982), no. 2, 235-247. https://doi.org/10.1007/BF01318907 Innami, N., Splitting theorems of Riemannian manifolds, Compositio Math. 47 (1982), no. 3, 237-247. http://www.numdam.org/item?id=CM_1982__47_3_237_0 Innami, N., On tori having poles, Invent. Math. 84 (1986), no. 2, 437-443. https://doi.org/10.1007/BF01388813 Innami, N., Families of geodesics which distinguish flat tori, Math. J. Okayama Univ. 28 (1986), 207-217. Innami, N., The n-plane with integral curvature zero and without conjugate points, Proc. Japan Acad. Ser. A Math. Sci. 62 (1986), no. 7, 282-284. http://projecteuclid.org/euclid.pja/1195514220 Innami, N., A note on nonfocality properties in compact manifolds, Arch. Math. (Basel) 48 (1987), no. 3, 277-280. https://doi.org/10.1007/BF01195361 Innami, N., Riemannian metrics having poles and nonpoles on surfaces, Mem. Fac. Sci. Kyushu Univ. Ser. A 46 (1992), no 1, 79-84. https://doi.org/10.2206/kyushumfs.46.79 Innami, N., Itokawa, Y., Nagano, T., and Shiohama, K., Axial straight lines in the covering surface of a Finsler surface, to appear in Nihonkai Math. J. (2021). Innami, N., and Kondo, K., Geodesics and geodesic circles in a geodesically convex surface: a sub-mixing property, Publ. Math. Debrecen 95 (2019), no. 3-4, 279-306. https://doi.org/10.5486/pmd.2019.8362 Shiohama, K., Shioya, T., and Tanaka, M., The geometry of total curvature on complete open surfaces, Cambridge Tracts in Mathematics, 159. Cambridge University Press, Cambridge, 2003. https://doi.org/10.1017/CBO9780511543159 · doi:10.1007/s12044-017-0343-6{\par}Bangert, |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
