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The ideal boundaries and global geometric properties of complete open surfaces

Published online by Cambridge University Press:  22 January 2016

Takashi Shioya
Takashi Shioya*
Affiliation:
Department of Mathematics, Faculty of Science, Fukuoka 812, Japan
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In this paper we study the ideal boundaries of surfaces admitting total curvature as a continuation of [Sy2] and [Sy3]. The ideal boundary of an Hadamard manifold is defined to be the equivalence classes of rays. This equivalence relation is the asymptotic relation of rays, defined by Busemann [Bu]. The asymptotic relation is not symmetric in general. However in Hadamard manifolds this becomes symmetric. Here it is essential that the manifolds are focal point free.

Type
Research Article
Information
Nagoya Mathematical Journal , Volume 120 , December 1990 , pp. 181 - 204
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 1990

References

[BGS] Ballmann, W., Gromov, M. and Schroeder, V., Manifolds of Nonpositive Curvature, Progress in Math. 61, Birkhäuser, Boston-Basel Stuttgart, 1985.Google Scholar
[Bu] Busemann, H., The geometry of geodesies, Academic Press, New York, 1955.Google Scholar
[Co1] Cohn-Vossen, S., Kürzeste Wege und Totalkrümmung auf Flächen, Composito Math., 2 (1935), 63–133.Google Scholar
[Co2] Cohn-Vossen, S., Totalkrümmung und geodätisehe Linien auf einfach zusammenhängenden offenen volständigen Flächenstücken, Recueil Math. Moscow, 43 (1936), 139–163.Google Scholar
[EO] Eberlein, P. and O’Neill, B., Visibility manifolds, Pac. J. Math., 46 (1973), 45–110.Google Scholar
[Fi] Fiala, F., Le problème isopérimètres sur les surface onvretes à courbure positive, Comment. Math. Helv., 13 (1941), 293–346.Google Scholar
[Ha] Hartman, P., Geodesic parallel coordinates in the large, Amer. J. Math., 86 (1964), 705–727.Google Scholar
[Ks] Kasue, A., A compactification of a manifold with asymptotically nonnegative curvature, Ann. scient. Éc. Norm. Sup., 21 (1988), 593–622.Google Scholar
[Md1] Maeda, M., A geometric significance of total curvature on complete open surfaces, Geometry of Geodesies and Related Topics, Advanced Studies in Pure Math., 3 (1984), 451–458, Kinokuniya, Tokyo, 1984.Google Scholar
[Md2] Maeda, M., On the total curvature of noncompact Riemannian manifolds I, Yokohama Math. J., 33 (1985), 93–101.Google Scholar
[Og] Oguchi, T., Total curvature and measure of rays, Proc. Fac. Sci. Tokai Univ., 21 (1986), 1–4.Google Scholar
[Ot] Ohtsuka, F., On a relation between total curvature and Tits metric, Bull. Fac. Sci. Ibaraki Univ., 20 (1988).Google Scholar
[Sg1] Shiga, K., On a relation between the total curvature and the measure of rays, Tsukuba J. Math., 6 (1982), 41–50.Google Scholar
[Sg2] Shiga, K., A relation between the total curvature and the measure of rays, II, Tôhoku Math. J., 36 (1984), 149–157.Google Scholar
[Sh1] Shiohama, K., Busemann function and total curvature, Invent. Math., 53 (1979), 281–297.Google Scholar
[Sh2] Shiohama, K., The role of total curvature on complete noncompact Riemannian 2-manifolds, Illinois J. Math., 28 (1984), 597–620.Google Scholar
[Sh3] Shiohama, K., Cut locus and parallel circles of a closed curve on a Riemannian plane admitting total curvature, Comment. Math. Helv., 60 (1985), 125–138.Google Scholar
[Sh4] Shiohama, K., Total curvatures and minimal areas of complete open surfaces, Proc. Amer. Math. Soc., 94 (1985), 310–316.Google Scholar
[Sh5] Shiohama, K., An integral formula for the measure of rays on complete open surfaces, J. Differential Geometry, 23 (1986), 197–205.Google Scholar
[SST] Shiohama, K., Shioya, T. and Tanaka, M., Mass of rays on complete open surfaces, Pac. J. Math., 143 (1990), 349–358.Google Scholar
[ST] Shiohama, K. and Tanaka, M., An isoperimetric problem for infinitely connected complete open surfaces, Geometry of Manifolds, Perspectives in Mathematics, 8 (1989), 317–343, Academic Press.Google Scholar
[Sy1] Shioya, T., On asymptotic behavior of the mass of rays, Proc. Amer. Math. Soc., 108 (1990), 495–505.Google Scholar
[Sy2] Shioya, T., The ideal boundaries of complete open surfaces, to appear in Tôhoku Math. J. Google Scholar
[Sy3] Shioya, T., The ideal boundaries of complete open surfaces admitting total curvature c(M)=−∞, Geometry of Manifolds, Perspectives in Mathematics, 8 (1989) 351–364, Academic Press.Google Scholar