The Cut Loci, Conjugate Loci and Poles in a Complete Riemannian Manifold
Tóm tắt
Let M be a complete Riemannian manifold. We first prove that there exist at least two geodesics connecting p and every point in M if the tangent cut locus of
$${p \in M}$$
is not empty and does not meet its tangent conjugate locus. It follows from this that if M admits a pole and
$${p \in M}$$
is not a pole, then the tangent conjugate and tangent cut loci of p have a point in common. Here we say that a point q in M is a pole if the exponential map from the tangent space T
q
M at q onto M is a diffeomorphism. Using this result, we estimate the size of the set of all poles in M having a pole whose sectional curvature is pinched by those of two von Mangoldt surfaces of revolution, meaning that their Gaussian curvatures are monotone and nonincreasing with respect to the distances to their vertices.
Tài liệu tham khảo
J. Cheeger and D.G. Ebin. Comparison Theorems in Riemannian Geometry. AMS Chelsea Publishing, Chelsea (2008).
Greene R.E., Wu. H.: Function Theory on Manifolds which possess a pole. Lecture Notes in Mathematics.. Springer, Berlin (1979)
Innami N.: Integral formulas for polyhedral and spherical billiards. Journal of the Mathematical Society of Japan. 2(50), 339–357 (1998)
H. von Mangoldt. Uber dijenigen Punkte auf positive gekrümmten Flähen, welche die Eigenshaft haben, das die von Ihnen ausgehenden geodatischen Linien nie aufhoren kurzeste Linien zu sein. Journal für die Reine und Angewandte Mathematik, 91 (1881), 23–53.
H. Rauch. Geodesics and Curvature in Differential Geometry in the Large. Yeshiva Univ. Press, New York (1959).
Shiohama K., Shioya T., Tanaka M.: The Geometry of Total Curvature on Complete Open Surfaces, Cambridge Tracts in Mathematics. Cambridge University Press, Cambridge (2003)
T. Soga Remarks on the set of poles on a pointed complete surface. Nihonkai Mathematical Journal, (1)22 (2011).
Tanaka M.: On a characterization of a surface of revolution with many poles. Memoirs of the Faculty of Science. Kyushu University Series A 2(46), 251–268 (1992)
Tanaka M.: On the cut loci of a von Mangoldt’s surface of revolution. Journal of the Mathematical Society of Japan 4(44), 631–641 (1992)
Weinstein A.: The cut locus and conjugate locus of a riemannian manifold. Annals of Mathematics 2(87), 29–41 (1968)