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The Cut Loci, Conjugate Loci and Poles in a Complete Riemannian Manifold

Content Provider	Springer Nature Link
Author	Soga, Toshiro Shiohama, Katsuhiro Innami, buhiro
Copyright Year	2012
Abstract	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.
Ending Page	1406
Page Count	7
Starting Page	1400
File Format	PDF
ISSN	1016443X
e-ISSN	14208970
Journal	Geometric and Functional Analysis
Issue Number	5
Volume Number	22
Language	English
Publisher	SP Birkhäuser Verlag Basel
Publisher Date	2012-08-23
Publisher Place	Basel
Access Restriction	One Nation One Subscription (ONOS)
Subject Keyword	surface of revolution Global Riemannian geometry, including pinching geodesic Geodesics Analysis Pole cut point conjugate point
Content Type	Text
Resource Type	Article
Subject	Analysis Geometry and Topology

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