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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

On asymptotic behavior of the mass of rays


Author: Takashi Shioya
Journal: Proc. Amer. Math. Soc. 108 (1990), 495-505
MSC: Primary 53C20
DOI: https://doi.org/10.1090/S0002-9939-1990-0986652-X
MathSciNet review: 986652
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Abstract: We consider the measure of the set of all unit vectors tangent to rays emanating from a point $ p$ in a finitely connected complete open Riemannian $ 2$-manifold $ M$. If $ M$ with one end admits total curvature $ c(M)$, then this measure tends to $ \min \{ 2\pi \chi (M) - c(M),2\pi \} $ as $ p$ tends to infinity, where $ \chi (M)$ is the Euler characteristic.


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DOI: https://doi.org/10.1090/S0002-9939-1990-0986652-X
Keywords: Complete open manifolds, Gauss-Bonnet theorem, geodesies, rays, total curvature
Article copyright: © Copyright 1990 American Mathematical Society

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