Rays and souls in von Mangoldt planes

Belegradek, Igor; Choi, Eric H. C.; Innami, Nobuhiro

doi:10.2140/pjm.2012.259.279

“…As its application, some diameter sphere theorems have been proved in [Kondo 2007;Kondo and Ohta 2007;Lee 2005;Innami et al 2013b]. The geometry of geodesics on surfaces of revolution has been developed in [Belegradek et al 2012;Sinclair and Tanaka 2007;Tanaka 1992].…”

Section: An Examplementioning

confidence: 99%

Untitled

2015

Pacific J. Math.

0

View full text Add to dashboard Cite

show abstract

“…Von Mangoldt's surfaces of revolution have served as useful model surfaces. In fact, by bounding the sectional curvature of complete open Riemannian manifolds from below by the Gaussian curvature of a von Mangoldt's surface of revolution, various global structure theorems on the manifolds are obtained (see [2,14] for example.) Hence, generalized von Mangoldt surfaces can play a very important role as model surfaces in the global structure theorems mentioned above.…”

Section: Introductionmentioning

confidence: 99%

Generalized von Mangoldt surfaces of revolution and asymmetric two-spheres of revolution with simple cut locus structure

Tanaka¹,

Akamatsu²,

Sinclair³

et al. 2022

Preprint

0

View full text Add to dashboard Cite

It was proven in [24] that if the Gaussian curvature function along each meridian on a surface of revolution (R 2 , dr 2 + m(r) 2 dθ 2 ) is decreasing, then the cut locus of each point of θ −1 (0) is empty or a subarc of the opposite meridian θ −1 (π). Such a surface is called a von Mangoldt's surface of revolution in [24] (see also [17]). A surface of revolution (R 2 , dr 2 + m(r) 2 dθ 2 ) is called a generalized von Mangoldt surface of revolution if the cut locus of each point of θ −1 (0) is empty or a subarc of the opposite meridian θ −1 (π).For example, the surface of revolution (R 2 , dr 2 + m 0 (r) 2 dθ 2 ), where m 0 (x) = x/(1 + x 2 ), has the same cut locus structure as above and the cut locus of each point in r −1 ((0, ∞)) is nonempty. Note that the Gaussian curvature function is not decreasing along a meridian for this surface. In this article, we give sufficient conditions for a surface of revolution (R 2 , dr 2 + m(r) 2 dθ 2 ) to be a generalized von Mangoldt surface of revolution. Moreover, we prove that for any surface of revolution with finite total curvature c, there exists a generalized von Mangoldt surface of revolution with the same total curvature c such that the Gaussian curvature function along a meridian is not monotone on [a, ∞) for any a > 0.

show abstract

“…As its application, some diameter sphere theorems have been proved in [Kondo 2007;Kondo and Ohta 2007;Lee 2005;Innami et al 2013b]. The geometry of geodesics on surfaces of revolution has been developed in [Belegradek et al 2012;Sinclair and Tanaka 2007;Tanaka 1992].…”

Section: Introductionmentioning

confidence: 99%