The Geometry of Total Curvature on Complete Open Surfaces

Shiohama, Katsuhiro; Shioya, Takashi; Tanaka, Minoru

doi:10.1017/cbo9780511543159

Cited by 91 publications

(98 citation statements)

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“…The next lemma is concerned with the fundamental properties of Busemann functions (cf. [SST,Theorem 3.8.2]). We give proofs for completeness as our distance is nonsymmetric.…”

Section: Analysis Of Busemann Functionsmentioning

confidence: 99%

Splitting theorems for Finsler manifolds of nonnegative Ricci curvature

Ohta

2013

Journal Für Die Reine Und Angewandte Mathematik (Crelles Journal)

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We investigate the structure of a Finsler manifold of nonnegative weighted Ricci curvature including a straight line, and extend the classical Cheeger-GromollLichnerowicz splitting theorem. Such a space admits a diffeomorphic, measurepreserving splitting in general. As for a special class of Berwald spaces, we can perform the isometric splitting in the sense that there is a one-parameter family of isometries generated from the gradient vector field of the Busemann function. A Betti number estimate is also given for Berwald spaces.

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“…The next lemma is concerned with the fundamental properties of Busemann functions (cf. [SST,Theorem 3.8.2]). We give proofs for completeness as our distance is nonsymmetric.…”

Section: Analysis Of Busemann Functionsmentioning

confidence: 99%

Splitting theorems for Finsler manifolds of nonnegative Ricci curvature

Ohta

2013

Journal Für Die Reine Und Angewandte Mathematik (Crelles Journal)

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“…We briefly recall the general tools to handle the analysis of surfaces of revolution with applications to the ellipsoids [3], [11].…”

Section: Riemannian Metrics On Surfaces Of Revolutionmentioning

confidence: 99%

“…We study the deformation of the conjugate and cut loci for k ≥ 1 in Figs. [8][9][10][11]. The key point is: when k > 1, θ is not monotonous for all the trajectories.…”

Section: Numerical Computation Of Conjugate and Cut Locimentioning

confidence: 99%

Geometric and numerical techniques to compute conjugate and cut loci on Riemannian surfaces

Bonnard

Cots²,

Jassionnesse³

2014

Geometric Control Theory and Sub-Riemannian Geometry

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“…It is based on the work of [22] which uses the fact that the Jacobi equation is integrable and in particular the Jacobi fields can be estimated. The following proposition holds.…”

Section: (ϕ) If the First Return Mapping To The Equator Is Monotone mentioning

confidence: 99%

Conjugate-cut loci and injectivity domains on two-spheres of revolution

Bonnard¹,

Caillau²,

Janin³

2013

ESAIM: COCV

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to the period mapping of the ϕ-variable. One purpose of this article is to use this relation to evaluate the cut and conjugate loci for a family of metrics arising as a deformation of the round sphere and to determine the convexity properties of the injectivity domains of such metrics. These properties have applications in optimal control of space and quantum mechanics, and in optimal transport.Mathematics Subject Classification. 58B20, 49K15, 53C22.

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The Geometry of Total Curvature on Complete Open Surfaces

Cited by 91 publications

References 0 publications

Splitting theorems for Finsler manifolds of nonnegative Ricci curvature

Splitting theorems for Finsler manifolds of nonnegative Ricci curvature

Geometric and numerical techniques to compute conjugate and cut loci on Riemannian surfaces

Conjugate-cut loci and injectivity domains on two-spheres of revolution

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