Deformations of geodesic fields

Gluck, Herman; Singer, David A.

doi:10.1090/s0002-9904-1976-14107-9

Cited by 8 publications

(2 citation statements)

References 4 publications

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“…Note that the volume element, all of the functions H 0 appearing above, and both B and B can be computed from the Jacobi fields along the geodesics given by θ and θ. Theorem 12 tells us that the only contributions to the singular part come from points in P , which on M are places where locally the cut locus looks like a smooth hypersurface and the singular part of ∇ 2 E(x, y) is just given by the jump discontinuity of ∇E(x, y) across this hypersurface. While the cut locus itself can be quite complicated (for example, it may not be triangulable, as shown by Gluck and Singer [8]), the singular part of ∇ 2 E(x, y) is supported only at those points with the nicest local structure.…”

Section: Mollification Of Energymentioning

confidence: 99%

The small-time asymptotics of the heat kernel at the cut locus

Neel¹

2007

Communications in Analysis and Geometry

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Abstract. We study the small time asymptotics of the gradient and Hessian of the logarithm of the heat kernel at the cut locus, giving, in principle, complete expansions for both quantities. We relate the leading terms of the expansions to the structure of the cut locus, especially to conjugacy, and we provide a probabilistic interpretation in terms of the Brownian bridge. In particular, we show that the cut locus is the set of points where the Hessian blows up faster than 1/t. We also study the distributional asymptotics and use them to compute the distributional Hessian of the energy function (that is, one-half the distance function squared).

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Section: Mollification Of Energymentioning

confidence: 99%

The small-time asymptotics of the heat kernel at the cut locus

Neel¹

2007

Communications in Analysis and Geometry

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“…We claim that the orthogonal trajectories can be made to be the level curves of a distance-function r on a compact surface. We first blend together the field of tangent lines to the cusp with a field of geodesies radiating from a point m, by the method of Gluck and Singer [2]. The geodesic arc from n to the tip is minimizing in some neighborhood of the arc, and we can map this neighborhood diffeomorphically into a compact surface.…”

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confidence: 99%

Decomposition of Cut Loci

Bishop¹

1977

Proceedings of the American Mathematical Society

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Abstract. If p is a point in a complete riemannian manifold, the points of the cut locus of p are designated as singular or ordinary according to whether there is just one or more minimizing geodesies from p. It is proved that the ordinary cut-points are dense in the cut locus.1 Introduction. Throughout this paper we will consider a complete riemannian manifold M of dimension d. For m E M, a cut-point of m is a point/» such that every extension of a distance-minimizing segment from m to p beyond p is no longer minimizing. In the tangent space Mm, a vector x E Mm is a tangent cut-point if the ray t -» tx, 0 < / < 1, is mapped by expm to a minimizing segment from m to a cut-point expmx. If there are two or more minimizing segments from' m to p, we define p to be an ordinary cut-point. If there is just one minimizing segment, we call p a singular cut-point. Ordinary and singular tangent cut-points are defined so that they correspond to the same kind of points in M under expm.

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The radial part of Brownian motion II. Its life and times on the cut locus

Cranston

Kendall

March

1993

Probab. Th. Rel. Fields

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This paper is a sequel to Kendall (1987), which explained how the It6 formula for the radial part of Brownian motion X on a Riemannian manifold can be extended to hold for all time including those times at which X visits the cut locus. This extension consists of the subtraction of a correction term, a continuous predictable non-decreasing process L which changes only when X visits the cut locus. In this paper we derive a representation of L in terms of measures of local time of X on the cut locus. In analytic terms we compute an expression for the singular part of the Laplacian of the Riemannian distance function. The work uses a relationship of the Riemannian distance function to convexity, first described by Wu (1979) and applied to radial parts of F-martingales in Kendall (1993). (1991): 58G32, 60H10, 60J45 Mathematics Subject Classification

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Deformations of geodesic fields

Cited by 8 publications

References 4 publications

The small-time asymptotics of the heat kernel at the cut locus

The small-time asymptotics of the heat kernel at the cut locus

Decomposition of Cut Loci

The radial part of Brownian motion II. Its life and times on the cut locus

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