Aspects of global Riemannian geometry

Petersen, Peter

doi:10.1090/s0273-0979-99-00787-9

Cited by 15 publications

(10 citation statements)

References 120 publications

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“…Before continuing, we briefly mention some general references to Riemannian geometry [10,25,26]. As this paper is intended also for researchers working on applied sciences, we recall some standard notations and constructions in local coordinates, (x 1 , x 2 , .…”

Section: Background and Notationmentioning

confidence: 99%

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Recovering the isometry type of a Riemannian manifold from local boundary diffraction travel times

Hoop

Holman

Iversen

et al. 2015

Journal de Mathématiques Pures et Appliquées

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MSC: 35R30 86A15 53C21Keywords: Geometric inverse problems Riemannian manifold Shape operatorWe analyze the inverse problem, if a manifold and a Riemannian metric on it can be reconstructed from the sphere data. The sphere data consist of an open set U ⊂M and the pairs (t, Σ) where Σ ⊂ U is a smooth subset of a generalized metric sphere of radius t. This problem is an idealization of a seismic inverse problem, originally formulated by Dix [8], of reconstructing the wave speed inside a domain from boundary measurements associated with the single scattering of waves. In this problem, one considers a domain M with a varying and possibly anisotropic wave speed which we model as a Riemannian metric g. For our data, we assume that M contains a dense set of point diffractors and that in a subset U ⊂M , we can measure the wave fronts of the waves generated by these. The inverse problem we study is to recover the metric g in local coordinates anywhere on a set M ⊂M up to an isometry (i.e. we recover the isometry type of M ). To do this we show that the shape operators related to wave fronts produced by the point diffractors within M satisfy a certain system of differential equations which may be solved along geodesics of the metric. In this way, assuming that we know g as well as the shape operator of the wave fronts in the region U , we may recover g in certain coordinate systems (e.g. Riemannian normal coordinates centered at point diffractors). This generalizes the method of Dix to metrics which may depend on all spatial variables and be anisotropic. In particular, the novelty of this solution lies in the fact that it can be used to reconstruct the metric also in the presence of the caustics. (M.V. de Hoop), sean.holman@manchester.ac.uk (S.F. Holman), Einar.Iversen@norsar.com (E. Iversen), matti.lassas@helsinki.fi (M. Lassas), bjorn.ursin@ntnu.no (B. Ursin). http://dx.doi.org/10.1016/j.matpur.2014.09.003 0021-7824/© 2014 Elsevier Masson SAS. All rights reserved. M.V. de Hoop et al. / J. Math. Pures Appl. 103 (2015) 830-848 831 r é s u m éOn analyse un problème inverse, si une variété riemannienne peut être reconstruite à partir des données sphère. Les données sphère sont constituées d'un ensemble ouvert U ⊂M et les paires (t, Σ), où Σ ⊂ U set un sous-ensemble lisse d'une sphère métrique généralisée. Ce problème est une idéalisation d'un problème sismique inverse, à l'origine formulé par Dix [8], consistant à reconstruire la vitesse d'onde dans un domaine à partir des mesures aux frontières associées à la dispersion simple des ondes sismiques. On considère un domaine M avec une vitesse d'onde variable et éventuellement anisotrope modélisée par une métrique riemannienne g. On suppose que M contient une densité élevée de points diffractants et que dans un sous-ensemble U ⊂M , correspondant à un domaine contenant les instruments de mesure, on peut mesurer les fronts d'onde de la diffusion simple des ondes diffractées depuis les points diffractants. Le problème inverse étudié consiste à reconstruire la métrique g en coordon...

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Section: Background and Notationmentioning

confidence: 99%

“…(26), the t 1 there should be replaced by t 2 − t 1 in the second step). By the proof of Lemma 5 (28) can be bounded uniformly in terms of the directional curvature and a finite number of its derivatives, and therefore (28) can be bounded independent of t 1 .…”

Section: It Is Immediate Thatmentioning

confidence: 99%

Recovering the isometry type of a Riemannian manifold from local boundary diffraction travel times

Hoop

Holman

Iversen

et al. 2015

Journal de Mathématiques Pures et Appliquées

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“…(as a consequence of the Gauss Lemma; see Section 3.2 of [47]), which is an Eikonal equation for an isotropic medium, a special case of the Hamilton-Jacobi equation. Such equations have been studied extensively in the context of acoustic wave propagation in geophysics, where what we have called the distance function is known as the traveltime, fluid flow, computer vision, semiconductor fabrication (etching and deposition processes), and many more areas.…”

Section: Computing the Distance Map To Locate The Cut Locusmentioning

confidence: 99%

Jacobi’s last geometric statement extends to a wider class of Liouville surfaces

Sinclair

Tanaka

2006

Math. Comp.

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Abstract. Numerical evidence is presented which strongly suggests that "Jacobi's last geometric statement"-that the conjugate locus from a point has exactly four cusps and the corresponding cut locus consists of only one topological segment-holds for compact real analytic Liouville surfaces diffeomorphic to S 2 if the Gaussian curvature is everywhere positive and has exactly six critical points, these being two saddles, two global minima, and two global maxima (as is the case for an ellipsoid). Our experiments suggest that this is a sufficient rather than a necessary condition. Furthermore, for compact real analytic Liouville surfaces diffeomorphic to S 2 upon which the Gaussian curvature can be negative but has exactly six critical points, these being two saddles, two global minima, and two global maxima, it appears that the cut locus is always a subarc of a line given by x 1 = const or x 2 = const, where (x 1 , x 2 ) are canonical coordinates with respect to which the metric has the form (f 1 (x 1 ) + f 2 (x 2 ))(dx 2 1 + dx 2 2 ). In the case of an ellipsoid, these curves are lines of curvature.The point of this paper is to present a conjecture, already contained in the abstract, as a contribution to pure mathematical research in global Riemannian geometry. The overwhelming bulk of the paper will however necessarily be devoted to a description of the computational methods used to motivate the conjecture, and, in particular, the checks which have been performed to verify that the software (which has been written specifically for this study) is indeed performing correctly. Such checks are vital in any experimental work. This paper is not intended to be a contribution to computational science as such, since the algorithms we have used, although perhaps combined in an unusual manner, are standard.The conjugate locus from a point p (which we will often refer to as the starting point) on a Riemannian manifold is the envelope of the geodesics emanating from p. In two dimensions, it can also be defined analogously to the nodes of a vibrating string as follows. A geodesic passing through p can be uniquely defined by the angle θ it makes at p with one fixed (reference) geodesic also passing through p. Varying this angle by a small amount (δθ) will cause the geodesic to deviate from its original path. The distance between points on the geodesic with angle θ (say g θ ) and the geodesic curve with angle θ + δθ (write g θ+δθ ) can be written as a real function ξ δθ of arclength s along g θ , where s = 0 corresponds to the point p. Clearly ξ δθ (0) = 0, since we are only considering geodesics which pass through p. The limit of ξ δθ (s)/δθ as δθ approaches zero may have further zeros (the nodes of the string, where, to

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“…Zhongmin Qian proved that, under the assumption (1.7), M is compact and satisfies the diameter bound (1.8) (see [13]). In order to prove the above theorems, we use the index form I of a minimizing unit speed geodesic segment [4,10,12] and we also use elementary inequalities.…”

Section: Introductionmentioning

confidence: 99%

Modifications of the Ricci tensor and applications

Limoncu

2010

Arch. Math.

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Aspects of global Riemannian geometry

Abstract: Abstract. In this article we survey some of the developments in Riemannian geometry. We place special emphasis on explaining the relationship between curvature and topology for Riemannian manifolds with lower curvature bounds.

Cited by 15 publications

References 120 publications

Recovering the isometry type of a Riemannian manifold from local boundary diffraction travel times

Recovering the isometry type of a Riemannian manifold from local boundary diffraction travel times

Jacobi’s last geometric statement extends to a wider class of Liouville surfaces

Modifications of the Ricci tensor and applications

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