A general approach to Morse theory

Tromba, Anthony J.

doi:10.4310/jdg/1214433845

Cited by 53 publications

(31 citation statements)

References 13 publications

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“…We assume that M is a closed Riemannian submanifold of R n -a mild condition in virtue of Nash's isometric embedding theorem-since the required theory on the Palais metric is only available in this case [Tro77,Prop. 6.1].…”

Section: Our Approachmentioning

confidence: 99%

“…In this section, we exploit results of Palais [Pal63,§13] and Tromba [Tro77,§6] to define the domain Γ of the objective function E in such a way that the gradient of E with respect to the Palais metric is guaranteed to exist and be unique.…”

Section: Preliminariesmentioning

confidence: 99%

“…The derivative and gradient of E s,1 (2) can be readily deduced, e.g., from [Tro77,§6] or [KS06, Th. 1].…”

Section: Derivative Of E S1mentioning

confidence: 99%

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A Gradient-Descent Method for Curve Fitting on Riemannian Manifolds

et al. 2011

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Given data points p0, . . . , pN on a closed submanifold M of R n and time instants 0 = t0 < t1 < . . . < tN = 1, we consider the problem of finding a curve γ on M that best approximates the data points at the given instants while being as "regular" as possible. Specifically, γ is expressed as the curve that minimizes the weighted sum of a sum-of-squares term penalizing the lack of fitting to the data points and a regularity term defined, in the first case as the mean squared velocity of the curve, and in the second case as the mean squared acceleration of the curve. In both cases, the optimization task is carried out by means of a steepest-descent algorithm on a set of curves on M . The steepest-descent direction, defined in the sense of the first-order and second-order Palais metric, respectively, is shown to admit analytical expressions involving parallel transport and covariant integral along curves. Illustrations are given in R n and on the unit sphere.

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Section: Our Approachmentioning

confidence: 99%

Section: Preliminariesmentioning

confidence: 99%

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A Gradient-Descent Method for Curve Fitting on Riemannian Manifolds

et al. 2011

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“…The linear map L is assumed to have a spectrum which is bounded away from the imaginary axis and the index of x is defined to be the dimension of the contracting subspace of eu. Tromba has a more general definition in terms of vector fields [14]. The following theorem is due to Palais and Smale in the case of nondegenerate critical points, and the author and Tromba in the weaker cases [14], [15].…”

mentioning

confidence: 99%

“…Tromba has a more general definition in terms of vector fields [14]. The following theorem is due to Palais and Smale in the case of nondegenerate critical points, and the author and Tromba in the weaker cases [14], [15]. The dimensions of the handles correspond to the indices of the critical points.…”

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confidence: 99%

Morse Theory by Perturbation Methods with Applications to Harmonic Maps

Uhlenbeck¹

1981

Transactions of the American Mathematical Society

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Abstract. There are many interesting variational problems for which the PalaisSmale condition cannot be verified. In cases where the Palais-Smale condition can be verified for an approximating integral, and the critical points converge, a Morse theory is valid. This theory applies to a class of variational problems consisting of the energy integral for harmonic maps with a lower order potential.The abstract Morse theory of Palais and Smale [9] can be applied with success to many problems in the calculus of variations to get existence theorems for stationary or critical points of integrals from topological information. However, there is a natural limitation in these applications in that the conditions are stated in terms of a fixed function space (which will usually be a Sobolev space). The integral must both be differentiable and satisfy a norm convergence property, which Palais and Smale call condition C, in the same space of functions. In this article we introduce a perturbation method which circumvents this difficulty in some variational problems. This technique may be applied in particular to harmonic mappings, and we include this somewhat technical discussion in the last section. The author is grateful to G. K. Francis and R. S. Palais for a number of helpful conversations.

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Smale and Nonlinear Analysis: A Personal Perspective

Tromba¹

1993

From Topology to Computation: Proceedings of the Smalefest

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A general approach to Morse theory

Cited by 53 publications

References 13 publications

A Gradient-Descent Method for Curve Fitting on Riemannian Manifolds

A Gradient-Descent Method for Curve Fitting on Riemannian Manifolds

Morse Theory by Perturbation Methods with Applications to Harmonic Maps

Smale and Nonlinear Analysis: A Personal Perspective

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