On focal stability in dimension two

Peixoto, M. M.; Pugh, Charles

doi:10.1590/s0001-37652007000100001

Cited by 5 publications

(5 citation statements)

References 2 publications

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“…We refer to [10] for an overview of the interrelationships between the focal decomposition and physics, arithmetic and geometry. In [7,12], the notion of focal stability was introduced, which is a local notion, where the results imply that in dimension two, in the absence of conjugate points, generically in the strong C ∞ -Whitney topology, the focal decomposition is locally topologically stable.…”

Section: Definitions and Statement Of Resultsmentioning

confidence: 99%

Focal rigidity of hyperbolic surfaces

Kwakkel

Pura²,

Castorina³

2012

Ann. Acad. Sci. Fenn. Math.

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Abstract. In this note, we consider the rigidity of the focal decomposition of closed hyperbolic surfaces. We show that, generically, the focal decomposition of a closed hyperbolic surface does not allow for non-trivial topological deformations, without changing the hyperbolic structure of the surface. By classical rigidity theory this is also true in dimension n ≥ 3. Our current result extends a previous result that flat tori in dimension n ≥ 2 that are focally equivalent are isometric modulo rescaling.

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Section: Definitions and Statement Of Resultsmentioning

confidence: 99%

Focal rigidity of hyperbolic surfaces

Kwakkel

Pura²,

Castorina³

2012

Ann. Acad. Sci. Fenn. Math.

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“…In [7], the notion of focal stability is introduced for smooth Riemannian manifolds in dimension n. In analogy with structural stability for dynamical systems, this gives rise to the focal stability conjecture: given p ∈ M , the generic Riemannian structure is focally stable at p. In [13], if n = 2 and there are no conjugate points, the above conjecture has been shown to be true. The notion of focal stability in [7] is local.…”

Section: Definitions and Statement Of Resultsmentioning

confidence: 99%

Focal rigidity of flat tori

Kwakkel

Martens

Peixoto

2011

An. Acad. Bras. Ciênc.

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Given a closed Riemannian manifold (M, g), i.e. compact and boundaryless, there is a partition of its tangent bundle TM = i i called the focal decomposition of TM. The sets i are closely associated to focusing of geodesics of (M, g), i.e. to the situation where there are exactly i geodesic arcs of the same length joining points p and q in M. In this note, we study the topological structure of the focal decomposition of a closed Riemannian manifold and its relation with the metric structure of the manifold. Our main result is that flat n-tori, n ≥ 2, are focally rigid in the sense that if two flat tori are focally equivalent then the tori are isometric up to rescaling. The case n = 2 was considered before by F. Kwakkel.Key words: Riemannian manifolds, focal decomposition, rigidity. DEFINITIONS AND STATEMENT OF RESULTSIn general, topological characteristics of a Riemannian manifold do not determine its geometry, that is, its metric structure. However, starting in the 1960s, examples have been discovered for which such characteristics do determine the geometry. The manifolds can not be deformed without changing the characteristic. One speaks of rigidity. The prototype rigidity result is due to Mostow (Mostow 1968). MOSTOW'S RIGIDITY THEOREM. Two compact hyperbolic n-manifolds, n ≥ 3, with isomorphic fundamental groups are isometric.Given an analytic manifold M, we denote R ω (M) the class of analytic Riemannian structures on M.This class of structures is dense in the class R ∞ (M) of smooth Riemannian structures in the C ∞ strong Whitney topology, see (Hirsch 1976). Let (M, g) and ( M, g) be closed (compact and boundaryless) analytic manifolds. Two Riemannian manifolds are said to be isometric up to a rescaling if, up to a constant rescaling of the metric g, the manifolds (M, g) and ( M, g) are isometric.

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“…In my suitcase, I carried a first study of the note of Andronov and Leontovich on First Order Structural Stability [3], which later would be one of the fundamental elements of motivation for my Doctoral Thesis. Another, crucial, piece of motivation would come from the works of Peixoto [12], [13], [15]. The aforementioned note [3] consisted only of four pages, without any demonstration.…”

Section: An Interview With Prof Peixotomentioning

confidence: 99%

Reminiscences of a Mathematical Sojourn at San Marcos, 1959 - 61, and at IMPA, 1962.

Sotomayor¹

2020

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This evocative essay recounts aspects of the initiation of the author into Mathematics at the University of San Marcos, Lima, Peru, 1959-61, and at the NationalInstitute of Pure and Applied Mathematics, IMPA, Rio de Janeiro, Brazil, 1962. It links four previous mathematical essays: My Geometry Notebook [18]: DOI:10.13140/RG.2.2.18090.47043, Mathematical Encounters [21], On a list of Ordinary Differential Equations Problems, [22]: http://arxiv.org/abs/1808.10571,and On Mauricio M. Peixoto and the arrival of Structural Stability to Rio de Janeiro, 1955. [23] : https://doi.org/10.1590/0001-3765202020191219 .

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On focal stability in dimension two

Abstract: In Kupka et al. 2006 appears the Focal Stability Conjecture: the focal decomposition of the generic Riemann structure on a manifold M is stable under perturbations of the Riemann structure. In this paper, we prove the conjecture when M has dimension two, and there are no conjugate points.

Cited by 5 publications

References 2 publications

Focal rigidity of hyperbolic surfaces

Focal rigidity of hyperbolic surfaces

Focal rigidity of flat tori

Reminiscences of a Mathematical Sojourn at San Marcos, 1959 - 61, and at IMPA, 1962.

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