An Introduction to Differential Manifolds

Lafontaine, Jacques

doi:10.1007/978-3-319-20735-3

Cited by 19 publications

(17 citation statements)

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“…Proof. This is a variation of the standard Morse Lemma with the same proof, using Moser's trick, see [12], Theorem 3.44. It is sufficient to check that the solution provided by the proof has the properties required by the lemma.…”

Section: 1mentioning

confidence: 82%

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Six-point configurations in the hyperbolic plane and ergodicity of the mapping class group

Marché

Wolff

2019

Groups Geom. Dyn.

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Let X be the space of isometry classes of ordered sextuples of points in the hyperbolic plane such that the product of the six corresponding rotations of angle π is the identity. This space X is closely related to the PSL2(R)-character variety of the genus 2 surface Σ. In this article we study the topology and the natural symplectic structure on X, and we describe the action of the mapping class group of Σ on X. This completes the classification of the ergodic components of the character variety in genus 2 initiated in [13]. MSC Classification: 58D29, 57M05, 20H10, 30F60.

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Section: 1mentioning

confidence: 82%

“…As y 1 = y 6 , y 2 = y 3 and y 4 = y 5 this already gives B(z ′ ) ≤ A(z), by triangular inequalities. 12 2 for i = 1, 2 and d(x ′ i , y i ) ≤ a 56 2 for i = 5, 6 directly yield, by triangle inequalities, B(z ′ ) ≤ 5 6 A(z).…”

Section: Proofs Of the Lemmasmentioning

confidence: 89%

Six-point configurations in the hyperbolic plane and ergodicity of the mapping class group

Marché

Wolff

2019

Groups Geom. Dyn.

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“…In this section we present some standards results of differential geometry. An interested reader can find more on this subject in [29]. The Lemmas 1 and 3 are proved in Section 5.2.…”

Section: Submanifoldsmentioning

confidence: 99%

“…To prove Lemma 1 notice that by [29,Theorem 1.21] there is Ṽ Ă R d a neighborhood of zero, U 1 Ă U a neighborhood of y and φ : V Ñ U 1 a C p diffeomorphism such that φpx 1 , . .…”

Section: Proof Of Lemmas 1 Andmentioning

confidence: 99%

Stochastic Subgradient Descent on a Generic Definable Function Converges to a Minimizer

Schechtman¹

2021

Preprint

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It was previously shown by Davis and Drusvyatskiy that every Clarke critical point of a generic, semialgebraic (and more generally definable in an o-minimal structure), weakly convex function is lying on an active manifold and is either a local minimum or an active strict saddle. In the first part of this work, we show that when the weak convexity assumption fails a third type of point appears: a sharply repulsive critical point. Moreover, we show that the corresponding active manifolds satisfy the Verdier and the angle conditions which were introduced by us in our previous work. In the second part of this work, we show that, under a density-like assumption on the perturbation sequence, the stochastic subgradient descent (SGD) avoids sharply repulsive critical points with probability one. We show that such a density-like assumption could be obtained upon adding a small random perturbation (e.g. a nondegenerate Gaussian) at each iteration of the algorithm. These results, combined with our previous work on the avoidance of active strict saddles, show that the SGD on a generic definable (e.g. semialgebraic) function converges to a local minimum.

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“…Leaving aside the so-called 'trivial' manifolds diffeomorphic to S 3 , the remaining possibilities consist of an infinite list of nontrivial manifolds, each of which is diffeomorphic to a finite connected sum of the following form [5,6,25] (11) where k, l, and m are non-negative integers satisfying k + l + m ≥ 1 and if either k, l or m equals to zero, then terms of that type do not occur. Here each Γ k (for k ≥ 1) is a nontrivial ( = I) subgroup of SO(4) acting freely and properly discontinuously on S 3 and the resulting manifold is S 3 /Γ k (see [22] and [23] for free and properly discontinuous actions). S 2 × S 1 denotes the wormhole and supports only anisotropic (even locally) homogeneous metrics and therefore is not compatible with the cosmological observations.…”

Section: Adm Formalism With a Matter Sourcementioning

confidence: 99%

Asymptotic behavior of a matter filled universe with exotic topology

Mondal¹

2020

Class. Quantum Grav.

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The ADM formalism together with a constant mean curvature (CMC) temporal gauge is used to derive the monotonic decay of a weak Lyapunov function of the Einstein dynamical equations in an expanding universe with a positive cosmological constant and matter sources satisfying suitable energy conditions. While such a Lyapunov function does not, in general, represent a true Hamiltonian of the mattercoupled gravity dynamics (unlike in the vacuum case when it does), it can nevertheless be used to study the asymptotic behavior of the spacetimes. The Lyapunov function attains its infimum only in the limit that the matter sources be 'turned off' or, at least, become asymptotically negligible provided that the universe model does not re-collapse and form singularities. Later we specialize our result to the case of a perfect fluid which satisfies the desired energy conditions. However, even in this special case, we show using Shutz's velocity potential formalism cast into Hamiltonian form that unlike the vacuum spacetimes (with or without a positive cosmological constant), construction of a true Hamiltonian for the dynamics in constant mean curvature temporal gauge is difficult and therefore the Lyapunov function does not have a straightforward physical interpretation. Nevertheless, we show, for the fluid with equation of state P = (γ − 1)ρ (1 ≤ γ ≤ 2), that the general results obtained hold true and the infimum of the weak Lyapunov function can be related to the Sigma constant, a topological invariant of the manifold. Utilizing these results, some general conclusions are drawn regarding the asymptotic state of the universe and the dynamical control of the allowed spatial topologies in the cosmological models.

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An Introduction to Differential Manifolds

Cited by 19 publications

References 0 publications

Six-point configurations in the hyperbolic plane and ergodicity of the mapping class group

Six-point configurations in the hyperbolic plane and ergodicity of the mapping class group

Stochastic Subgradient Descent on a Generic Definable Function Converges to a Minimizer

Asymptotic behavior of a matter filled universe with exotic topology

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