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Local rigidity, contact homeomorphisms, and conformal factors
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In this paper, it was shown that if the image of a Legendrian submanifold under a contact homeomorphism is smooth then it is Legendrian, assuming only positive local lower bounds on the conformal factors of the approximating contactomorphisms.Abstract:
We show that if the image of a Legendrian submanifold under a contact homeomorphism (i.e. a homeomorphism that is a $C^0$-limit of contactomorphisms) is smooth then it is Legendrian, assuming only positive local lower bounds on the conformal factors of the approximating contactomorphisms. More generally the analogous result holds for coisotropic submanifolds in the sense of arXiv:1306.6367. This is a contact version of the Humiliere-Leclercq-Seyfaddini coisotropic rigidity theorem in $C^0$ symplectic geometry, and the proof adapts the author's recent re-proof of that result in arXiv:1912.13043 based on a notion of local rigidity of points on locally closed subsets. We also provide two different flavors of examples showing that a contact homeomorphism can map a submanifold that is transverse to the contact structure to one that is smooth and tangent to the contact structure at a point.read more
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Chekanov's dichotomy in contact topology
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TL;DR: In this article, the main submanifolds of contact coisotropic manifolds were studied and a Chekanov type pseudo-metric was defined on the orbit space of a fixed sub-manifold of a contact manifold.
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The Herglotz Principle and Vakonomic Dynamics
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$C^0$-limits of Legendrian Submanifolds
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Local rigidity, symplectic homeomorphisms, and coisotropic submanifolds
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Remarks on the oscillation energy of Legendrian isotopies
TL;DR: In this article , the authors constructed non-compact contact manifolds containing compact Legendrians which can be displaced from their Reebow with arbitrarily small oscillation energy, and used this to show that the Shelukhin-Chekanov-Hofer pseudo-metric considered by [RZ20] is degenerate on the isotopy class of the constructed Legendrians.
References
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Book
Foundations of mechanics
TL;DR: In this article, Ratiu and Cushman introduce differential theory calculus on manifolds and derive an overview of qualitative and topological properties of differentiable properties of topological dynamics.
Book
Introduction to symplectic topology
Dusa McDuff,Dietmar Salamon +1 more
TL;DR: In this article, the authors present a survey of the history of classical and modern manifold geometry, from classical to modern, including linear and almost complex structures, and the Arnold conjecture of the group of symplectomorphisms.
Book
The Convenient Setting of Global Analysis
Andreas Kriegl,Peter W. Michor +1 more
TL;DR: Calculus of smooth mappings Calculus of holomorphic and real analytic mappings Partitions of unity Smoothly real compact spaces Extensions and liftings of mappings Infinite dimensional manifolds Calculus on infinite dimensional manifold, infinite dimensional differential geometry Manifolds of Mappings Further applications References as mentioned in this paper.
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The foundations of mechanics.
TL;DR: Find the secret to improve the quality of life by reading this foundations of mechanics, which can be your favorite book to read after having this book.
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An introduction to symplectic topology
TL;DR: In this article, the authors show that any symplectic vector space has even dimension and any isotropic subspace is contained in a Lagrangian subspace and Lagrangians have dimension equal to half the dimension of the total space.
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