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Concept# Contact geometry

Summary

In mathematics, contact geometry is the study of a geometric structure on smooth manifolds given by a hyperplane distribution in the tangent bundle satisfying a condition called 'complete non-integrability'. Equivalently, such a distribution may be given (at least locally) as the kernel of a differential one-form, and the non-integrability condition translates into a maximal non-degeneracy condition on the form. These conditions are opposite to two equivalent conditions for 'complete integrability' of a hyperplane distribution, i.e. that it be tangent to a codimension one foliation on the manifold, whose equivalence is the content of the Frobenius theorem.
Contact geometry is in many ways an odd-dimensional counterpart of symplectic geometry, a structure on certain even-dimensional manifolds. Both contact and symplectic geometry are motivated by the mathematical formalism of classical mechanics, where one can consider either the even-dimensional phase space of a mechanical system or constant-energy hypersurface, which, being codimension one, has odd dimension.
Like symplectic geometry, contact geometry has broad applications in physics, e.g. geometrical optics, classical mechanics, thermodynamics, geometric quantization, integrable systems and to control theory. Contact geometry also has applications to low-dimensional topology; for example, it has been used by Kronheimer and Mrowka to prove the property P conjecture, by Michael Hutchings to define an invariant of smooth three-manifolds, and by Lenhard Ng to define invariants of knots. It was also used by Yakov Eliashberg to derive a topological characterization of Stein manifolds of dimension at least six.
A contact structure on an odd dimensional manifold is a smoothly varying family of codimension one subspaces of each tangent space of the manifold, satisfying a non-integrability condition. The family may be described as a section of a bundle as follows:
Given an n-dimensional smooth manifold M, and a point p ∈ M, a contact element of M with contact point p is an (n − 1)-dimensional linear subspace of the tangent space to M at p.

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Contact geometry

In mathematics, contact geometry is the study of a geometric structure on smooth manifolds given by a hyperplane distribution in the tangent bundle satisfying a condition called 'complete non-integrability'. Equivalently, such a distribution may be given (at least locally) as the kernel of a differential one-form, and the non-integrability condition translates into a maximal non-degeneracy condition on the form. These conditions are opposite to two equivalent conditions for 'complete integrability' of a hyperplane distribution, i.

Floer homology

In mathematics, Floer homology is a tool for studying symplectic geometry and low-dimensional topology. Floer homology is a novel invariant that arises as an infinite-dimensional analogue of finite-dimensional Morse homology. Andreas Floer introduced the first version of Floer homology, now called Lagrangian Floer homology, in his proof of the Arnold conjecture in symplectic geometry. Floer also developed a closely related theory for Lagrangian submanifolds of a symplectic manifold.

Frobenius theorem (differential topology)

In mathematics, Frobenius' theorem gives necessary and sufficient conditions for finding a maximal set of independent solutions of an overdetermined system of first-order homogeneous linear partial differential equations. In modern geometric terms, given a family of vector fields, the theorem gives necessary and sufficient integrability conditions for the existence of a foliation by maximal integral manifolds whose tangent bundles are spanned by the given vector fields.

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This thesis deals with applications of Lie symmetries in differential geometry and dynamical systems. The first chapter of the thesis studies the singular reduction of symmetries of cosphere bundles, the conservation properties of contact systems and their reduction. We generalise the results of [15] to the singular case making a complete topological and geometrical analysis of the reduced space. Applying the general theory of contact reduction developed by Lerman and Willett in [33] and [57], one obtains contact stratified spaces that lose all information of the internal structure of the cosphere bundle. Based on the cotangent bundle reduction theorems, both in the regular and singular case, as well as regular cosphere bundle reduction, one expects additional bundle-like structure for the contact strata. The cosphere bundle projection to the base manifold descends to a continuous surjective map from the reduced space at zero to the orbit quotient of the configuration space, but it fails to be a morphism of stratified spaces if we endow the reduced space with its contact stratification and the base space with the customary orbit type stratification defined by the Lie group action. In this chapter we introduce a new stratification of the contact quotient at zero, called the C-L stratification (standing for the coisotropic or Legendrian nature of its pieces) which solves the above mentioned two problems. Its main features are the following. First, it is compatible with the contact stratification of the quotient and the orbit type stratification of the configuration orbit space. It is also finer than the contact stratification. Second, the natural projection of the C-L stratified quotient space to its base space, stratified by orbit types, is a morphism of stratified spaces. Third, each C-L stratum is a bundle over an orbit type stratum of the base and it can be seen as a union of C-L pieces, one of them being open and dense in its corresponding contact stratum and contactomorphic to a cosphere bundle. The other strata are coisotropic or Legendrian submanifolds in the contact components that contain them. We also describe the relation between contact vector fields and the time dependent Hamilton-Jacobi equation. The reduction of contact systems and time dependent Hamiltonians is mentioned. In the second chapter we study geometric properties of Sasakian and Kähler quotients. We construct a reduction procedure for symplectic and Kähler manifolds using the ray preimages of the momentum map. More precisely, instead of taking as in point reduction the preimage of a momentum value μ, we take the preimage of ℝ+μ, the positive ray of μ. We have two reasons to develop this construction. One is geometric: non zero Kähler point reduction is not always well defined. The problem is that the complex structure may not leave invariant the horizontal distribution of the Riemannian submersion πμ : J-1(μ) → Mμ. The solution proposed in the literature is correct only in the case of totally isotropic momentum (i.e. Gμ = G). The other reason is that it provides invariant submanifolds for conformal Hamiltonian systems. They are usually non-autonomous mechanical systems with friction whose integral curves preserve, in the case of symmetries, the ray pre-images of the momentum map. We extend the class of conformal Hamiltonian systems already studied and complete the existing Lie Poisson reduction with the general ray one. As examples of symplectic (Kähler) and contact (Sasakian) ray reductions we treat the case of cotangent and cosphere bundles and we show that they are universal for ray reductions. Using techniques of A. Futaki, we prove that, under appropriate hypothesis, ray quotients of Kähler-Einstein or Sasaki-Einstein manifolds remain Kähler or Sasaki-Einstein. Note that it suffices to prove the Kähler case and the compatibility of ray reduction with the Boothby-Wang fibration. In the last chapter, we prove a stratification theorem for proper groupoids. First we find an equivalent way of describing the same result for a proper Lie group action, way which uses the theory of foliations and can be adapted to the language of Lie groupoids. We treat separately the case of free and proper groupoids. The orbit foliation of a proper Lie groupoid is a singular Riemannian foliation and we show this explicitly.

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A gravity-driven droplet will rapidly flow down an inclined substrate, resisted only by stresses inside the liquid. If the substrate is compliant, with an elastic modulus G < 100 kPa, the droplet will markedly slow as a consequence of viscoelastic braking. This phenomenon arises due to deformations of the solid at the moving contact line, enhancing dissipation in the solid phase. Here, we pattern compliant surfaces with textures and probe their interaction with droplets. We show that the superhydrophobic Cassie state, where a droplet is supported atop air-immersed textures, is preserved on soft textured substrates. Confocal microscopy reveals that every texture in contact with the liquid is deformed by capillary stresses. This deformation is coupled to liquid pinning induced by the orientation of contact lines atop soft textures. Thus, compared to flat substrates, greater forcing is required for the onset of drop motion when the soft solid is textured. Surprisingly, droplet velocities down inclined soft or hard textured substrates are indistinguishable; the textures thus suppress viscoelastic braking despite substantial fluid-solid contact. High-speed microscopy shows that contact line velocities atop the pillars vastly exceed those associated with viscoelastic braking. This velocity regime involves less deformation, thus less dissipation, in the solid phase. Such rapid motions are only possible because the textures introduce a new scale and contact-line geometry. The contact-line orientation atop soft pillars induces significant deflections of the pillars on the receding edge of the droplet; calculations confirm that this does not slow down the droplet.

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