Contact Structures & Codimension-one Symplectic Foliations

10.33540/119 ◽  
2021 ◽  
Author(s):  
◽  
Lauran Evariste Toussaint
Keyword(s):  
Contact Structures ◽  
Codimension One

On submersions of the complex indicatrix

Filomat ◽  
2017 ◽  
Vol 31 (16) ◽  
pp. 5081-5092
Author(s):  
Elena Popovicia
Keyword(s):  
Fixed Point ◽  
Complex Projective Space ◽  
Finsler Space ◽  
Hermitian Manifold ◽  
Almost Hermitian Manifold ◽  
Codimension One ◽  
Real Submanifold ◽  
Fundamental Equations ◽  
Complex Projective ◽  
Extrinsic Sphere

In this paper we study the complex indicatrix associated to a complex Finsler space as an embedded CR - hypersurface of the holomorphic tangent bundle, considered in a fixed point. Following the study of CR - submanifolds of a K?hler manifold, there are investigated some properties of the complex indicatrix as a real submanifold of codimension one, using the submanifold formulae and the fundamental equations. As a result, the complex indicatrix is an extrinsic sphere of the holomorphic tangent space in each fibre of a complex Finsler bundle. Also, submersions from the complex indicatrix onto an almost Hermitian manifold and some properties that can occur on them are studied. As application, an explicit submersion onto the complex projective space is provided.

Autonomous Geometrical Mechanics

2018 ◽  
Author(s):  
Peter Mann
Keyword(s):  
Phase Space ◽  
Contact Structure ◽  
Invariant Tori ◽  
Time Dependent ◽  
Contact Structures ◽  
Natural Setting ◽  
Adiabatic Invariance ◽  
Jet Bundle ◽  
Integrability Theorem ◽  
Extended Phase

This chapter examines the structure of the phase space of an integrable system as being constructed from invariant tori using the Arnold–Liouville integrability theorem, and periodic flow and ergodic flow are investigated using action-angle theory. Time-dependent mechanics is formulated by extending the symplectic structure to a contact structure in an extended phase space before it is shown that mechanics has a natural setting on a jet bundle. The chapter then describes phase space of integrable systems and how tori behave when time-dependent dynamics occurs. Adiabatic invariance is discussed, as well as slow and fast Hamiltonian systems, the Hannay angle and counter adiabatic terms. In addition, the chapter discusses foliation, resonant tori, non-resonant tori, contact structures, Pfaffian forms, jet manifolds and Stokes’s theorem.

scholarly journals Contact structures and reducible surgeries

2015 ◽  
Vol 152 (1) ◽  
pp. 152-186 ◽  
Author(s):  
Tye Lidman ◽  
Steven Sivek
Keyword(s):  
Contact Structures ◽  
Surgery Theory ◽  
Contact Topology ◽  
Exceptional Surgery ◽  
Genus 2 ◽  
Cabling Conjecture

We apply results from both contact topology and exceptional surgery theory to study when Legendrian surgery on a knot yields a reducible manifold. As an application, we show that a reducible surgery on a non-cabled positive knot of genus$g$must have slope$2g-1$, leading to a proof of the cabling conjecture for positive knots of genus 2. Our techniques also produce bounds on the maximum Thurston–Bennequin numbers of cables.

scholarly journals Affine cones over cubic surfaces are flexible in codimension one

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Alexander Perepechko
Keyword(s):  
Automorphism Group ◽  
Open Subset ◽  
Pezzo Surface ◽  
Ample Divisor ◽  
Transitive Action ◽  
Del Pezzo Surface ◽  
Codimension One ◽  
Cubic Surfaces ◽  
Special Automorphism ◽  
Del Pezzo

AbstractLet Y be a smooth del Pezzo surface of degree 3 polarized by a very ample divisor that is not proportional to the anticanonical one. Then the affine cone over Y is flexible in codimension one. Equivalently, such a cone has an open subset with an infinitely transitive action of the special automorphism group on it.

scholarly journals Microfluidic Optical Shutter Flexibly x-y Actuated via Electrowetting-on-Dielectrics with <20 ms Response Time

2017 ◽  
Vol 2017 ◽  
pp. 1-5
Author(s):  
Henning Fouckhardt ◽  
Johannes Strassner ◽  
Carina Heisel ◽  
Dominic Palm ◽  
Christoph Doering
Keyword(s):  
Optical Properties ◽  
Response Time ◽  
Visible Spectral Range ◽  
Contact Structures ◽  
Light Path ◽  
Cell Type ◽  
Electrically Conductive ◽  
Droplet Movement ◽  
In Series ◽  
Optical Shutter

Tunable microoptics deals with devices of which the optical properties can be changed during operation without mechanically moving solid parts. Often a droplet is actuated instead, and thus tunable microoptics is closely related to microfluidics. One such device/module/cell type is an optical shutter, which is moved in or out of the path of the light. In our case the transmitting part comprises a moving transparent and electrically conductive water droplet, embedded in a nonconductive blackened oil, that is, an opaque emulsion with attenuation of 30 dB at 570 nm wavelength over the 250 μm long light path inside the fluid (15 dB averaged over the visible spectral range). The insertion loss of the cell is 1.5 dB in the “open shutter” state. The actuation is achieved via electrowetting-on-dielectrics (EWOD) with rectangular AC voltage pulses of 2·90 V peak-to-peak at 1 kHz. To flexibly allow for horizontal, vertical, and diagonal droplet movement in the upright x-y plane, the contact structures are prepared such that four possible stationary droplet positions exist. The cell is configured as two capacitors in series (along the z axis), such that EWOD forces act symmetrically in the front and back of the 60 nl droplet with a response time of <20 ms.

Unstable dimension variability and codimension-one bifurcations of two-dimensional maps

2004 ◽  
Vol 321 (4) ◽  
pp. 244-251 ◽  
Author(s):  
Ricardo L. Viana ◽  
José R.R. Barbosa ◽  
Celso Grebogi
Keyword(s):  
Two Dimensional ◽  
Codimension One
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