The tubular neighborhood theorem in contact geometry

This chapter describes the topology of a group action. It proves some topological facts about the fixed point set and the stabilizers of a continuous or a smooth action. The chapter also introduces the equivariant tubular neighborhood theorem and the equivariant Mayer–Vietoris sequence. A tubular neighborhood of a submanifold S in a manifold M is a neighborhood that has the structure of a vector bundle over S. Because the total space of a vector bundle has the same homotopy type as the base space, in calculating cohomology one may replace a submanifold by a tubular neighborhood. The tubular neighborhood theorem guarantees the existence of a tubular neighborhood for a compact regular submanifold. The Mayer–Vietoris sequence is a powerful tool for calculating the cohomology of a union of two open subsets. Both the tubular neighborhood theorem and the Mayer–Vietoris sequence have equivariant counterparts for a G-manifold where G is a compact Lie group.

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Weinstein Tubular Neighborhood Theorem

Lecture Notes in Mathematics - Lectures on Symplectic Geometry ◽

10.1007/978-3-540-45330-7_9 ◽

2008 ◽

pp. 61-66

Author(s):

Ana Cannas da Silva

Keyword(s):

Tubular Neighborhood ◽

Neighborhood Theorem

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The Approximate Tubular Neighborhood Theorem

Annals of Mathematics ◽

10.2307/3597284 ◽

2002 ◽

Vol 156 (3) ◽

pp. 867 ◽

Cited By ~ 2

Author(s):

Bruce Hughes

Keyword(s):

Tubular Neighborhood ◽

Neighborhood Theorem

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Optimization of the back contact geometry for high efficiency solar cells

Conference Record of the Twenty Third IEEE Photovoltaic Specialists Conference - 1993 (Cat. No.93CH3283-9) ◽

10.1109/pvsc.1993.347164 ◽

2002 ◽

Cited By ~ 1

Author(s):

M. Schofthaler ◽

U. Rau ◽

W. Fussel ◽

J.H. Werner

Keyword(s):

Solar Cells ◽

High Efficiency ◽

Contact Geometry ◽

Back Contact ◽

High Efficiency Solar Cells

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A novel three-dimensional wheel–rail contact geometry method in the switch panel considering variable cross-sections and yaw angle

Vehicle System Dynamics ◽

10.1080/00423114.2021.1941140 ◽

2021 ◽

pp. 1-24

Author(s):

Yu Chen ◽

Jian Wang ◽

Jiayin Chen ◽

Ping Wang ◽

Jingmang Xu ◽

...

Keyword(s):

Cross Sections ◽

Three Dimensional ◽

Contact Geometry ◽

Variable Cross ◽

Geometry Method ◽

Yaw Angle

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Contact geometry influence on the rail/wheel surface stress distribution

Procedia Engineering ◽

10.1016/j.proeng.2010.03.241 ◽

2010 ◽

Vol 2 (1) ◽

pp. 2249-2257 ◽

Cited By ~ 31

Author(s):

Juraj Gerlici ◽

Tomáš Lack

Keyword(s):

Stress Distribution ◽

Surface Stress ◽

Contact Geometry ◽

Wheel Surface

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Contact-geometry engineering in a circular light-emitting diode (LED) for improved electrical and optical performances

Semiconductor Science and Technology ◽

10.1088/0268-1242/28/1/015006 ◽

2012 ◽

Vol 28 (1) ◽

pp. 015006 ◽

Cited By ~ 2

Author(s):

Yun Soo Park ◽

Hwan Gi Lee ◽

Chung-Mo Yang ◽

Dong-Seok Kim ◽

Jin-Hyuk Bae ◽

...

Keyword(s):

Light Emitting Diode ◽

Contact Geometry ◽

Light Emitting

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Topology of ambient manifolds of non-singular Morse – Smale flows with three periodic orbits

Izvestiya VUZ Applied Nonlinear Dynamics ◽

10.18500/0869-6632-2021-29-6-863-868 ◽

2021 ◽

Vol 29 (6) ◽

pp. 863-868

Author(s):

Danila Shubin ◽

◽

Keyword(s):

Periodic Orbits ◽

Three Dimensional ◽

Solid Torus ◽

Lens Space ◽

Tubular Neighborhood ◽

Lens Spaces ◽

Ambient Manifold ◽

Orientable Manifold ◽

Smale Flows ◽

Published Result

The purpose of this study is to establish the topological properties of three-dimensional manifolds which admit Morse – Smale flows without fixed points (non-singular or NMS-flows) and give examples of such manifolds that are not lens spaces. Despite the fact that it is known that any such manifold is a union of circular handles, their topology can be investigated additionally and refined in the case of a small number of orbits. For example, in the case of a flow with two non-twisted (having a tubular neighborhood homeomorphic to a solid torus) orbits, the topology of such manifolds is established exactly: any ambient manifold of an NMS-flow with two orbits is a lens space. Previously, it was believed that all prime manifolds admitting NMS-flows with at most three non-twisted orbits have the same topology. Methods. In this paper, we consider suspensions over Morse – Smale diffeomorphisms with three periodic orbits. These suspensions, in turn, are NMS-flows with three periodic trajectories. Universal coverings of the ambient manifolds of these flows and lens spaces are considered. Results. In this paper, we present a countable set of pairwise distinct simple 3-manifolds admitting NMS-flows with exactly three non-twisted orbits. Conclusion. From the results of this paper it follows that there is a countable set of pairwise distinct three-dimensional manifolds other than lens spaces, which refutes the previously published result that any simple orientable manifold admitting an NMS-flow with at most three orbits is lens space.

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