scholarly journals The Approximate Tubular Neighborhood Theorem

2002 ◽  
Vol 156 (3) ◽  
pp. 867 ◽  
Author(s):  
Bruce Hughes
Keyword(s):  
Tubular Neighborhood ◽  
Neighborhood Theorem

The Topology of a Group Action

2020 ◽  
pp. 205-212
Author(s):  
Loring W. Tu
Keyword(s):  
Fixed Point ◽  
Vector Bundle ◽  
Group Action ◽  
Lie Group ◽  
Tubular Neighborhood ◽  
Compact Lie Group ◽  
Fixed Point Set ◽  
Smooth Action ◽  
Point Set ◽  
Neighborhood Theorem

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.

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

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.

scholarly journals Distance Bounds for Graphs with Some Negative Bakry-Émery Curvature

2019 ◽  
Vol 7 (1) ◽  
pp. 1-14
Author(s):  
Shiping Liu ◽  
Florentin Münch ◽  
Norbert Peyerimhoff ◽  
Christian Rose
Keyword(s):  
Upper Bound ◽  
Tubular Neighborhood ◽  
Exceptional Set ◽  
Positively Curved

Abstract We prove distance bounds for graphs possessing positive Bakry-Émery curvature apart from an exceptional set, where the curvature is allowed to be non-positive. If the set of non-positively curved vertices is finite, then the graph admits an explicit upper bound for the diameter. Otherwise, the graph is a subset of the tubular neighborhood with an explicit radius around the non-positively curved vertices. Those results seem to be the first assuming non-constant Bakry-Émery curvature assumptions on graphs.

scholarly journals ALL ORDER COVARIANT TUBULAR EXPANSION

2014 ◽  
Vol 26 (01) ◽  
pp. 1350019 ◽  
Author(s):  
PARTHA MUKHOPADHYAY
Keyword(s):  
Riemannian Manifold ◽  
Closed Form ◽  
Tubular Neighborhood ◽  
Normal Coordinate ◽  
Integral Theorem ◽  
Normal Coordinates ◽  
Expansion Coefficients

We consider tubular neighborhood of an arbitrary submanifold embedded in a (pseudo-) Riemannian manifold. This can be described by Fermi normal coordinates (FNC) satisfying certain conditions as described by Florides and Synge in [15]. By generalizing the work of Muller et al. in [54] on Riemann normal coordinate expansion, we derive all order FNC expansion of vielbein in this neighborhood with closed form expressions for the curvature expansion coefficients. Our result is shown to be consistent with certain integral theorem for the metric proved in [15].

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