Topology of Manifolds

2016 ◽  
pp. 231-270
Keyword(s):  
Topology Of Manifolds

Geometry and Topology of Manifolds

2020 ◽  
Author(s):  
Vicente Muñoz ◽  
Ángel González-Prieto ◽  
Juan Rojo
Keyword(s):  
And Topology ◽  
Topology Of Manifolds

scholarly journals On a theorem of Ambrose

2006 ◽  
Vol 81 (2) ◽  
pp. 149-152 ◽  
Author(s):  
David J. Wraith
Keyword(s):  
Ricci Curvature ◽  
Alternative Proof ◽  
And Topology ◽  
Riccati Inequality ◽  
Topology Of Manifolds

AbstractA Riccati inequality involving the Ricci curvature can be used to deduce many interesting results about the geometry and topology of manifolds. In this note we use it to present a short alternative proof to a theorem of Ambrose.

scholarly journals On Sectional Curvature of Metric Connection with Vectorial Torsion

2020 ◽  
pp. 124-127
Author(s):  
E.D. Rodionov ◽  
V.V. Slavsky ◽  
O.P. Khromova
Keyword(s):  
Sectional Curvature ◽  
Curvature Tensor ◽  
Sufficient Conditions ◽  
Civita Connection ◽  
Modern Physics ◽  
Metric Connection ◽  
Levi Civita Connection ◽  
The Difference ◽  
Topology Of Manifolds ◽  
Conformal Deformations

Papers of many mathematicians are devoted to the study of semisymmetric connections or metric connections with vector torsion on Riemannian manifolds. This type of connectivity is one of the three main types discovered by E. Cartan and finds its application in modern physics, geometry, and topology of manifolds. Geodesic lines and the curvature tensor of a given connection were studied by I. Agricola, K. Yano, and other mathematicians. In particular, K. Yano proved an important theorem on the connection of conformal deformations and metric connections with vector torsion. Namely: a Riemannian manifold admits a metric connection with vector torsion and the curvature tensor being equal to zero if and only if it is conformally flat. Although the curvature tensor of a hemisymmetric connection has a smaller number of symmetries compared to the Levi-Civita connection, it is still possible to define the concept of sectional curvature in this case. The question naturally arises about the difference between the sectional curvature of a semisymmetric connection and the sectional curvature of a Levi-Civita connection.This paper is devoted to the study of this issue, and the authors find the necessary and sufficient conditions for the sectional curvature of the semisymmetric connection to coincide with the sectional curvature of the Levi-Civita connection. Non-trivial examples of hemisymmetric connections are constructed when possible.

scholarly journals On Embedding of the Morse-Smale Diffeomorphisms in a Topological Flow

2020 ◽  
Vol 66 (2) ◽  
pp. 160-181
Author(s):  
V. Z. Grines ◽  
E. Ya. Gurevich ◽  
O. V. Pochinka
Keyword(s):  
Invariant Manifolds ◽  
Three Dimensional ◽  
Sufficient Conditions ◽  
Periodic Points ◽  
Topological Classification ◽  
Higher Dimension ◽  
Topological Information ◽  
Topology Of Manifolds ◽  
Necessary And Sufficient

This review presents the results of recent years on solving of the Palis problem on finding necessary and sufficient conditions for the embedding of Morse-Smale cascades in topological flows. To date, the problem has been solved by Palis for Morse-Smale diffeomorphisms given on manifolds of dimension two. The result for the circle is a trivial exercise. In dimensions three and higher new effects arise related to the possibility of wild embeddings of closures of invariant manifolds of saddle periodic points that leads to additional obstacles for Morse-Smale diffeomorphisms to embed in topological flows. The progress achieved in solving of Paliss problem in dimension three is associated with the recently obtained complete topological classification of Morse-Smale diffeomorphisms on three-dimensional manifolds and the introduction of new invariants describing the embedding of separatrices of saddle periodic points in a supporting manifold. The transition to a higher dimension requires the latest results from the topology of manifolds. The necessary topological information, which plays key roles in the proofs, is also presented in the survey.

Basic notions

2020 ◽  
pp. 3-20
Author(s):  
Piotr T. Chruściel
Keyword(s):  
Moving Frames ◽  
Lorentzian Manifolds ◽  
Civita Connection ◽  
Levi Civita Connection ◽  
Basic Facts ◽  
Topology Of Manifolds

The aim of this chapter is to set the ground for the remainder of this work. We present our conventions and notations in Section 1.1. We review some basic facts about the topology of manifolds in Section 1.2. Lorentzian manifolds and spacetimes are introduced in Section 1.3. Elementary facts concerning the Levi-Civita connection and its curvature are reviewed in Section 1.4. Some properties of geodesics, as needed in the remainder of this book, are presented in Section 1.5. The formalism of moving frames is outlined in Section 1.6, where it is used to calculate the curvature of metrics of interest.

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