Radial vector fields and the Poincaré-Hopf theorem

2000 ◽  
pp. 25-30
Author(s):  
Jean-Paul Brasselet
Keyword(s):  
Vector Fields ◽  
Hopf Theorem

scholarly journals On Sha's Secondary Chern–Euler Class

2012 ◽  
Vol 55 (3) ◽  
pp. 586-596
Author(s):  
Zhaohu Nie
Keyword(s):  
Vector Field ◽  
Cohomology Class ◽  
Vector Fields ◽  
Euler Class ◽  
Boundary Term ◽  
Manifold With Boundary ◽  
Normal Vectors ◽  
Tangential Projection ◽  
Euler Form ◽  
Hopf Theorem

AbstractFor a manifold with boundary, the restriction of Chern's transgression form of the Euler curvature form over the boundary is closed. Its cohomology class is called the secondary Chern– Euler class and was used by Sha to formulate a relative Poincaré–Hopf theorem under the condition that the metric on the manifold is locally product near the boundary. We show that the secondary Chern–Euler form is exact away from the outward and inward unit normal vectors of the boundary by explicitly constructing a transgression form. Using Stokes’ theorem, this evaluates the boundary term in Sha's relative Poincaré–Hopf theorem in terms of more classical indices of the tangential projection of a vector field. This evaluation in particular shows that Sha's relative Poincaré–Hopf theorem is equivalent to the more classical law of vector fields.

scholarly journals TOPOLOGICAL QUANTUM NUMBERS AND CURVATURE — EXAMPLES AND APPLICATIONS

2009 ◽  
Vol 06 (03) ◽  
pp. 533-553 ◽  
Author(s):  
JERZY SZCZȨSNY ◽  
MAREK BIESIADA ◽  
MAREK SZYDŁOWSKI
Keyword(s):  
Riemannian Geometry ◽  
Vector Fields ◽  
Elementary Proof ◽  
Dimension Vector ◽  
Smooth Mapping ◽  
Quantum Numbers ◽  
Topological Quantum ◽  
Bonnet Theorem ◽  
Hopf Theorem

Using the idea of the degree of a smooth mapping between two manifolds of the same dimension we present here the topological (homotopical) classification of the mappings between spheres of the same dimension, vector fields, monopole and instanton solutions. Starting with a review of the elements of Riemannian geometry we also present an original elementary proof of the Gauss–Bonnet theorem and also one of the Poincaré–Hopf theorem.

Normal Forms and Bifurcation of Planar Vector Fields

1994 ◽  
Author(s):  
Shui-Nee Chow ◽  
Chengzhi Li ◽  
Duo Wang
Keyword(s):  
Vector Fields ◽  
Normal Forms ◽  
Planar Vector Fields ◽  
Planar Vector

Parallel Computation of Complex Antennas around the Coated Object Using Iterative Vector Fields Technique

2014 ◽  
Vol E97.C (7) ◽  
pp. 661-669
Author(s):  
Ying YAN ◽  
Xunwang ZHAO ◽  
Yu ZHANG ◽  
Changhong LIANG ◽  
Zhewang MA
Keyword(s):  
Parallel Computation ◽  
Vector Fields

Differential p-forms and q-vector fields with constant coefficients

2020 ◽  
pp. 1
Author(s):  
Jaime Muñoz Masqué ◽  
Luis M. Pozo Coronado ◽  
M. Eugenia Rosado
Keyword(s):  
Vector Fields ◽  
Constant Coefficients ◽  
Q Vector

Applying The Poincare-Hopf Theorem

2018 ◽  
Author(s):  
Michael Nestler
Keyword(s):  
Hopf Theorem
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