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On the energy of unit vector fields with isolated singularities

100%
Brito F. G. B., Walczak P. G.
Annales Polonici Mathematici
|
2000
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tom 73
|
nr 3
269-274
EN
We consider the energy of a unit vector field defined on a compact Riemannian manifold M except at finitely many points. We obtain an estimate of the energy from below which appears to be sharp when M is a sphere of dimension >3. In this case, the minimum of energy is attained if and only if the vector field is totally geodesic with two singularities situated at two antipodal points (at the 'south and north pole').
2
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Limit cycles for vector fields with homogeneous components

100%
Cima A., Gasukk A., Mañosas F.
Applicationes Mathematicae
|
1996-1997
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tom 24
|
nr 3
281-287
EN
We study planar polynomial differential equations with homogeneous components. This kind of equations present a simple and well known dynamics when the degrees (n and m) of both components coincide. Here we consider the case $n\ne m$ and we show that the dynamics is more complicated. In fact, we prove that such systems can exhibit periodic orbits only when nm is odd. Furthermore, for nm odd we give examples of such differential equations with at least (n+m)/2 limit cycles.
3
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Ordinary differential equations and their exponentials

75%
Kock A., Reyes G.
Open Mathematics
|
2006
|
tom 4
|
nr 1
64-81
EN
In the context of Synthetic Differential Geometry, we discuss vector fields/ordinary differential equations as actions; in particular, we exploit function space formation (exponential spaces) in the category of actions.
4
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On Lie algebras of vector fields related to Riemannian foliations

75%
Rybicki T.
Annales Polonici Mathematici
|
1993
|
tom 58
|
nr 2
111-122
EN
Riemannian foliations constitute an important type of foliated structures. In this note we prove two theorems connecting the algebraic structure of Lie algebras of foliated vector fields with the smooth structure of a Riemannian foliation.
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