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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
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2000
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tom 73
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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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Tangent Lines and Lipschitz Differentiability Spaces

100%
Cavalletti F., Rajala T.
Analysis and Geometry in Metric Spaces
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2016
|
tom 4
|
nr 1
EN
We study the existence of tangent lines, i.e. subsets of the tangent space isometric to the real line, in tangent spaces of metric spaces.We first revisit the almost everywhere metric differentiability of Lipschitz continuous curves. We then show that any blow-up done at a point of metric differentiability and of density one for the domain of the curve gives a tangent line. Metric differentiability enjoys a Borel measurability property and this will permit us to use it in the framework of Lipschitz differentiability spaces.We show that any tangent space of a Lipschitz differentiability space contains at least n distinct tangent lines, obtained as the blow-up of n Lipschitz curves, where n is the dimension of the local measurable chart. Under additional assumptions on the space, such as curvature lower bounds, these n distinct tangent lines span an n-dimensional part of the tangent space.
3
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Angles between Curves in Metric Measure Spaces

88%
Han B.-X., Mondino A.
Analysis and Geometry in Metric Spaces
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2017
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tom 5
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nr 1
47-68
EN
The goal of the paper is to study the angle between two curves in the framework of metric (and metric measure) spaces. More precisely, we give a new notion of angle between two curves in a metric space. Such a notion has a natural interplay with optimal transportation and is particularly well suited for metric measure spaces satisfying the curvature-dimension condition. Indeed one of the main results is the validity of the cosine formula on RCD*(K, N) metric measure spaces. As a consequence, the new introduced notions are compatible with the corresponding classical ones for Riemannian manifolds, Ricci limit spaces and Alexandrov spaces.
4
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Torsional rigidity on compact Riemannian manifolds with lower Ricci curvature bounds

88%
Gamara N., Hasnaoui A., Makni A.
Open Mathematics
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2015
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tom 13
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nr 1
EN
In this article we prove a reverse Hölder inequality for the fundamental eigenfunction of the Dirichlet problem on domains of a compact Riemannian manifold with lower Ricci curvature bounds. We also prove an isoperimetric inequality for the torsional ridigity of such domains
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