The Volume of Vector Fields on Riemannian Manifolds by Olga Gil-Medrano

Author:Olga Gil-Medrano
Language: eng
Format: epub
ISBN: 9783031368578
Publisher: Springer Nature Switzerland

The study of the volume of unit vector fields on a compact semisimple Lie group has been done by M. Salvai in [87], using the characterisation of the minimality in Proposition 3.19 and the structure and properties of semisimple Lie groups, in a development where the roots of the Lie algebra play a central role. To describe the results we need some definitions; for more details the reader is referred for instance to [59].

Let be the Lie algebra of a compact semisimple Lie group G. Let be a maximal abelian subalgebra, the corresponding root system, C a Weyl chamber and the associated basis of . This means that and that the closure of the Weyl chamber is the simplex .

Given , there is a unique such that for all , and . Each vector is a vertex of the simplex

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