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On the volume of unit vector fields on a compact semisimple Lie group. (English) Zbl 1028.22015
Let $$G$$ be a compact connected semisimple Lie group endowed with a bi-invariant Riemannian metric. A unit vector $$v$$ in its Lie algebra is said to be maximal singular if the adjoint orbits satisfy $$\dim \text{Ad}(G)v< \dim \text{Ad}(G)w$$ for all unit vectors $$w$$ near $$v$$. Let $$V$$ be a left or right invariant vector field on $$G$$. We say that $$V$$ is maximal singular if its value at the identity is maximal singular. The volume of $$V$$ is defined by the volume of the submanifold it determines on the tangent bundle of $$G$$. The paper proves that if $$V$$ is a maximal singular vector field, then its volume is minimal among the volumes of the unit vector fields. Let $${\mathfrak t}$$ be the Lie algebra of a maximal torus of $$G$$. Given a choice of simple roots $$\{\alpha\}$$, let $$\{v_\alpha\}\subset{\mathfrak t}$$ be the unit vectors on the edges of the closed Weyl chamber, namely $$\alpha(v_\alpha)> 0$$ and $$\beta(v_\alpha)= 0$$ for $$\beta\neq \alpha$$. The paper proves that in the shortest arc joining distinct $$v_\alpha$$ and $$v_\beta$$, there lies some $$v$$ such that the corresponding left invariant vector field is minimal. The paper also gives a lower bound estimate for the number of nonequivalent minimal unit vector fields on $$G$$, and works out some examples such as when $$G$$ is the orthogonal group.
##### MSC:
 22E46 Semisimple Lie groups and their representations 53C30 Differential geometry of homogeneous manifolds 53C35 Differential geometry of symmetric spaces 53C42 Differential geometry of immersions (minimal, prescribed curvature, tight, etc.)
##### Keywords:
semisimple Lie group; minimal unit vector field
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