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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.)
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