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On the set of homogeneous geodesics of a left-invariant metric. (English) Zbl 1047.53026

A geodesic \(\gamma\) in a Riemannian manifold \(M\) is said to be homogeneous or stationary if it is an orbit of a \(1\)-parameter group of isometries of \(M\). Then \(\dot{\gamma}(0)\) is called a geodesic vector. The author studies the set of homogeneous geodesics of a compact semi-simple Lie group \(G\) endowed with a left-invariant Riemannian metric. If \(G\) has rank \(\geq 2\), there are infinitely many homogeneous geodesics starting at the identity element in \(G\) [see the author, ibid. 1245, Acta Math. 38, 99–103 (2000; Zbl 1008.53040].
Let \(\mathcal{H}\) be a Cartan subalgebra of the Lie algebra \(\mathcal{G}\) of \(G\). It is shown that the set of geodesic vectors of \(\mathcal{H}\) is either trivial, or there is a subspace \(\mathcal{S}\subset\mathcal{H}\), \(\dim{\mathcal S}\geq 1\), of geodesic vectors containing all regular geodesic vectors of \(\mathcal{H}\). Let the scalar product \(A\) on the Lie algebra \({\mathcal G}\) be related to the Cartan-Killing form \(K\) by \(A(x,y)=-K(\kappa x,y)\) with a symmetric endomorphism \(\kappa\) of \(\mathcal{G}\). Suppose that \(\kappa\) has only \(1\)-dimensional eigenspaces. Then the author shows that an eigenvector corresponding to the maximal eigenvalue of \(\kappa\) is an isolated geodesic vector in its adjoint orbit.

MSC:

53C30 Differential geometry of homogeneous manifolds
53C22 Geodesics in global differential geometry

Citations:

Zbl 1008.53040
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