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Conjugate points of left invariant metrics on Lie groups. (English. Russian original) Zbl 0722.53049

Sov. Math. 34, No. 11, 32-44 (1990); translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1990, No. 11(342), 27-37 (1990).
The author studies one-parameter subgroups \(\exp(ta)\) of a Lie group \(G\) that are geodesics of a given left-invariant Riemannian metric \(g\) on \(G\). The existence of such geodesic one-parameter subgroups is proved for an arbitrary pair \((G,g)\). More precisely, it is proved that a one-parameter subgroup generated by an element a of the Lie algebra \(\mathfrak g\) of \(G\) is a geodesic if one of the following conditions is fulfilled: (i) \(a\) is \(g\)-orthogonal to the commutant \([\mathfrak g,\mathfrak g]\), (ii) \(a\) is a central element of \(\mathfrak g\), (iii) \(a\) is an eigenvector of the operator \(g^{-1}\circ B\) with non-zero eigenvalues, where \(B\) is the Cartan-Killing form of \(\mathfrak g\). In case (ii), the geodesic \(\exp(ta)\) has no conjugate points iff \(\mathbb Ra\) is a direct summand of the Lie algebra \(\mathfrak g\).
This implies the following result: Let \(G\) be a Lie group with the Lie algebra \(\mathfrak g\). Assume that \(\mathfrak g\) has no commutative direct summand and the center \(Z(\mathfrak g)\neq 0\). Then any left-invariant metric \(g\) on \(G\) has a geodesic with a conjugate point. The author also proves that a \(3\)-dimensional Lie group \(G\) admits a left-invariant metric \(g\) such that all geodesic one-parameter subgroups have no conjugate points iff \(G\) admits a left-invariant metric \(g'\) of non-positive curvature and he classifies all such groups.

MSC:

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