P.N. Klepikov, D.N. Oskorbin, E.D. Rodionov
On Curvature Operators Spectra of Some Four-dimensional Lie Groups with Left-invariant Riemannian Metrics
Curvature operators, such as the Ricci operator, a one-dimensional curvature operator and a sectional curvature operator, are important in a study of Riemannian manifolds. Investigation of their properties is interesting for understanding the geometrical and topological structure of homogeneous Riemannian manifolds. In particular, it is interesting to find spectra of the curvature operators. The Ricci operator and its spectrum on Lie groups and homogeneous spaces was studied by J. Milnor, V.N. Berestovskii, A.G. Kremlev and Yu.G. Nikonorov, and spectra of one-dimensional and sectional curvature operators were studied by D.N. Oskorbin, E.D. Rodionov, O.P. Khromova. However, the number of problems associated with the spectra of curvatures on metric Lie groups are still not solved in dimension not less than 4. For example, there are no exact formulas for calculating the spectrum of Ricci operator on metric Lie groups in dimension 4. The problem of spectra calculation for curvature operators of left-invariant Riemannian metrics on given Lie group is local since the curvature operators are defined on Lie algebra of Lie group. It is natural to reformulate the problem in terms of metric Lie algebras. Namely, to calculate the spectra of Ricci, one-dimensional and sectional curvature operators for various scalar products on a given Lie algebra in terms of its structure constants.
Key words: Lie groups, Lie algebras, curvature operators, left-invariant Riemannian metric, J. Milnor’s generalized bases
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