Negative Ricci curvature on some non-solvable Lie groups
A construction of new invariant Riemannian metrics on Lie groups (solvable and non-solvable), which have negative Ricci curvature, are given. Non-solvable Lie groups here are the semidirect products \(L= \mathrm{U}(2) \times _{\phi} V\), where \(\phi\) are non-trivial representations and the center of \(\mathrm{U}(2)\) acts on \(V\) by scalar multiplication (as it is noted at the end of this paper, some generalization of this action is possible). The main steps of the construction are connected with two kinds of irreducible linear representations of the compact Lie group \(\mathrm{SU}(2)\). The aforementioned metrics on \(L\) are constructed with aid of metrics with negative Ricci curvature on some solvable Lie groups, which are obtained from \(L\) by the construction of a degeneration.
invariant Riemannian metric, solvable Lie group
scientific article; zbMATH DE number 6686315
scientific article