Abstract
We show that for any non-trivial representation \((V, \pi )\) of \(\mathfrak {u}(2)\) with the center acting as multiples of the identity, the semidirect product \(\mathfrak {u}(2) \ltimes _\pi V\) admits a metric with negative Ricci curvature that can be explicitly obtained. It is proved that \(\mathfrak {u}(2) \ltimes _\pi V\) degenerates to a solvable Lie algebra that admits a metric with negative Ricci curvature. An n-dimensional Lie group with compact Levi factor \(\mathrm {SU}(2)\) admitting a left invariant metric with negative Ricci is therefore obtained for any \(n \ge 7\).
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This research was partially supported by Grants from CONICET, FONCYT and SeCyT (Universidad Nacional de Córdoba).
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Will, C.E. Negative Ricci curvature on some non-solvable Lie groups. Geom Dedicata 186, 181–195 (2017). https://doi.org/10.1007/s10711-016-0185-x
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DOI: https://doi.org/10.1007/s10711-016-0185-x