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Asian Journal of Mathematics
Volume 25 (2021)
A class of cubic hypersurfaces and quaternionic Kähler manifolds of co‑homogeneity one
Pages: 1 – 30
We classify all complete projective special real manifolds with reducible cubic potential, obtaining four series. For two of the series the manifolds are homogeneous, for the two others the respective automorphism group acts with co‑homogeneity one. Complete projective special real manifolds give rise to complete quaternionic Kähler manifolds via the supergravity $q$‑map, which is the composition of the supergravity $c$‑map and $r$‑map. We develop curvature formulas for manifolds in the image of the $q$‑map. Applying the $q$‑map to one of the above series of projective special real manifolds, we obtain a series of complete quaternionic Kähler manifolds, which are shown to be inhomogeneous (of co‑homogeneity one) based on our curvature formulas.
projective special real manifolds, projective special Kähler manifolds, quaternionic Kähler manifolds, co‑homogeneity one
2010 Mathematics Subject Classification
Primary 53C26. Secondary 22F50, 53A15.
Received 23 January 2020
Accepted 13 March 2020
Published 30 September 2021