Journal of Mathematical Physics, Analysis, Geometry 2021, vol. 17, No 4, pp. 407-421   https://doi.org/10.15407/mag17.04.407     ( to contents , go back )

Conformal Geometry of Semi-Direct Extensions of the Heisenberg Group

Giovanni Calvaruso

Dipartimento di Matematica e Fisica "E. De Giorgi", Università del Salento, Prov. Lecce-Arnesano, 73100 Lecce, Italy
E-mail: giovanni.calvaruso@unisalento.it

Amirhesam Zaeim

Department of Mathematics, Payame Noor University (PNU), P.O. Box 19395-3697, Tehran, Iran
E-mail: zaeim@pnu.ac.ir

Received September 15, 2020, revised April 24, 2021.

Abstract

We consider the general semi-direct extension $G_S=H$ ⋊ $_S \mathbb R$ of the Heisenberg Lie group $H$, as defined in [10] by any $S \in \mathfrak{sp}(1,\mathbb R)$, and equipped with a family of left-invariant metrics $g_a$ $(a^2 \neq 1)$. This construction is a natural generalization of the oscillator group. We completely determine the conformally Einstein examples.

Mathematics Subject Classification 2010: 53C20, 53C50, 53C44
Key words: Heisenberg group, semi-direct extensions, oscillator group, Bach-flat metrics, conformally Einstein metrics