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arXiv · 2005.06430

An Interpolation from Sol to Hyperbolic Space

Matei P. Coiculescu
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Abstract

We study a one-parameter family of nonisomorphic solvable Lie groups, which, when equipped with canonical left-invariant metrics, $$ds^2=e^{-2z}dx^2+e^{2\alpha z}dy^2+dz^2$$ becomes an interpolation from a model of the Sol geometry to a model of Hyperbolic Space, with a stop at $\mathbb{H}^2\times \mathbb{R}$. These Lie groups are also Bianchi groups of Type VI with orthogonal coordinates. As a continuation of joint work with Richard Schwartz on Sol, we primarily analyze those Lie groups in our interpolation with some positive sectional curvature. Our main result is a characterization of the cut locus at the identity of the group that maximizes scalar curvature.

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BibTeXRIS

Matei P. Coiculescu. 2020-05-13. An Interpolation from Sol to Hyperbolic Space. https://doi.org/10.1080/10586458.2021.1980454

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