On nilpotent and solvable quasi-Einstein manifolds

Abstract: In this talk, I will discuss the classification of quasi-Einstein metrics on nilpotent and unimodular solvable Lie groups. Focusing on quasi-Einstein metrics $(M,g,X)$ for which the metric $g$ and the vector field $X$ are left-invariant—what we call totally left-invariant quasi-Einstein metrics—I will first present a complete classification in the nilpotent case. In particular, I will show that a nilpotent Lie group admits such a metric if and only if it is Heisenberg.

I will then turn to unimodular solvable Lie groups and show that the existence of a non-flat totally left-invariant quasi-Einstein metric imposes strong structural restrictions, forcing the center of the group to be one-dimensional. Under the additional assumption that the adjoint action is given by a normal derivation, I will describe a full classification: the Lie group must be standard and its nilradical necessarily Heisenberg. As an application, I will explain why the only near-horizon geometries arising on nilmanifolds are quotients $\Gamma \backslash H_n$ of the Heisenberg group.

differential geometrygeometric topologymetric geometry

Audience: researchers in the topic

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Virtual seminar on geometry with symmetries

Series comments: Description: Research seminar in Lie group actions in Differential geometry.

The seminar meets every other Wednesday. To accommodate most time zones, the time rotates. The Zoom link is sent to the mailing list around 24 hours before each talk. To subscribe to the mailing list, fill the following form: docs.google.com/forms/d/e/1FAIpQLSdKrJ-nivgjr7ZVJmIY0qkN-VbzTl5NHHNyg6nNsCqjhB-4WA/viewform?usp=sf_link.

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