Isometries of 3-Dimensional Semi-Riemannian Lie Groups

Abstract: Let G be a connected, simply connected three-dimensional Lie group (unimodular or non-unimodular) equipped with a left-invariant (Riemannian or Lorentzian) met- ric g. By definition, the isometry group Isom(G, g) contains G itself, acting by left translations. It turns out that, generically, Isom(G, g) is actually equal to G, and the natural question then becomes to classify those special metrics for which this is not the case. Using Lie-theoretical methods, we present a unified approach to obtain all pairs (G, g) whose full isometry group Isom(G, g) has dimension greater than or equal to four. As a consequence, we determine, for every pair (G, g), up to automor- phism and scaling, the dimension of Isom(G, g), which can be three, four, or six. (Joint work with S. Chaib and A. Zeghib).

differential geometry

Audience: researchers in the topic

Virtual seminar on geometry with symmetries

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

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