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Regularity and completeness of half-Lie groups (2302.01631v3)

Published 3 Feb 2023 in math.DG

Abstract: Half Lie groups exist only in infinite dimensions: They are smooth manifolds and topological groups such that right translations are smooth, but left translations are merely required to be continuous. The main examples are groups of $Hs$ or $Ck$ diffeomorphisms and semidirect products of a Lie group with kernel an infinite dimensional representation space. Here, we investigate mainly Banach half-Lie groups, the groups of their $Ck$-elements, extensions, and right invariant strong Riemannian metrics on them: surprisingly the full Hopf--Rinow theorem holds, which is not the case in general even for Hilbert manifolds.

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