Abstract
Motivated by known results in locally conformal symplectic geometry, we study different classes of G 2 -structures defined by a locally conformal closed 3-form. In particular, we provide a complete characterization of invariant exact locally conformal closed G 2 -structures on simply connected Lie groups, and we present examples of compact manifolds with different types of locally conformal closed G 2 -structures.
Highlights
The study of smooth manifolds endowed with geometric structures defined by a differential form which is locally conformal to a closed one has attracted a great deal of attention
Particular consideration has been devoted to locally conformal Kähler (LCK) structures and their non-metric analogous, locally conformal symplectic (LCS) structures
LCK structures belong to the pure class W4 of Gray–Hervella’s celebrated 16 classes of almost
Summary
The study of smooth manifolds endowed with geometric structures defined by a differential form which is locally conformal to a closed one has attracted a great deal of attention. G2 -structures fulfilling Equation (1) correspond to the class W4 in Fernández–Gray’s classification [10], and they are called locally conformal parallel (LCP), as being closed and coclosed for a G2 -form φ is equivalent to being parallel with respect to the associated Levi Civita connection (see [10]) It was proved by Ivanov, Parton, and Piccinni in [11] (Theorem A) that a compact LCP G2 -manifold is a mapping torus bundle over S1 with fiber a connected nearly Kähler manifold of dimension six and finite structure group. Unlike the LCS case, there exist exact LCC structures on unimodular Lie algebras that are not of the first kind (see Remark 6)
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