Locally conformally symplectic and Kähler geometry

Abstract

The goal of this note is to give an introduction to locally conformally symplectic and Kähler geometry. In particular, Sections 1 and 3 aim to provide the reader with enough mathematical background to appreciate this kind of geometry. The reference book for locally conformally Kähler geometry is [36] by Sorin Dragomir and Liviu Ornea. Many progresses in this field, however, were accomplished after the publication of this book, hence are not contained there -see the introduction of [97]. On the other hand, there is no book on locally conformally symplectic geometry and many recent advances lie scattered in the literature. Sections 2 and 4 would like to demonstrate how these geometries can be used to give precise mathematical formulations to ideas deeply rooted in classical and modern Physics.

Time-dependent Hamiltonian mechanics on a locally conformal symplectic manifold

“…One can notice though that recently, the topic of lcs manifolds has become increasingly popular. For an extended description of the revival of lcs geometry and a comprehensive discussion of its relations to other topics, we refer the reader to [10].…”

Special Types of Locally Conformal Closed G2-Structures

“…Over the last years, 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, see [3,12,28,31] and the references therein. In both cases, the condition of being locally conformal closed concerns a suitable non-degenerate 2-form ω, and is encoded in the equation dω = θ ∧ ω, where θ is a closed 1-form called the Lee form.…”

Locally conformal symplectic structures on Lie algebras of type I and their solvmanifolds

“…The fact that non-degeneracy is a local condition implies that the definition of the Hamiltonian vector field is local and therefore locally conformal symplectic manifolds provide an adequate and more general context for Hamiltonian mechanics. It can be seen that the cotangent bundle T * M admits a canonical exact locally conformal symplectic structure (see for instance [13], [22]).…”