2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/58938We introduce a global Cauchy-Riemann($CR$)-invariant and discuss its behavior on the moduli space of $CR$-structures. We argue that this study is related to the Smale conjecture in 3-topology and the problem of counting complex structures. Furthermore, we propose a contact-analogue of Ray-Singer's analytic torsion. This ``contact torsion'' is expected to be able to distinguish among ``contact lens'' spaces. We also propose the study of a certain kind of monopole equation associated with a contact structure.22 pages,review paperDifferential GeometrySymplectic Geometry57R17;32V05,53D35text