Contact geometry in superconductors and New Massive Gravity

Thurston Geometries in Three-Dimensional New Massive Gravity

“…Let us mention that every Sasakian manifold is εcontact and K-contact. Thus, by our previous results [2], the η-Einstein geometries described above are vacuum solutions of NMG. Let us discuss this in further detail starting with the Nil geometry which is described by the metric…”

“…In Ref. [2] we specialized in a Lorentzian Nil geometry [3]. However, our results are independent of the metric's signature.…”

“…A class of contact structures has recently emerged [1], arising in the context of six-dimensional supergravity, three-dimensional massive gravity and two-dimensional superconductors [2]. They have been dubbed ε-contact manifold, where the sign of ε indicates if a distinguished vector field in the manifold is timelike, spacelike or null.…”

“…The canonical Lorentzian geometry on the three-sphere is not Sasakian nor para-Sasakian, however, it is εcontact. Indeed, the relations in (28) are contained within the results of [2] as a special case. There a Wick rotated Nil geometry was studied and found to be a para-Sasakian analog of the Nil geometry.…”

The duality of Fermat and Huygens principles through contact transformations

“…In recent years, contact geometry has evolved into a unifying framework for various physical theories, eg. it provides a solid foundation for thermodynamics, nonconservative classical mechanics, electromagnetic fields among other applications [18][19][20][21][22][23][24]. In particular, it has been used to show an explicit correspondence between Fermat's and Huygens' principles [25].…”