Hamiltonian Fields and Energy in Contact Manifolds

Abstract

To the contact distribution of a contact manifold we associate Hamiltonian type vector fields, called contact Hamiltonian fields. Their properties are investigated and the existence of such vector fields nowhere tangent to a given submanifold is proved. Time-depending contact Hamiltonian vector fields allow us to define the contact energy whose properties are studied. A class of submanifolds in relation to the study of contact Hamiltonian fields is also analyzed.

Capacities and Displacement Energy in Contact Hofer Geometry

“…Now, if N is normal in M then N × S 1 is normal in M × T * S 1 with the above contact structure and since the Euler characteristic of N ×S 1 is zero then, assuming that N is non-Legendrian, by [16,Theorem 5.6], we have E c (N × S 1 ) = 0 and then E c s (A) ≤ E c s (N ) = 0. (c) Since N is the Legendrian lift of the Lagrangian submanifold L of P , the restriction π |N : N → L is a diffeomorphism (see [14, pp.…”

“…Then there exists a globally defined associate contact form η, i.e. such that ker η = D. Also, denote by The contact Hamiltonian field of H ∈ H c B (I × M ) ( [3,16]) is the unique vector field X H (or X Ht in order to show its dependence on the parameter t), tangent to the contact distribution D and such that (M, D), is a group.…”

“…For H ∈ H c B (I × M ) we define the contact Hofer distance H (see for instance [16,17] and also [3] for a different interesting definition) by …”

“…(b) First we recall that the n-dimensional submanifold N of M is normal, [16] if N × R is normal in the manifold M × R endowed with the almost complex structure naturally induced by the almost contact structure associate to η (see for instance [4, p. 79]). Now, if N is normal in M then N × S 1 is normal in M × T * S 1 with the above contact structure and since the Euler characteristic of N ×S 1 is zero then, assuming that N is non-Legendrian, by [16,Theorem 5.6], we have E c (N × S 1 ) = 0 and then E c s (A) ≤ E c s (N ) = 0.…”

Spectral geometry of eta-Einstein Sasakian manifolds

“…Tanno [18] showed that Sasaki metric on the unit tangent sphere bundle of any sphere S n is η-Einstein and D-homothetic deformation of this metric produces a homogeneous Einstein metric on T 1 S n . We refer to [5,10,16,21] for related work and some physical applications. In this paper, we shall extend our study in the Riemannian setting to the case of the contact geometry setting.…”