Hofer type geometry on the space of Legendrian submanifolds of a contact manifold (Q2836438)

In the past two decades, the Hofer geometry on the space of Lagrangian submanifolds of a symplectic manifold has been intensively and successfully studied. The fundamental argument for this study is its importance for intersection theorems and for Floer homology. The contact version of this theory was also developed. Like in the symplectic case, the starting point of intersection theorems for contact manifolds was the contact version of Arnold's conjecture concerning the number of intersection points of a Lagrangian submanifold with its Hamiltonian deformation.NEWLINENEWLINEIn this paper, the author study some facts from a Hofer type geometry on the space of Legendrian submanifolds of a contact manifold. First, the group \(\text{Ham}^c(M)\) of Hamiltonian contactomorphisms of a contact manifold is analyzed. It is endowed with a natural pseudodistance and, in a particular case, it reduces to the contact Hofer distance defined by the author in [Int. J. Geom. Methods Mod. Phys. 5, No. 1, 63--77 (2008; Zbl 1151.53066)]. Also, some properties of contact fields on a contact manifold and their relation with contact Hamiltonian fields are reviewed. Next, the author studies the Hofer geometry of the space \({\mathcal L}(M,N_0)\) of Legendrian submanifolds of a contact manifold \(M\). Using contact Hofer energy, the author defines a pseudometric \({\mathbf d}_H\) on this space and finds sufficient conditions that this be nondegenerate. Finally, it is proved that an ample class of distances on \({\mathcal L}(M,N_0)\) reduces to multiples of \({\mathbf d}_H\). Some examples of subsets with positive contact displacement energy are given.