Operations on Legendrian submanifolds

Abstract

We focus on Legendrian submanifolds of the space of one-jets of functions, $J^1({\mathbb{R}}^n,\mathbb{R})$. We are interested in processes - operations - that build new Legendrian submanifolds from old ones. We introduce in particular two operations, namely the sum and the convolution, which in some sense lift to $J^1({\mathbb{R}}^n,\mathbb{R})$ the operations sum and infimal-convolution on functions that belong to convex analysis. We show that these operations fit well with the classical theory of generating functions. Finally, we refine this theory so that the min-max selector of generating functions plays its natural role.

31 pages, 10 figures. We complete the previous version with the second Fourier-type which was missing, together with minor modifications

References in corpus (2)

• Legendrian Products
• Subtleties of the minmax selector

Cited by in corpus (2)

• Nonequilibrium thermodynamic process with hysteresis and metastable states -- A contact Hamiltonian with unstable and stable segments of a Legendre submanifold
• On Legendrian cobordisms and generating functions