The Maslov index in PDEs geometry

TL;DR

This paper demonstrates how the Maslov index naturally appears in PDEs geometry, providing new formulas for bordism groups, a novel proof of global solutions for Navier-Stokes, and exploring related geometric structures.

Contribution

It introduces a new framework linking Maslov index with PDE solutions, offering formulas for bordism groups and a novel proof of Navier-Stokes solutions.

Findings

Maslov index characterized in PDEs geometry

New formulas for bordism groups of submanifolds

Proof of global Navier-Stokes solutions on 

Abstract

It is proved that the Maslov index naturally arises in the framework of PDEs geometry. The characterization of PDE solutions by means of Maslov index is given. With this respect, Maslov index for Lagrangian submanifolds is given on the ground of PDEs geometry. New formulas to calculate bordism groups of $(n-1)$-dimensional compact sub-manifolds bording via $n$-dimensional Lagrangian submanifolds of a fixed $2n$-dimensional symplectic manifold are obtained too. As a by-product it is given a new proof of global smooth solutions existence, defined on all $\mathbb{R}^3$, for the Navier-Stokes PDE. Further, complementary results are given in Appendices concerning Navier-Stokes PDE and Legendrian submanifolds of contact manifolds.