  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.
