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Many interesting geometric structures can be described as regular infinitesimal flag structures, which occur as the underlying structures of parabolic geometries. Among these structures we have for instance conformal structures, contact structures, certain types of generic distributions and partially integrable almost CR-structures of hypersurface type. The aim of this article is to develop for a large class of (semi-) linear overdetermined systems of partial differential equations on regular infinitesimal flag manifolds M a conceptual method to rewrite these systems as systems of the form , where is a linear connection on some vector bundle V over M and C : V → T* M ⊗ V is a (vector) bundle map. In particular, if the overdetermined system is linear, will be a linear connection on V and hence the dimension of its solution space is bounded by the rank of V. We will see that the rank of V can be easily computed using representation theory.

In this paper, we establish nonlinear ellipticity of the equation of contact instantons with Legendrian boundary condition on punctured Riemann surfaces by proving the a priori elliptic coercive estimates for the contact instantons with Legendrian boundary condition, and prove an asymptotic exponential $C^{\infty }$-convergence result at a puncture under the uniform $C^1$ bound. We prove that the asymptotic charge of contact instantons at the punctures under the Legendrian boundary condition vanishes. This eliminates the phenomenon of the appearance of spiraling cusp instanton along a Reeb core, which removes the only remaining obstacle towards the compactification and the Fredholm theory of the moduli space of contact instantons in the open string case, which plagues the closed string case. Leaving the study of $C^1$-estimates and details of Gromov-Floer-Hofer style compactification of contact instantons to [27], we also derive an index formula which computes the virtual dimension of the moduli space. These results are the analytic basis for the sequels [27–29] and [36] containing applications to contact topology and contact Hamiltonian dynamics.

The formulation of covariant brackets on the space of solutions to a variational problem is analyzed in the framework of contact geometry. It is argued that the Poisson algebra on the space of functionals on fields should be read as a Poisson subalgebra within an algebra of functions equipped with a Jacobi bracket on a suitable contact manifold.

The analysis of the covariant brackets on the space of functions on the solutions to a variational problem in the framework of contact geometry initiated in the companion letter${}^{19}$ is extended to the case of the multisymplectic formulation of the free Klein–Gordon theory and of the free Schrödinger equation.

We show that Jacobi algebras (Poisson algebras respectively) can be defined only as Lie–Rinehart–Jacobi algebras (as Lie–Rinehart–Poisson algebras respectively). Also we show that contact manifolds, locally conformal symplectic manifolds (symplectic manifolds respectively) can be defined only as symplectic Lie–Rinehart–Jacobi algebras (only as symplectic Lie–Rinehart–Poisson algebras respectively). We define symplectic Lie algebroids.

We describe some natural relations connecting contact geometry, classical Monge–Ampère equations (MAEs) and theory of singularities of solutions to nonlinear PDEs. They reveal the hidden meaning of MAEs and sheds new light on some aspects of contact geometry.

The characterization of curves plays an important role in both geometry and topology of almost contact manifolds. Olszak found the equation $2\alpha \beta +\xi [\beta ]=0$ on normal almost contact manifolds. The pair $(\alpha ,\beta )$ denotes the type of these manifolds. In this study, we obtained the curvatures of non-geodesic Frenet curves on $3$-dimensional normal almost contact manifolds without neglecting $\alpha$ and $\beta$, and provided the results of their characterization. We exemplified these results with examples.

The generalization of the virial theorem, introduced by Clausius in statistical mechanics, has recently been carried out in the framework of geometric approaches to Hamiltonian and Lagrangian theories and it has been formulated in an intrinsic way. It is here revisited not only in its more general situation of a generic vector field but mainly from the perspective of contact and cosymplectic geometry. The previous generalizations allowing virial like relations from one-parameter groups of non-strictly canonical transformations and the rôle of Killing and conformal Killing vector fields for Lagrangians of a mechanical type are here completed with the theory for contact Hamiltonian, as well as gradient and evolution, vector fields. The corresponding theories in the framework of cosymplectic geometry and the particular case of time-dependent vector fields are also developed.

In [Y.-G. Oh and R. Wang, Analysis of contact Cauchy-Riemann maps I: A priori $C^k$ estimates and asymptotic convergence, Osaka J. Math. 55(4) (2018) 647–679; Y.-G. Oh and R. Wang, Analysis of contact Cauchy-Riemann maps II: Canonical neighborhoods and exponential convergence for the Morse-Bott case, Nagoya Math. J. 231 (2018) 128–223], the authors studied the nonlinear elliptic system

The extended Heisenberg algebra for a contact manifold has a symbolic calculus that accommodates both Heisenberg pseudodifferential operators as well as classical pseudodifferential operators. We derive here a formula for the index of Fredholm operators in this extended calculus. This formula incorporates in a single expression the Atiyah–Singer formula for elliptic operators, as well as Boutet de Monvel's Toeplitz index formula.