Integral Geometry and Cohomology in Field Theory on the Space-Time as Complex Riemannian Manifold

Francisco Bulnes

doi:10.5772/intechopen.92969

Abstract

The study of the relationships between the integration invariants and the different classes of operators, as well as of functions inside the context of the integral geometry, establishes diverse homologies in the dual space of the functions. This is given in the class of cohomology of the integral operators that give solution to certain class of differential equations in field theory inside a holomorphic context. By this way, using a cohomological theory of appropriate operators that establish equivalences among cycles and cocycles of closed submanifolds, line bundles and contours can be obtained by a cohomology of general integrals, useful in the evaluation and measurement of fields, particles, and physical interactions of diverse nature that occurs in the space-time geometry and phenomena. Some of the results applied through this study are the obtaining of solutions through orbital integrals for the tensor of curvature R μν , of Einstein’s equations, and using the imbedding of cycles in a complex Riemannian manifold through the duality: line bundles with cohomological contours and closed submanifolds with cohomological functional. Concrete results also are obtained in the determination of Cauchy type integral for the reinterpretation of vector fields.

Highlights

Obtaining an integral cohomology of general integral operators that determine complex analytic solutions through classes of cohomology born of the ∂‐cohomology is necessary to use a holomorphic language with the purpose of obtaining the holomorphic forms that involve exact forms
The following question arises, how to establish isomorphisms of cohomological classes for functions, functional, and vector fields inside the holomorphic context possible? How to determine a cohomological theory of integral operators that establish equivalences among these objects and the geometric objects of closed submanifolds, bundles of lines, and Feynman diagrams? How everything can decrease to a single cohomology of general integrals on contours or a cohomology of generalized functionals?
The idea to obtain an integral operator cohomology is develop a theory through integral invariants, that is to say, explore the complex Riemannian manifolds though the value of its integrals along the cycles and the corresponding cocycles of the manifold

Summary

Introduction

Obtaining an integral cohomology of general integral operators that determine complex analytic solutions through classes of cohomology born of the ∂‐cohomology is necessary to use a holomorphic language with the purpose of obtaining the holomorphic forms that involve exact forms. The holomorphic structure that constitutes these complexes induces (in the corresponding integral manifold) a conformal generalized structure of integral submanifolds where the arches γ, are local parts of integral curves of the fibers of the vector sheaf of lines.

Results

Conclusion