My Scientific Biography...

"The jet formalism, when I
first met it, was quite developed in application to theory of differential
operators and differential equations, differential geometry, and also, as I
have already mentioned, in main aspects to Lagrangian formalism. It seemed that
I as a theoretician should only apply it to the particular field models: gauge
theory, gravitation theory, etc. However, I had to do develop a number of its
basic issues: geometry of composite bundles, Lagrangian theory in formalism of a
variational bicomplex, Noether identities and the second Noether theorem.

A composition of fibre bundles

*Y->S->X*is called the composite bundle. They arise in a number of models of field theory and mechanics. In mechanics, these are models with parameters described by sections of a fibre bundle*S->X*. In field theory, they are systems with a background field and models with spontaneous symmetry breaking, e.g., gravitation theory, when sections of a fibre bundle*S->X*are Higgs fields. A key point is that, if*h*is a section of a fibre bundle*S->X,*then the restriction of*Y->S*to a submanifold*h(X)*of*S*is a subbundle*h*Y->X*of a fibre bundle*Y->X*, describing a system in the presence of a background field (or a parametric function)*h(X)*.
Using a relation between jet
manifolds of fibre bundles 1991 in
the article [64] and was already used in the book [9] for description of spinors in a
gravitational field. Subsequently, I have used it in different models of field
theory and mechanics. One of them, the key to construct the gauge gravitation theory,
is classical field theory with spontaneous symmetry breaking."

*Y->X**, Y->S*and*S->X*, I obtained that between connections on these bundles and, most importantly, the new differential operator on sections of a fibre bundle*Y->S*, called the vertical covariant differential determined by a connection*A*on*Y->S*. The fact is that, being restricted to*h(X)*, this operator coincides with the familiar covariant differential yielded by the restriction of a connection A onto*h*Y->X*. Thus, this vertical covariant differential should appear in description of the dynamics of field systems on a composite bundle. This result was published in**References:**

*WikipediA*: Connection (composite bundle)*,*

**G. Sardanashvily****Advanced Differential Geometry for Theoreticians. Fiber bundles, jet manifolds and Lagrangian theory**(2013).