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 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 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."
References:
WikipediA: Connection (composite bundle)
G. Sardanashvily, Advanced Differential Geometry for Theoreticians. Fiber bundles, jet manifolds and Lagrangian theory (2013).
