Geometry of classical mechanics and field theory is mainly differential geometry of finite-dimensional smooth manifolds, fibre bundles and Lie groups.

The key point why geometry plays a prominent role in classical field theory lies in the fact that it enables one to deal with invariantly defined objects. Gauge theory has shown clearly that this is a basic physical principle. At first, a pseudo-Riemannian metric has been identified to a gravitational field in the framework of Einstein's General Relativity. In 60 70th, one has observed that connections on a principal bundle provide the mathematical model of classical gauge potentials [120; 284; 442]. Furthermore, since the characteristic classes of principal bundles are expressed in terms of the gauge strengths, one can also describe the topological phenomena in classical gauge models [142]. Spontaneous symmetry breaking and Higgs fields have been explained in terms of reduced G-structures [341]. A gravitational field seen as a pseudo-Riemannian metric exemplifies such a Higgs field [230]. In a general setting, differential geometry of smooth fibre bundles gives the adequate mathematical formulation of classical field theory, where fields are represented by sections of fibre bundles and their dynamics is phrased in terms of jet manifolds [169].

Autonomous classical mechanics speaks the geometric language of symplectic and Poisson manifolds [1; 279; 426]. Non-relativistic time-dependent mechanics can be formulated as a particular field theory on fibre bundles over ? [294].

At the same time, the standard mathematical language of quantum mechanics and perturbative field theory, except gravitation theory, has been...