Some applications of Riemannian submersions in physics

Many results on the Riemannian submersions are relevant in various areas of mathematical physics as Kaluza-Klein theories, Yang-Mills equations, strings and supergravity. Interest in higher dimensional theories has been ignited once again in recent years due largely to the discovery that the underlying symmetry of the fundamental interactions is geometrical, local and gauged. A current trend in modern physics is the search for a theory which provides a unification of gravity with the other fundamental forces of nature. One of the early possibilities for such a unification was suggested by Kaluza and later expanded upon by Klein. It was shown within a five dimensional extension of Einstein's theory of general relativity how both gravity and electromagnetism could be treated on a similar footing. Both interactions were described as part of the five dimensional metric. The fifth coordinate was made invisible through a 'cylindrical condition': it was assumed that in the fifth direction, the world curled up into a cylinder of very small radius (10-33 cm, i.e. Planck's length). A natural generalization of the original Kaluza-Klein idea which incorporates non-Abelian gauge fields is to consider a higher than five dimensional theory in which the gauge fields become part of the metric in the same way as the electromagnetic field did in Kaluza's theory. We describe a compactification scheme for the Kaluza-Klein theory triggered by a scalar sector in the form of a non-linear sigma model. The final part of the paper is devoted to the Kaluza-Klein monopole in connection with the Hopf maps. In physics, the Hopf maps represent systems with nontrivial topological properties, e.g. the Z2 kink or sine-Gordon soliton, U(1) magnetic monopole or vortex in a superconducting sheet, SU(2) instanton, etc. Other physical realizations of the Hopf maps are possible. After a brief presentation of the formalism of the magnetic charges, the Dirac monopole is described in terms of Hopf maps. Finally the Kaluza-Klein monopole is constructed by embedding the Taub-NUT gravitational instanton into five-dimensional Kaluza-Klein theory. Let us note also that the same object has re-emerged in the study of monopole scattering. In the long-distance limit, neglecting radiation, the relative motion of two monopoles is described by the geodesics of the Taub-NUT space. (authors