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We consider the Berry phase for quantum systems with dynamical symmetry and nondegenerate spectrum. It is shown that the corresponding phase factor is the holonomy group element of the group fibre bundle G(G/S) where G is the dynamical symmetry group and S is its maximal commutative subgroup. The connection between such elements and Cartan structure 1-forms of the bundle is established. The particular case of SU(3) symmetry group is completely investigated.

The parallel transport governing Berry’s phase [1], [2], and the Wilzcek and Zee [3] holonomy for degenerate states extends naturally to density operators [4] (and, up to the pecularities of infinite dimensional analysis, at least partly to state spaces of certain *-algebras [5].) Our approach is consistent, with the point of view of [6], to which it reduces by restricting to the pure states case.

I would like to report some results on an extension of electrodynamics, and in particular of the magnetic pole, in which the vector potential is replaced by a totally antisymmetric tensor with p indices, a p-form. The comments will be brief. Those interested may find more on the subject in Refs. 1–4.

The material discussed at the conference consisted of two applications of Kahler differential geometry nonlinear σ-models [1,2]. Since this material will be published in the standard literature, there is little point in interesting it again here. However, in the course of our work we found that Kahler geometry is a beautiful subject not well-known to most theoretical physicists and the essential part not as accessible as one might hope from the many mathematical textbooks. Kahler geometry not only has applications in supersymmetric σ-models, as was shown first by Zumino [3] and then in our work, but also in quantum gravity where every self-dual gravitational instantation is a Kahler manifold [4].

The Grinding Process of Three Plates (GTP), which is discussed in the philosophy of science chiefly by Dingier, Lorenzen and the protophysicists, is defined mathematically within the frame of Riemannian geometry. For this the notions of geodesic rigidity at a point which generalizes that of isometric rigidity, and of a dingler surface which describes the surfaces generated by GTP, are introduced. It is proved that GTP can be carried out if and only if the space has constant curvature. Furthermore, in this case the generated surfaces are just the totally geodesic hypersurfaces.

It is pointed out that the gauge principle which by now plays a decisive role in many domains of physics is based on a most natural mathematical, or, better geometrical view-point. In fact, all fundamental interactions known so far are due to this principle. It is stressed, in particular, that, according to our recent attempt, even nuclear physics may be directly incorporated into this general geometrical view-point.