Petko Nikolov | Searcharxiv

arXiv2008

General Notion of Curvature in Catastrophe Theory Terms

We introduce a new notion of a curvature of a superconnection, different from the one obtained by a purely algebraic analogy with the curvature of a linear connection. The naturalness of this new notion of a curvature of a superconnection comes from the study of singularities of smooth sections of vector bundles (Catastrophe Theory). We demonstrate that the classical examples of obstructions to a local equivalence: exterior differential for 2-forms, Riemannian tensor, Weil tensor, curvature of a linear connection and Nijenhuis tensor can be treated in terms of one general approach. This approach, applied to the superconnection leads to a new notion of a curvature (proposed in this paper) of a superconnection.

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arXiv2006

Description of all conformally invariant differential operators, acting on scalar functions

We give an algorithm to write down all conformally invariant differential operators acting between scalar functions on Minkowski space. All operators of order k are nonlinear and are functions on a finite family of functionally independent invariant operators of order up to k. The independent differential operators of second order are three and we give an explicit realization of them. The applied technique is based on the jet bundle formalism, algebraization of the the differential operators, group action and dimensional reduction. As an illustration of this method we consider the simpler case of differential operators between analytic functions invariant under the modular group. We give a power series generating explicitly all the functionally independent invariant operators of an arbitrary order.

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