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4:00 pm Wednesday, August 22, 2012 Geometry-Analysis Seminar: Equivalence of nonholonomic vector distributions via geometry of curves of symplectic flagsby Igor Zelenko (Texas A & M ) in HB227- ABSTRACT: My talk is devoted to local equivalence problem for vector distributions(subbundles of tangent bundles) on manifolds with respect to the action of the group of diffeomorphisms. Vector distributions appear naturally in Geometric Control Theory (as control systems linear with respect to control parameters) and Geometric Theory of Differential Equations (as natural distributions on submanifolds of jet spaces). The general way to solve such equivalence problems is to assign to a geometric structure the (co)frame (or the structure of absolute parallelism) on some (fiber) bundle over the ambient manifold in a canonical way. In my talk first I will review the classical approaches to this problem, making special emphasis to the algebraic version of Cartanâ€™s method of equivalence developed by N. Tanaka in 1970s. The central object in the Tanaka approach is the notion of a symbol of a distributions at a point, which is a graded nilpotent Lie algebra. The prolongation procedure (i.e. the procedure of getting a canonical frame) can be described in terms of natural algebraic operation in the category of graded Lie algebras. Through this review of Tanaka theory I will motivate the recent approach of B. Doubrov and myself to this problem reducing the original problem to the problems of equivalence of curves of symplectic flags. The latter problem is simpler in many respects than the original one. Our approach is a combination of a cerain symplectification of the problem (taking its origin in Pontryagin theory in Optimal Control) and various novel Tanaka type prolongations. This approach allowed us to make a unified construction of canonical frames for distribution of arbitrary rank independently of their Tanaka symbols, avoiding the problem of classification of graded nilpotent Lie algebras with given number of generators, which is important for the application of the Tanaka theory. Our approach significantly extends the set of distributions for which the canonical frame can be explicitly constructed.
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