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Mathematics 2016, 4(1), 1; doi:10.3390/math4010001

On Diff(M)-Pseudo-Differential Operators and the Geometry of Non Linear Grassmannians

Lyc’ee Jeanne d’Arc, Avenue de Grande Bretagne, F-63000 Clermont-Ferrand, France
Academic Editor: Lei Ni
Received: 7 August 2015 / Revised: 16 December 2015 / Accepted: 17 December 2015 / Published: 25 December 2015
View Full-Text   |   Download PDF [331 KB, uploaded 25 December 2015]

Abstract

We consider two principal bundles of embeddings with total space E m b ( M , N ) , with structure groups D i f f ( M ) and D i f f + ( M ) , where D i f f + ( M ) is the groups of orientation preserving diffeomorphisms. The aim of this paper is to describe the structure group of the tangent bundle of the two base manifolds: B ( M , N ) = E m b ( M , N ) / D i f f ( M ) and B + ( M , N ) = E m b ( M , N ) / D i f f + ( M ) from the various properties described, an adequate group seems to be a group of Fourier integral operators, which is carefully studied. It is the main goal of this paper to analyze this group, which is a central extension of a group of diffeomorphisms by a group of pseudo-differential operators which is slightly different from the one developped in the mathematical litterature e.g. by H. Omori and by T. Ratiu. We show that these groups are regular, and develop the necessary properties for applications to the geometry of B ( M , N ) . A case of particular interest is M = S 1 , where connected components of B + ( S 1 , N ) are deeply linked with homotopy classes of oriented knots. In this example, the structure group of the tangent space T B + ( S 1 , N ) is a subgroup of some group G L r e s , following the classical notations of (Pressley, A., 1988). These constructions suggest some approaches in the spirit of one of our previous works on Chern-Weil theory that could lead to knot invariants through a theory of Chern-Weil forms. View Full-Text
Keywords: Fourier-Integral operators; non-liner Grassmannian; Chern-Weil forms; infinite dimensional frame bundle Fourier-Integral operators; non-liner Grassmannian; Chern-Weil forms; infinite dimensional frame bundle
This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. (CC BY 4.0).

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Magnot, J.-P. On Diff(M)-Pseudo-Differential Operators and the Geometry of Non Linear Grassmannians. Mathematics 2016, 4, 1.

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