On Diff(M)-Pseudo-Differential Operators and the Geometry of Non Linear Grassmannians
Abstract
:1. Introduction
2. Preliminaries on Algebras and Groups of Operators
2.1. Differential and Pseudodifferential Operators on a Manifold M
- the multiplication operators: for we define the multiplication operator:
- the vector fields on M: for a vector field , we define the differentiation operator:
2.2. Fourier Integral Operators
2.3. Topological Structures and Regular Lie Groups of Operators
2.4. and
- the projection induces a map with kernel
2.5. Diffeomorphisms and kernel operators
2.6. Renormalized Traces on Diff(M)-Pseudodifferential Operators
- is a weight of the same order as Q
3. Splittings on the Set of Fourier Integral Operators
3.1. The Group and the Diffeomorphism Group
3.2. Its Formal Symbol and the Splitting of
- -
- its kernel , made of constant maps
- -
- the vector space spanned by eigenvectors related to positive eigenvalues
- -
- the vector space spanned by eigenvectors related to negative eigenvalues.
- (i)
- (ii)
- where , with .
- (iii)
- , where is the sign of D, since
- (iv)
- Let (resp. ) be the projection on (resp. ), then and
3.3. The Case of Non Trivial (Real) Vector Bundle Over
3.4. The Splitting Read on the Phase Function
3.5. The Schwinger Cocycle on When E Is a Real Vector Bundle
4. Sets of Fourier Integral Operators
4.1. The set
4.2. Yet Some Subgroups of
- is a diffeomorphism of such that
- if u is a smooth section of E such that then
- Let be a regular Lie group of based operators, that contains the space of based invertible multiplication operators, with Lie algebra
- Let be a regular Lie subgroup of based diffeomorphisms, with regular Lie algebra
5. Manifolds of Embeddings
- -
- the product bundle, of basis M, with typical fiber ;
- -
- the space of on M with values in E, that is, the set of smooth maps that are fiberwise k-linear and skew-symmetric for any . If , we note the space of k-forms instead of .
5.1. as a Principal Bundle
5.2. Almost Complex Structure on Based Oriented Knots
6. Chern-Weil Forms on Principal Bundle of Embeddings and Homotopy Invariants
6.1. Chern Forms in Infinite Dimensional Setting
- (i)
- it is vanishing on vertical vectors and defines a closed form on M.
- (ii)
- the cohomology class of this form does not depend on the choice of the chosen connexion θ on P.
- -
- by the action of G in the vertical directions
- -
- setting the vectors fields constant on , where U is a local chart on M around .
- (i)
- it is vanishing on vertical vectors and defines a closed form on M.
- (ii)
- The cohomology class of this form does not depend on the choice of the chosen connexion θ on P.
6.2. Application to
Knot Invariant Through Kontsevich and Vishik Trace
7. Conclusions
Conflicts of Interest
Appendix
Renormalized Traces of PDOs
- (i)
- Given two (classical) pseudo-differential operators A and B, given a weight Q,
- (ii)
- Given a differentiable family of pseudo-differential operators, given a differentiable family of weights of constant order q,
- (i)
- is a subalgebra of with unit.
- (ii)
- Let , where is the parametrix.
- (iii)
- Let , and , then
- (iv)
- For , is an algebra on which the renormalized trace is a trace (i.e., vanishes on the brackets).
- (i)
- if .
- (ii)
- when , if A and B are classical pseudo-differential operators, if A is compact and B is of order 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. https://doi.org/10.3390/math4010001
Magnot J-P. On Diff(M)-Pseudo-Differential Operators and the Geometry of Non Linear Grassmannians. Mathematics. 2016; 4(1):1. https://doi.org/10.3390/math4010001
Chicago/Turabian StyleMagnot, Jean-Pierre. 2016. "On Diff(M)-Pseudo-Differential Operators and the Geometry of Non Linear Grassmannians" Mathematics 4, no. 1: 1. https://doi.org/10.3390/math4010001