10. Euler-Like Vector Fields, Deformation Spaces and Manifolds with Filtered Structure

  • Ahmad Reza Haj Saeedi Sadegh Dept. of Mathematics, Penn State University, University Park, PA 16802, USA
  • Nigel Higson Dept. of Mathematics, Penn State University, University Park, PA 16802, USA
Keywords: Deformation to the normal cone, Euler-like vector field, Tangent groupoid

Abstract

Let $M$ be a smooth submanifold of a smooth manifold $V$.  Bursztyn, Lima and Meinrenken   defined a concept of Euler-like vector field on $V$ associated to the embedding of $M$ into $V$, and proved that  there is a bijection between germs of tubular neighborhoods of $M$  and germs of Euler-like   vector fields. We shall present a new view of this result by characterizing Euler-like vector fields algebraically and examining their relation to the  deformation to the normal cone from algebraic geometry.  Then we shall extend our algebraic point  of view to   smooth manifolds that are equipped with Lie filtrations, and define deformations to the normal cone and  Euler-like vector fields in that context.     Our algebraic construction of the deformation to the normal cone  gives a new approach to  Connes' tangent groupoid and its generalizations to  filtered manifolds.  In addition, Euler-like vector fields   give rise to preferred coordinate systems on filtered manifolds.

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Published
2018-05-14
Section
Unassigned Articles