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


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.


A.~Agrachev, D.~Barilari, and U.~Boscain.
\newblock Introduction to {R}iemannian and sub-{R}iemannian geometry.
\newblock Book draft, 2016.
\newblock {Available at people.sissa.it/$\sim$agrachev}.
zbl 1362.53001

\newblock The tangent space in sub-{R}iemannian geometry.
\newblock In {\em Sub-{R}iemannian geometry}, volume 144 of {\em Progr. Math.},
pages 1--78. Birkh\"auser, Basel, 1996.
zbl 0862.53031;

R.~Beals and P.~Greiner.
\newblock {\em Calculus on {H}eisenberg manifolds}, volume 119 of {\em Annals
of Mathematics Studies}.
\newblock Princeton University Press, Princeton, NJ, 1988.
DOI 10.1515/9781400882397;
zbl 0654.58033;

H.~Bursztyn, H.~Lima, and E.~Meinrenken.
\newblock Splitting theorems for {P}oisson and related structures.
\newblock Preprint, 2016.
\newblock {arXiv:1605.05386}.

A.~Connes and H.~Moscovici.
\newblock The local index formula in noncommutative geometry.
\newblock {\em Geom. Funct. Anal.}, 5(2):174--243, 1995.
DOI 10.1007/BF01895667;
zbl 0960.46048;

\newblock {\em Noncommutative geometry}.
\newblock Academic Press, Inc., San Diego, CA, 1994.

W.~Choi and R.~Ponge.
\newblock Privileged coordinates and tangent groupoid for {C}arnot manifolds.
\newblock Preprint, 2015.
\newblock {arXiv:1510.05851}.

\newblock A {S}chwartz type algebra for the tangent groupoid.
\newblock In {\em {$K$}-theory and noncommutative geometry}, EMS Ser. Congr.
Rep., pages 181--199. Eur. Math. Soc., Z\"urich, 2008.
zbl 1165.58003;
arxiv 0802.3596

A.~{\v{C}}ap and J.~Slov{\'a}k.
\newblock {\em Parabolic geometries. {I}}, volume 154 of {\em Mathematical
Surveys and Monographs}.
\newblock American Mathematical Society, Providence, RI, 2009.
zbl 1183.53002;

E.~van Erp.
\newblock {\em The {A}tiyah-{S}inger index formula for subelliptic operators on
contact manifolds}.
\newblock ProQuest LLC, Ann Arbor, MI, 2005.
\newblock Thesis (Ph.D.)--The Pennsylvania State University.

E.~van Erp.
\newblock The {A}tiyah-{S}inger index formula for subelliptic operators on
contact manifolds. {P}art {I}.
\newblock {\em Ann. of Math. (2)}, 171(3):1647--1681, 2010.
DOI 10.4007/annals.2010.171.1647;
zbl 1206.19004;
arxiv 0804.2490v1

E.~van Erp.
\newblock The {A}tiyah-{S}inger index formula for subelliptic operators on
contact manifolds. {P}art {II}.
\newblock {\em Ann. of Math. (2)}, 171(3):1683--1706, 2010.
DOI 10.4007/annals.2010.171.1683;
zbl 1206.19005;
arxiv 0804.2492v2

E.~van Erp and R.~Yuncken.
\newblock A groupoid approach to pseudodifferential operators.
\newblock Preprint, 2015.
\newblock {arXiv:1511.01041}.

E.~van Erp and R.~Yuncken.
\newblock On the tangent groupoid of a filtered manifold.
\newblock Preprint, 2016.
\newblock {arXiv:1611.01081}.
DOI 10.1112/blms.12096;
zbl 06826678

\newblock {\em Intersection theory}, volume~2 of {\em Ergebnisse der Mathematik
und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics
[Results in Mathematics and Related Areas. 3rd Series. A Series of Modern
Surveys in Mathematics]}.
\newblock Springer-Verlag, Berlin, second edition, 1998.
zbl 0885.14002;

\newblock The tangent groupoid and the index theorem.
\newblock In {\em Quanta of maths}, volume~11 of {\em Clay Math. Proc.}, pages
241--256. Amer. Math. Soc., Providence, RI, 2010.
zbl 1230.58016;

G.~P. Hochschild.
\newblock {\em Basic theory of algebraic groups and {L}ie algebras}, volume~75
of {\em Graduate Texts in Mathematics}.
\newblock Springer-Verlag, New York-Berlin, 1981.
zbl 0589.20025;

\newblock {L}ie filtrations and pseudo-differential operators.
\newblock Unpublished manuscript, 1982.

I.~Moerdijk and J.~Mr\v{c}un.
\newblock {\em Introduction to foliations and {L}ie groupoids}, volume~91 of
{\em Cambridge Studies in Advanced Mathematics}.
\newblock Cambridge University Press, Cambridge, 2003.
zbl 1029.58012;

\newblock Geometric structures on filtered manifolds.
\newblock {\em Hokkaido Math. J.}, 22(3):263--347, 1993.
DOI 10.14492/hokmj/1381413178;
zbl 0801.53019;

\newblock {\em Calcul hypoelliptique sur les vari\'et\'es de Heisenberg,
r\'esidu non commutatif et g\'eom\'etrie pseudo-hermitienne}.
\newblock 2000.
\newblock Thesis (Ph.D.)--University of Paris-Sud (Orsay).

\newblock The tangent groupoid of a {H}eisenberg manifold.
\newblock {\em Pacific J. Math.}, 227(1):151--175, 2006.
DOI 10.2140/pjm.2006.227.151;
zbl 1133.58010;
arxiv math/0404174

\newblock On differential systems, graded {L}ie algebras and pseudogroups.
\newblock {\em J. Math. Kyoto Univ.}, 10:1--82, 1970.
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