Beyond Holonomy: Constrained Structure Equations. IHP conference "Relativity and Geometry" in memorial to André Lichnerowicz (in honor of his 100th Birthday). Institut Henri Poincaré, Paris, France. December 14, 2015

Invited Talk

The problem of determining the possible holonomy groups of $G$-structures with prescribed conditions has had a long and distinguished history, and the works of A. Lichnerowicz have played a fundamental role in this story. Many of these questions, at least at the local level, can be understood as problems of determining whether a given set of \emph{structure equations} (in the sense of \'Elie Cartan) has a solution. More explicitly, given a vector space~$V$ of dimension~$n$ and a submanifold~$A\subset V\otimes\Lambda^2(V^\ast)$, one wishes to know when there exists an $n$-manifold~$M$, a $V$-valued coframing~$\omega:TM\to V$, and a mapping~$a:M\to A$ satisfying $$ \mathrm{d}\omega = a (\omega\wedge\omega). $$ More generally, one wishes to know how to classify or describe such data~$(M,\omega,a)$ up to local equivalence. (The case when $A$ is a single point is, of course, resolved by the third fundamental theorem of Lie.) This problem and its natural generalizations (to be discussed in the talk) encompasses a vast array of problems in differential geometry. I will discuss analysis of this problem via the tools of Cartan-K\"ahler theory and exterior differential systems and will describe some of its applications to problems in holonomy as well as problems involving geometrically natural PDE that arise in various contexts, including mathematical relativity.

Service Performed By

Date

  • December 14, 2015

Service or Event Name

  • IHP conference "Relativity and Geometry" in memorial to André Lichnerowicz (in honor of his 100th Birthday)

Host Organization

  • Institut Henri Poincaré, Paris, France

Location or Venue

  • Institut Henri Poincaré, Paris, France