Equivalences and symmetries of k-nondegenerate CR manifolds Wider research context / theoretical framework
Cauchy-Riemann Geometry (shortly: CR-geometry) is an area of mathematics going back to the research of H.Poincaré and E.Cartan, and lying on the border of several fundamental mathematical disciplines, such as Complex Analysis and Differential Geometry. From the point of view of Complex Analysis, CR geometry is a tool for studying holomorphic functions in several variables; from the point of view of Differential Geometry, CR-geometry is a model geometry in the framework of Cartan’s moving frame machinery and Tanaka prolongation machinery.
Hypotheses / research questions / objectives
Starting with work of Henri Poincarè, the equivalence problem for nondegenerate CR Manifolds was solved by the celebrated papers of Cartan, Tanaka, Chern and Moser. The equivalence problem for degenerate CR Manifolds is still widely open. However, in the last decade, the methods of Cartan, Tanaka, Chern and Moser were successfully applied to solve the equivalence problem for many types 2—nondegenerate CR manifolds. The aim of this project is to solve the equivalence problem for uniformly k—nondegenerate CR manifolds. In particular, to generalize the Tanaka's prolongation procedure and construction of normal form for the case of uniformly k—nondegenerate CR manifolds.
Approach / Methods
The approach to solving research problems lies in combination of normal forms methods together with Tanaka prolongation methods. The normal form approach to uniformly 2-nondegenerate CR manifolds has straightforward generalization to uniformly k-nondegenerate CR manifolds, while the Tanaka's prolongation approach provides explicit conditions that the normal form of uniformly k-nondegenerate CR manifolds has to satisfy.
Level of originality / innovation
The suggested approach provides a new methods to determine models of uniformly k-nondegenerate CR manifolds and to solve their equivalence problem by generalizing normal form constructions and Tanaka's prolongation methods.
Primary researchers involved
The project team will include the PI, one postdoc, international scientific collaborations with M. Kolar and D. Sykes and collaborating colleagues from TU Wien.
