Geometry of rank 2 distributions with nonzero Wilczynski invariants*

Variational Approach to Differential Invariants of Rank 2 Vector Distributions

In the present paper we construct differential invariants for generic rank 2 vector distributions on n-dimensional manifold. In the case n = 5 (the first case containing functional parameters) E. Cartan found in 1910 the covariant fourth-order tensor invariant for such distributions, using his ”reduction-prolongation” procedure (see [12]). After Cartan’s work the following questions remained op...

Geometry of Rank 2

A notion of a Frobenius manifold with a nice real structure was introduced by Hertling. It is called CDV structure (Cecotti-Dubrovin-Vafa structure). In this paper, we introduce a “positivity condition” on CDV structures and show that any Frobenius manifold of rank two with real spectrum can be equipped with a positive CDV structure. We extend naturally the symmetries of Frobenius structures gi...

L 2 -invariants and Rank Metric

We introduce a notion of rank completion for bi-modules over a finite tracial von Neumann algebra. We show that the functor of rank completion is exact and that the category of complete modules is abelian with enough projective objects. This leads to interesting computations in the L 2-homology for tracial algebras. As an application, we also give a new proof of a Theorem of Gaboriau on invaria...

Sphere geometry and invariants

A finite abstract simplicial complex G defines two finite simple graphs: the Barycentric refinement G1, connecting two simplices if one is a subset of the other and the connection graph G′, connecting two simplices if they intersect. We prove that the Poincaré-Hopf value i(x) = 1−χ(S(x)), where χ(S(x)) is the Euler characteristics of the unit sphere S(x) of a vertex x in G1, agrees with the Gre...

Rigidity of Integral Curves of Rank 2 Distributions