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Register a new userThe bigraded Rumin complex via differential forms
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Authors
Jeffrey S. Case
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We give a new CR invariant treatment of the bigraded Rumin complex and related cohomology groups via differential forms. We also prove related Hodge decomposition theorems. Among many applications, we give a sharp upper bound on the dimension of the Kohn--Rossi groups $H^{0,q}(M^{2n+1})$, $1leq qleq n-1$, of a closed strictly pseudoconvex manifold with a contact form of nonnegative pseudohermitian Ricci curvature; we prove a sharp CR analogue of the Frolicher inequalities in terms of the second page of a natural spectral sequence; and we generalize the Lee class $mathcal{L}in H^1(M;mathscr{P})$ -- whose vanishing is necessary and sufficient for the existence of a pseudo-Einstein contact form -- to all nondegenerate orientable CR manifolds.
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We consider contact manifolds equipped with Carnot-Caratheodory metrics, and show that the Rumin complex is respected by Sobolev mappings: Pansu pullback induces a chain mapping between the smooth Rumin complex and the distributional Rumin complex. As a consequence, the Rumin flat complex -- the analog of the Whitney flat complex in the setting of contact manifolds -- is bilipschitz invariant. We also show that for Sobolev mappings between general Carnot groups, Pansu pullback induces a chain mapping when restricted to a certain differential ideal of the de Rham complex. Both results are applications of the Pullback Theorem from our previous paper.
In this paper an alternative definition of the Rumin complex $(E_0^bullet,d_c)$ is presented, one that relies on a different concept of weights of forms. In this way, the Rumin complex can be constructed on any nilpotent Lie group equipped with a Carnot-Caratheodory metric. Moreover, this construction allows for the direct application of previous non-vanishing results of $ell^{q,p}$ cohomology to all nilpotent Lie groups that admit a positive grading.
We study the equivalence problem of classifying second order ordinary differential equations $y_{xx}=J(x,y,y_{x})$ modulo fibre-preserving point transformations $xlongmapsto varphi(x)$, $ylongmapsto psi(x,y)$ by using Mosers method of normal forms. We first compute a basis of the Lie algebra ${frak{g}}_{{{y_{xx}=0}}}$ of fibre-preserving symmetries of $y_{xx}=0$. In the formal theory of Mosers method, this Lie algebra is used to give an explicit description of the set of normal forms $mathcal{N}$, and we show that the set is an ideal in the space of formal power series. We then show the existence of the normal forms by studying flows of suitable vector fields with appropriate corrections by the Cauchy-Kovalevskaya theorem. As an application, we show how normal forms can be used to prove that the identical vanishing of Hsu-Kamran primary invariants directly imply that the second order differential equation is fibre-preserving point equivalent to $y_{xx}=0$.
We express the dual forms of squares of Nijenhuis tensor in terms of the second order component derivatives of the exterior derivative on differential forms and give new vanishing results for the squares of Nijenhuis tensor.
We establish a relation between smooth 2-functors defined on the path 2-groupoid of a smooth manifold and differential forms on this manifold. This relation can be understood as a part of a dictionary between fundamental notions from category theory and differential geometry. We show that smooth 2-functors appear in several fields, namely as connections on (non-abelian) gerbes, as curvatures of smooth functors and as critical points in BF theory. We demonstrate further that our dictionary provides a powerful tool to discuss the transgression of geometric objects to loop spaces.
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