Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userDual Forms of the Squares of Nijenhuis Tensor
128
0
0.0
(
0
)
Authors
Jun Ling
Ask ChatGPT about the research
No Arabic abstract
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.
rate research
Read More
We give the strong form and weak form of the square of Nijenhuis tensor, study their properties, and give some applications.
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.
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 $y_{xx}=0$.
The space of vector-valued forms on any manifold is a graded Lie algebra with respect to the Frolicher-Nijenhuis bracket. In this paper we consider multiplicative vector-valued forms on Lie groupoids and show that they naturally form a graded Lie subalgebra. Along the way, we discuss various examples and different characterizations of multiplicative vector-valued forms.
We introduce a notion of compatibility between (almost) Dirac structures and (1,1)-tensor fields extending that of Poisson-Nijenhuis structures. We study several properties of the Dirac-Nijenhuis structures thus obtained, including their connection with holomorphic Dirac structures, the geometry of their leaves and quotients, as well as the presence of hierarchies. We also consider their integration to Lie groupoids, which includes the integration of holomorphic Dirac structures as a special case.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
