THE CARTAN-WEYL CONFORMAL GEOMETRY OF A PAIR OF SECOND-ORDER PARTIAL-DIFFERENTIAL EQUATIONS by
نویسنده
چکیده
We explore the conformal geometric structures of a pair of second-order partial-differential equations. In particular, we investigate the conditions under which this geometry is conformal to the vacuum Einstein equations of general relativity. Furthermore, we introduce a new version of the conformal Einstein equations, which are used in the analysis of the conformal geometry of the PDE’s.
منابع مشابه
Recurrent metrics in the geometry of second order differential equations
Given a pair (semispray $S$, metric $g$) on a tangent bundle, the family of nonlinear connections $N$ such that $g$ is recurrent with respect to $(S, N)$ with a fixed recurrent factor is determined by using the Obata tensors. In particular, we obtain a characterization for a pair $(N, g)$ to be recurrent as well as for the triple $(S, stackrel{c}{N}, g)$ where $stackrel{c}{N}$ is the canonical ...
متن کاملSolving high-order partial differential equations in unbounded domains by means of double exponential second kind Chebyshev approximation
In this paper, a collocation method for solving high-order linear partial differential equations (PDEs) with variable coefficients under more general form of conditions is presented. This method is based on the approximation of the truncated double exponential second kind Chebyshev (ESC) series. The definition of the partial derivative is presented and derived as new operational matrices of der...
متن کاملConformal mappings preserving the Einstein tensor of Weyl manifolds
In this paper, we obtain a necessary and sufficient condition for a conformal mapping between two Weyl manifolds to preserve Einstein tensor. Then we prove that some basic curvature tensors of $W_n$ are preserved by such a conformal mapping if and only if the covector field of the mapping is locally a gradient. Also, we obtained the relation between the scalar curvatures of the Weyl manifolds r...
متن کاملConformal Geometry and 3-plane Fields on 6-manifolds
The purpose of this note is to provide yet another example of the link between certain conformal geometries and ordinary differential equations, along the lines of the examples discussed by Nurowski [3]. In this particular case, I consider the equivalence problem for 3-plane fields D ⊂ TM on a 6-manifold M satisfying the nondegeneracy condition that D + [D, D] = TM . I give a solution of the eq...
متن کاملConformal Invariants and Partial Differential Equations
Our goal is to study quantities in Riemannian geometry which remain invariant under the “conformal change of metrics”–that is, under changes of metrics which stretch the length of vectors but preserve the angles between any pair of vectors. We call such a quantity “conformally invariant”. In conjunction with the study of conformal invariants, we are also interested in studying “conformally cova...
متن کامل