Modules of third-order differential operators on a conformally flat manifold

2001 ◽  
Vol 37 (3) ◽  
pp. 251-261 ◽  
Author(s):  
S.E. Loubon Djounga
Keyword(s):  
Differential Operators ◽  
Conformally Flat ◽  
Flat Manifold ◽  
Third Order ◽  
Conformally Flat Manifold

scholarly journals REMARKS ON BIANCHI SUMS AND PONTRJAGIN CLASSES

2014 ◽  
Vol 97 (3) ◽  
pp. 365-382 ◽  
Author(s):  
MOHAMMED LARBI LABBI
Keyword(s):  
Riemannian Curvature ◽  
Conformally Flat ◽  
Flat Manifold ◽  
Equality Case ◽  
Conformally Flat Manifold

AbstractWe use the exterior and composition products of double forms together with the alternating operator to reformulate Pontrjagin classes and all Pontrjagin numbers in terms of the Riemannian curvature. We show that the alternating operator is obtained by a succession of applications of the first Bianchi sum and we prove some useful identities relating the previous four operations on double forms. As an application, we prove that for a $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}k$-conformally flat manifold of dimension $n\geq 4k$, the Pontrjagin classes $P_i$ vanish for any $i\geq k$. Finally, we study the equality case in an inequality of Thorpe between the Euler–Poincaré characteristic and the $k{\rm th}$ Pontrjagin number of a $4k$-dimensional Thorpe manifold.

Hypersurfaces in a conformally flat space

2021 ◽  
pp. 2150067
Author(s):  
Sabina Eyasmin
Keyword(s):  
Space Form ◽  
Sufficient Conditions ◽  
Shape Operator ◽  
Flat Space ◽  
Conformally Flat ◽  
Flat Manifold ◽  
Conformal Curvature ◽  
Conformal Curvature Tensor ◽  
Conformally Flat Manifold ◽  
Proper Generalization

The hypersurface of a space is one of the most important objects in a space. Many authors studied the various geometric aspects of hypersurfaces in a space form. The notion of conformal flatness is one of the most primitive concepts in differential geometry. Again, conformally flat space is a proper generalization of a space form. In this paper, we study the geometry of hypersurfaces in a conformally flat manifold. Then we have investigated some sufficient conditions imposed on the shape operator for which the hypersurface satisfies various pseudosymmetric-type conditions imposed on its conformal curvature tensor.

KOBAYASHI CONFORMAL METRIC ON MANIFOLDS, CHERN-SIMONS AND η-INVARIANTS

1991 ◽  
Vol 02 (04) ◽  
pp. 361-382 ◽  
Author(s):  
BORIS N. APANASOV
Keyword(s):  
Conformal Structure ◽  
Conformally Flat ◽  
Flat Manifold ◽  
Conformal Invariant ◽  
Conformal Class ◽  
Conformal Metric ◽  
Conformally Flat Manifold ◽  
Euclidean Metrics ◽  
Chern Simons

The main aim of this paper is to present a canonical Riemannian smooth metric on a given uniformized conformal manifold (conformally flat manifold) which is compatible with the conformal structure. This metric is related to the Kobayashi construction for complex-analytic manifolds and gives a new conformal invariant. As an application, the paper studies the Chern-Simons functional and the η-invariant associated with the conformal class of conformally-Euclidean metrics on a closed hyperbolic 3-manifold.

Evolutionary heuristic with Gudermannian neural networks for the nonlinear singular models of third kind

2021 ◽  
Author(s):  
Zulqurnain Sabir ◽  
Hafiz Abdul Wahab
Keyword(s):  
Neural Networks ◽  
Differential Operators ◽  
Layer Structure ◽  
Research Work ◽  
Merit Function ◽  
Third Order ◽  
Intelligent Computing ◽  
Computing Platform ◽  
Hybrid Heuristics ◽  
Hidden Layer

Abstract The presented research work articulates a new design of heuristic computing platform with artificial intelligence algorithm by exploitation of modeling with feed-forward Gudermannian neural networks (FFGNN) trained with global search viability of genetic algorithms (GA) hybrid with speedy local convergence ability of sequential quadratic programing (SQP) approach, i.e., FFGNN-GASQP for solving the singular nonlinear third order Emden-Fowler (SNEF) models. The proposed FFGNN-GASQP intelligent computing solver Gudermannian kernel unified in the hidden layer structure of FFGNN systems of differential operators based on the SNEF that are arbitrary connected to represent the error-based merit function. The optimization objective function is performed with hybrid heuristics of GASQP. Three problems of the third order SNEF are used to evaluate the correctness, robustness and effectiveness of the designed FFGNN-GASQP scheme. Statistical assessments of the performance of FFGNN-GASQP are used to validate the consistent accuracy, convergence and stability.

scholarly journals Exact solutions in Weyl-cubed gravity

2019 ◽  
Vol 34 (05) ◽  
pp. 1950036
Author(s):  
Mohammad A. Ganjali
Keyword(s):  
Exact Solutions ◽  
Order Term ◽  
Equations Of Motion ◽  
Classical Solutions ◽  
Conformally Flat ◽  
Third Order ◽  
Gravitational Action

In this paper, we will use a unitary gravitational action up to third-order of curvature with respect to the holographic a-theorem. In particular, its third-order term has a Weyl-cubed term. In this paper, we study this Weyl-cubed theory and find some of its exact classical solutions. We show that the theory admits conformally flat, Lifshitz, Schrödinger and also hyperscaling-violating backgrounds as the solutions of the equations of motion. Our analysis has been done for the pure Weyl-cubed gravity, Einstein plus Weyl-cubed term and gravity with matter.

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