scholarly journals Reductions of Gauss-Codazzi Equations

2016 ◽  
Vol 137 (3) ◽  
pp. 306-327 ◽  
Author(s):  
Robert Conte ◽  
A. Michel Grundland
Keyword(s):  
Codazzi Equations

scholarly journals On the Stability in Bonnet's Theorem of the Surface Theory

2007 ◽  
Vol 14 (3) ◽  
pp. 543-564
Author(s):  
Yuri G. Reshetnyak
Keyword(s):  
Differential Operators ◽  
Sobolev Class ◽  
Integrability Condition ◽  
Complete Integrability ◽  
Integral Representations ◽  
Completely Integrable Systems ◽  
Frobenius Theorem ◽  
Fundamental Tensor ◽  
The Stability ◽  
Codazzi Equations

Abstract In the space , 𝑛-dimensional surfaces are considered having the parametrizations which are functions of the Sobolev class with 𝑝 > 𝑛. The first and the second fundamental tensor are defined. The Peterson–Codazzi equations for such functions are understood in some generalized sense. It is proved that if the first and the second fundamental tensor of one surface are close to the first and, respectively, to the second fundamental tensor of the other surface, then these surfaces will be close up to the motion of the space . A difference between the fundamental tensors and the nearness of the surfaces are measured with the help of suitable 𝑊-norms. The proofs are based on a generalization of Frobenius' theorem about completely integrable systems of the differential equations which was proved by Yu. E. Borovskiĭ. The integral representations of functions by differential operators with complete integrability condition are used, which were elaborated by the author in his other works.

Braneworld mimetic f(R) gravity

2019 ◽  
Vol 16 (03) ◽  
pp. 1950042 ◽  
Author(s):  
Kourosh Nozari ◽  
Naser Sadeghnezhad
Keyword(s):  
Scalar Field ◽  
Recent Work ◽  
Friedmann Equation ◽  
Equation Of Motion ◽  
Einstein’S Field Equations ◽  
Field Equations ◽  
Einstein's Field Equations ◽  
Bulk Scalar Field ◽  
Codazzi Equations ◽  
Mimetic Gravity

Following our recent work on braneworld mimetic gravity, in this paper, we study an extension of braneworld mimetic gravity to the case that the gravitational sector on the brane is modified in the spirit of [Formula: see text] theories. We assume the physical 5D bulk metric in the Randall–Sundrum II braneworld scenario consists of a 5D scalar field (which mimics the dark sectors on the brane) and an auxiliary 5D metric. We find the 5D Einstein’s field equations and the 5D equation of motion of the bulk scalar field in this setup. By using the Gauss–Codazzi equations, we obtain the induced Einstein’s field equations on the brane. Finally, by adopting the FRW background, we find the Friedmann equation on the brane in this [Formula: see text] mimetic braneworld setup.

scholarly journals On Subspaces of an Almost -Lagrange Space

2012 ◽  
Vol 2012 ◽  
pp. 1-14
Author(s):  
P. N. Pandey ◽  
Suresh K. Shukla
Keyword(s):  
Coupling Coefficients ◽  
Nonlinear Connection ◽  
Connection Coefficients ◽  
Codazzi Equations

We discuss the subspaces of an almost -Lagrange space (APL space in short). We obtain the induced nonlinear connection, coefficients of coupling, coefficients of induced tangent and induced normal connections, the Gauss-Weingarten formulae, and the Gauss-Codazzi equations for a subspace of an APL-space. Some consequences of the Gauss-Weingarten formulae have also been discussed.

scholarly journals On the Gauss-Codazzi equations

1990 ◽  
Vol 19 (2) ◽  
pp. 189-213 ◽  
Author(s):  
Eiji KANEDA
Keyword(s):  
Codazzi Equations

scholarly journals An Intrinsic Characterization of Bonnet Surfaces Based on a Closed Differential Ideal

Geometry ◽  
2014 ◽  
Vol 2014 ◽  
pp. 1-11 ◽  
Author(s):  
Paul Bracken
Keyword(s):  
Differential Equations ◽  
Coordinate System ◽  
Ordinary Differential Equations ◽  
Dimensional Manifold ◽  
Two Dimensional ◽  
Differential Ideal ◽  
Intrinsic Characterization ◽  
Structure Equations ◽  
Codazzi Equations

The structure equations for a two-dimensional manifold are introduced and two results based on the Codazzi equations pertinent to the study of isometric surfaces are obtained from them. Important theorems pertaining to isometric surfaces are stated and a theorem due to Bonnet is obtained. A transformation for the connection forms is developed. It is proved that the angle of deformation must be harmonic, and that the differentials of many of the important variables generate a closed differential ideal. This implies that a coordinate system exists in which many of the variables satisfy particular ordinary differential equations, and these results can be used to characterize Bonnet surfaces.

scholarly journals Quaternionic representation of the moving frame for surfaces in Euclidean three-space and Lax pair

2004 ◽  
Vol 2004 (15) ◽  
pp. 755-762 ◽  
Author(s):  
Paul Bracken
Keyword(s):  
Mean Curvature ◽  
Constant Mean Curvature ◽  
Lax Pair ◽  
Moving Frame ◽  
Lax Representation ◽  
Euclidean Three Space ◽  
Three Space ◽  
Codazzi Equations

The moving frame and associated Gauss-Codazzi equations for surfaces in three-space are introduced. A quaternionic representation is used to identify the Gauss-Weingarten equation with a particular Lax representation. Several examples are given, such as the case of constant mean curvature.

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