The Gauss Equation in Capillarity

1977 ◽  
Vol 105 (5_6) ◽  
pp. 225-235 ◽  
Author(s):  
Sun-Tak Hwang
Keyword(s):  
Gauss Equation

scholarly journals On the solution of a class of second-order quasi-linear PDEs and the Gauss equation

2001 ◽  
Vol 42 (3) ◽  
pp. 312-323
Author(s):  
A. R. Selvaratnam ◽  
M. Vlieg-Hulstman ◽  
B. van-Brunt ◽  
W. D. Halford
Keyword(s):  
Gaussian Curvature ◽  
Gordon Equation ◽  
Second Order ◽  
Bäcklund Transformations ◽  
Coordinate Systems ◽  
Gauss Equation ◽  
Sine Gordon Equation ◽  
Partial Differential ◽  
Metric Tensor ◽  
Backlund Transformations

AbstractGauss' Theorema Egregium produces a partial differential equation which relates the Gaussian curvature K to components of the metric tensor and its derivatives. Well-known partial differential equations (PDEs) such as the Schrödinger equation and the sine-Gordon equation can be derived from Gauss' equation for specific choices of K and coördinate systems. In this paper we consider a class of Bäcklund Transformations which corresponds to coördinate transformations on surfaces with a given Gaussian curvature. These Bäcklund Transformations lead to the construction of solutions to certain classes of non-linear second order PDEs of hyperbolic type by identifying these PDEs as the Gauss equation in some coördinate system. The possibility of solving the Cauchy Problem has also been explored for these classes of equations.

scholarly journals Binary Image Segmentation through the Carl Friedrich Gauss Equation

2018 ◽  
Vol 179 (41) ◽  
pp. 1-9
Author(s):  
Oppong-Twum Francis ◽  
Frimpong Twum ◽  
J. B.
Keyword(s):  
Image Segmentation ◽  
Binary Image ◽  
Gauss Equation

Real hypersurfaces in complex two-plane Grassmannians with parallel Ricci tensor

2012 ◽  
Vol 142 (6) ◽  
pp. 1309-1324 ◽  
Author(s):  
Young Jin Suh
Keyword(s):  
Curvature Tensor ◽  
Ricci Tensor ◽  
Real Hypersurface ◽  
Real Hypersurfaces ◽  
Gauss Equation ◽  
Full Expression ◽  
New Formula ◽  
Parallel Ricci Tensor

We introduce the full expression of the curvature tensor of a real hypersurface M in complex two-plane Grassmannians G2(ℂm+2) from the Gauss equation. We then derive a new formula for the Ricci tensor of M in G2(ℂm+2). Finally, we prove that there does not exist any Hopf real hypersurface in complex two-plane Grassmannians G2(ℂm+2) with parallel Ricci tensor.

scholarly journals Gauss Equation And Injectivity Radii For Subspaces in Spaces of Curvature Bounded Above

2006 ◽  
Vol 117 (1) ◽  
pp. 65-84 ◽  
Author(s):  
Stephanie B. Alexander ◽  
Richard L. Bishop
Keyword(s):  
Gauss Equation ◽  
Injectivity Radii

scholarly journals On steady motions of an ideal fibre-reinforced fluid in a curved stratum. Geometry and integrability

2021 ◽  
Author(s):  
Dmitry K Demskoi ◽  
Wolfgang Karl Schief
Keyword(s):  
Differential Equation ◽  
Gaussian Curvature ◽  
Coordinate Systems ◽  
Gauss Equation ◽  
Third Order ◽  
Constant Gaussian Curvature ◽  
Pseudospherical Surfaces ◽  
Coordinate Lines ◽  
Steady Motions ◽  
Orthogonal Coordinate Systems

Abstract It is shown that the kinematic equations governing steady motions of an ideal fibre-reinforced fluid in a curved stratum may be expressed entirely in terms of the intrinsic Gauss equation, which assumes the form of a partial differential equation of third order, for the surface representing the stratum. In particular, the approach adopted here leads to natural non-classical orthogonal coordinate systems on surfaces of constant Gaussian curvature with one family of coordinate lines representing the fibres. Integrable cases are isolated by requiring that the Gauss equation be compatible with another third-order hyperbolic differential equation. In particular, a variant of the integrable Tzitz\'eica equation is derived which encodes orthogonal coordinate systems on pseudospherical surfaces. This third-order equation is related to the Tzitz\'eica equation by an analogue of the Miura transformation for the (modified) Korteweg-de Vries equation. Finally, the formalism developed in this paper is illustrated by focussing on the simplest ``fluid sheets'' of constant Gaussian curvature, namely the plane, sphere and pseudosphere.

scholarly journals Dynamical heredity from f(R)-bulk to braneworld: Curvature dynamical constraint and f(R)-unimodular gravity

2018 ◽  
Vol 33 (26) ◽  
pp. 1850149
Author(s):  
André Martorano Kuerten
Keyword(s):  
Stress Tensor ◽  
Additional Stress ◽  
Gauss Equation ◽  
Dark Radiation ◽  
Radiation Correction

Recently, Borzou et al. (BSSY) generalized the Shiromizu–Maeda–Sasaki (SMS) formulation to [Formula: see text]-bulks. BSSY brane projected equation carries an additional stress tensor, besides SMS correction for Einstein’s theory on the brane. If we change this perspective, by requiring BSSY tensor in the geometrical side, acting as [Formula: see text]-brane generator, it is possible to relate [Formula: see text]-brane/bulk theories, by using curvature dynamical constraint (CDC), a concept that we developed. Since brane and bulk are [Formula: see text], 5D/4D scalar curvatures also play a dynamical role and, thus, a dynamical version to Gauss equation trace, or CDC, must be offered. We will work yet in a specific case to obtain [Formula: see text]-unimodular gravity, formally identical with that obtained by Nojiri et al. (NOO). Therefore, two applications which consider cosmological scenarios in [Formula: see text]-unimodular gravity with dark radiation correction will be offered.

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