Codazzi Equations Research Articles

Braneworld teleparallel gravity

We study the gravity in the context of a braneworld teleparallel scenario. The geometrical setup is assumed to be Randall–Sundrum II model where a single positive tension brane is embedded in an infinite AdS bulk. We derive the equivalent of the Gauss–Codacci equations in teleparallel gravity and junction conditions in this setup. Using these results we derive the induced teleparallel field equations on the brane. We show that compared to general relativity, the induced field equations in teleparallel gravity contain two extra terms arising from the extra degrees of freedom in the teleparallel Lagrangian. The term carrying the effects of the bulk to the brane is also calculated and its implications are discussed.

The Cauchy Problems for Einstein Metrics and Parallel Spinors

We show that in the analytic category, given a Riemannian metric $g$ on a hypersurface $M\subset \Z$ and a symmetric tensor $W$ on $M$, the metric $g$ can be locally extended to a Riemannian Einstein metric on $Z$ with second fundamental form $W$, provided that $g$ and $W$ satisfy the constraints on $M$ imposed by the contracted Codazzi equations. We use this fact to study the Cauchy problem for metrics with parallel spinors in the real analytic category and give an affirmative answer to a question raised in B\"ar, Gauduchon, Moroianu (2005). We also answer negatively the corresponding questions in the smooth category.

A conformal approach for surface inpainting

We address the problem of surface inpainting, which aims to fill in holes or missing regions on a Riemann surface based on its surface geometry. In practical situation, surfaces obtained from range scanners often have holes where the 3D models are incomplete. In order to analyze the 3D shapes effectively, restoring the incomplete shape by filling in the surface holes is necessary. In this paper, we propose a novel conformal approach to inpaint surface holes on a Riemann surface based on its surface geometry. The basic idea is to represent the Riemann surface using its conformal factor and mean curvature. According to Riemann surface theory, a Riemann surface can be uniquely determined by its conformal factor and mean curvature up to a rigid motion. Given a Riemann surface $S$, its mean curvature $H$ and conformal factor $\lambda$ can be computed easily through its conformal parameterization. Conversely, given $\lambda$ and $H$, a Riemann surface can be uniquely reconstructed by solving the Gauss-Codazzi equation on the conformal parameter domain. Hence, the conformal factor and the mean curvature are two geometric quantities fully describing the surface. With this $\lambda$-$H$ representation of the surface, the problem of surface inpainting can be reduced to the problem of image inpainting of $\lambda$ and $H$ on the conformal parameter domain. Once $\lambda$ and $H$ are inpainted, a Riemann surface can be reconstructed which effectively restores the 3D surface with missing holes. Since the inpainting model is based on the geometric quantities $\lambda$ and $H$, the restored surface follows the surface geometric pattern. We test the proposed algorithm on synthetic data as well as real surface data. Experimental results show that our proposed method is an effective surface inpainting algorithm to fill in surface holes on an incomplete 3D models based their surface geometry.

Foliations of lightlike hypersurfaces and their physical interpretation

Abstract This paper deals with a family of lightlike (null) hypersurfaces (H u) of a Lorentzian manifold M such that each null normal vector ℓ of H u is not entirely in H u, but, is defined in some open subset of M around H u. Although the family (H u) is not unique, we show, subject to some reasonable condition(s), that the involved induced objects are independent of the choice of (H u) once evaluated at u = constant. We use (n+1)-splitting Lorentzian manifold to obtain a normalization of ℓ and a well-defined projector onto H, needed for Gauss, Weingarten, Gauss-Codazzi equations and calculate induced metrics on proper totally umbilical and totally geodesic H u. Finally, we establish a link between the geometry and physics of lightlike hypersurfaces and a variety of black hole horizons.

Linear Surface Reconstruction from Discrete Fundamental Forms on Triangle Meshes

AbstractWe present a linear algorithm to reconstruct the vertex coordinates for a surface mesh given its edge lengths and dihedral angles, unique up to rotation and translation. A local integrability condition for the existence of an immersion of the mesh in 3D Euclidean space is provided, mirroring the fundamental theorem of surfaces in the continuous setting (i.e. Gauss's equation and the Mainardi–Codazzi equations) if we regard edge lengths as the discrete first fundamental form and dihedral angles as the discrete second fundamental form. The resulting sparse linear system to solve for the immersion is derived from the convex optimization of a quadratic energy based on a lift from the immersion in the 3D Euclidean space to the 6D rigid motion space. This discrete representation and linear reconstruction can benefit a wide range of geometry processing tasks such as surface deformation and shape analysis. A rotation‐invariant surface deformation through point and orientation constraints is demonstrated as well.

Soliton surfaces via a zero-curvature representation of differential equations

The main aim of this paper is to introduce a new version of the Fokas–Gel’fand formula for immersion of soliton surfaces in Lie algebras. The paper contains a detailed exposition of the technique for obtaining exact forms of 2D surfaces associated with any solution of a given nonlinear ordinary differential equation which can be written in the zero-curvature form. That is, for any generalized symmetry of the zero-curvature condition of the associated integrable model, it is possible to construct soliton surfaces whose Gauss–Mainardi–Codazzi equations are equivalent to infinitesimal deformations of the zero-curvature representation of the considered model. Conversely, it is shown (proposition 1) that for a given immersion function of a 2D soliton surface in a Lie algebra, it is possible to derive the associated generalized vector field in the evolutionary form which characterizes all symmetries of the zero-curvature condition. The theoretical considerations are illustrated via surfaces associated with the Painlevé equations P1, P2 and P3, including transcendental functions, the special cases of the rational and Airy solutions of P2 and the classical solutions of P3.

Multi-linear Formulation of Differential Geometry and Matris Regularizations

We prove that many aspects of the differential geometry of embedded Riemannian manifolds can be formulated in terms of multi-linear algebraic structures on the space of smooth functions. In particular, we find algebraic expressions for Weingartens formula, the Ricci curvature, and the Codazzi-Mainardi equations. For matrix analogues of embedded surfaces, we define discrete curvatures and Euler characteristics, and a non-commutative Gauss-Bonnet theorem is shown to follow. We derive simple expressions for the discrete Gauss curvature in terms of matrices representing the embedding coordinates, and explicit examples are provided. Furthermore, we illustrate the fact that techniques from differential geometry can carry over to matrix analogues by proving that a bound on the discrete Gauss curvature implies a bound on the eigenvalues of the discrete Laplace operator.

On Subspaces of an Almost -Lagrange Space

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.

Vacuum Spacetimes of Embedding Class Two

Doubt is cast on the much quoted results of Yakupov that the torsion vector in embedding class two vacuum space-times is necessarily a gradient vector and that class 2 vacua of Petrov type III do not exist. The first result is equivalent to the fact that the two second fundamental forms associated with the embedding necessarily commute and has been assumed in most later investigations of class 2 vacuum space-times. Yakupov stated the result without proof, but hinted that it followed purely algebraically from his identity: RijklCkl = 0 where Cij is the commutator of the two second fundamental forms of the embedding.From Yakupov's identity, it is shown that the only class two vacua with non-zero commutator Cij must necessarily be of Petrov type III or N. Several examples are presented of non-commuting second fundamental forms that satisfy Yakupovs identity and the vacuum condition following from the Gauss equation; both Petrov type N and type III examples occur. Thus it appears unlikely that his results could follow purely algebraically. The results obtained so far do not constitute definite counter-examples to Yakupov's results as the non-commuting examples could turn out to be incompatible with the Codazzi and Ricci embedding equations. This question is currently being investigated.

Twisted hierarchies associated with the generalized sine-Gordon equation

Twisted U- and twisted U/K-hierarchies are soliton hierarchies introduced by Terng to find higher flows of the generalized sine-Gordon equation. Twisted \documentclass[12pt]{minimal}\begin{document}$\frac{O(J,J)}{O(J)\times O(J)}$\end{document}O(J,J)O(J)×O(J)-hierarchies are among the most important classes of twisted hierarchies. In this paper, we derive explicit interesting first and higher flows of twisted \documentclass[12pt]{minimal}\begin{document}$\frac{O(J,J)}{O(J)\times O(J)}$\end{document}O(J,J)O(J)×O(J)-hierarchies, justify that the one-dimensional systems of twisted \documentclass[12pt]{minimal}\begin{document}$\frac{O(J,J)}{O(J)\times O(J)}$\end{document}O(J,J)O(J)×O(J)-hierarchies for J = Iq, n − q(1 ⩽ q ⩽ n − 1), called the generalized sinh-Gordon equations, are the Gauss-Codazzi equations for n-dimensional timelike submanifolds with constant sectional curvature 1 and index q in pseudo-Euclidean (2n − 1)-dimensional space \documentclass[12pt]{minimal}\begin{document}$\mathbb {R}^{2n-1}_{2q-1}$\end{document}R2q−12n−1 with index 2q − 1. Furthermore, a unified treatment of the inverse scattering theory for twisted \documentclass[12pt]{minimal}\begin{document}$\frac{O(J,J)}{O(J)\times O(J)}$\end{document}O(J,J)O(J)×O(J)-hierarchies is provided.

Pseudo-Hermitian Geometry in 3-D

Pseudo-hermitian geometry is the study of contact form in conjunction with an almost complex structure on the contact planes. In this two part survey, we first discuss the theory of surfaces in 3-D pseudo-hermitian manifolds. The local geometry of the surface is governed by the p-mean curvature equation. In analogy with the Gauss and Codazzi equation of a surface in Euclidean three space, we develop their analogue of these two equations. Due to the degeneracy of the p-mean curvature equation, the regularity theory is quite interesting. In the second part of this article, we describe a CR-invariant condition for the embeddability question. In CR geometry in 3-D, there is no local integrability condition for the almost-complex structure, so the condition to impose is global in nature. The condition involves two geometrically defined linear operators which have their counterparts in conformal geometry. We also formulate a notion of CR mass and discuss a positive mass theorem to characterize the Heisenberg space. The solvability of the \({\bar\partial}\) Neumann problem is also a crucial ingredient in the positivity of the CR mass.

Conformally flat submanifolds in spheres and integrable systems

É. Cartan proved that conformally flat hypersurfaces in $S^{n+1}$ for $n>3$ have at most two distinct principal curvatures and locally envelop a one-parameter family of $(n-1)$-spheres. We prove that the Gauss-Codazzi equation for conformally flat hypersurfaces in $S^4$ is a soliton equation, and use a dressing action from soliton theory to construct geometric Ribaucour transforms of these hypersurfaces. We describe the moduli of these hypersurfaces in $S^4$ and their loop group symmetries. We also generalise these results to conformally flat $n$-immersions in $(2n-2)$-spheres with flat and non-degenerate normal bundle.

SUBMANIFOLDS OF AN ALMOST r-PARACONTACT RIEMANNIAN MANIFOLD ENDOWED WITH A SEMI-SYMMETRIC METRIC CONNECTION

We define a semi-symmetric metric connection in an almost r-paracontact Riemannian manifold and we consider submanifolds of an almost r-paracontact Riemannian manifold endowed with a semi-symmetric metric connection and obtain Gauss and Codazzi equations, Weingarten equation and curvature tensor for submanifolds of an almost r-paracontact Riemannian manifold endowed with a semi-symmetric metric connection.

THE FUNDAMENTAL FORMULAS OF FINSLER SUBMANIFOLDS

Let ' : (M n ,F) ! (M n +p ,F) be an isometric immersion from a Finsler manifold to a Finsler manifold. In this paper, we shall obtain the Gauss and Codazzi equations with respect to the Chern con- nection on submanifolds M, by which we prove that if M is a weakly totally geodesic submanifold of M, then flag curvature of M equals flag curvature of M.

THE HIERARCHY PROBLEM, RADION MASS, LOCALIZATION OF GRAVITY AND 4D EFFECTIVE NEWTONIAN POTENTIAL IN STRING THEORY ON S1/Z2

In this paper, we present a systematical study of braneworlds of string theory on S1/Z2. In particular, starting with the toroidal compactification of the Neveu–Schwarz/Neveu–Schwarz sector in D + d dimensions, we first obtain an effective D-dimensional action, and then compactify one of the D - 1 spatial dimensions by introducing two orbifold branes as its boundaries. We divide the whole set of the gravitational and matter field equations into two groups, one holds outside the two branes, and the other holds on them. By combining the Gauss–Codacci and Lanczos equations, we write down explicitly the general gravitational field equations on each of the two branes, while using distribution theory we express the matter field equations on the branes in terms of the discontinuities of the first derivatives of the matter fields. Afterwards, we address three important issues: (i) the hierarchy problem; (ii) the radion mass; and (iii) the localization of gravity, the four-dimensional Newtonian effective potential and the Yukawa corrections due to the gravitational high-order Kaluza–Klein (KK) modes. The mechanism of solving the hierarchy problem is essentially the combination of the large extra dimension and warped factor mechanisms together with the tension coupling scenario. With very conservative arguments, we find that the radion mass is of the order of 10-2 GeV. The gravity is localized on the visible brane, and the spectrum of the gravitational KK modes is discrete and can be of the order of TeV. The corrections to the four-dimensional Newtonian potential from the higher order of gravitational KK modes are exponentially suppressed and can be safely neglected in current experiments. In an appendix, we also present a systematical and pedagogical study of the Gauss–Codacci equations and Israel's junction conditions across a (D - 1)-dimensional hypersurface, which can be either spacelike or timelike.

The Codazzi equation for surfaces

In this paper we study the classical Codazzi equation in space forms from an abstract point of view, and use it as an analytic tool to derive global results for surfaces in different ambient spaces. In particular, we study the existence of holomorphic quadratic differentials, the uniqueness of immersed spheres in geometric problems, height estimates, and the geometry and uniqueness of complete or properly embedded Weingarten surfaces.

New integrability conditions of derivational equations of a submanifold in a generalized Riemannian space

The present work is a continuation of [5] and [6]. In [5] we have obtained derivational equations of a submanifold XM of a generalized Riemannian space GRN. Since the basic tensor in GRN is asymmetric and in this way the connection is also asymmetric, in a submanifold the connection is generally asymmetric too. By reason of this, we define 4 kinds of covariant derivative and obtain 4 kinds of derivational equations. In [6] we have obtained integrability conditions and Gauss-Codazzi equations using the 1st and the 2st kind of covariant derivative. The present work deals in the cited matter, using the 3rd and the 4th kind of covariant derivative. One obtains three new integrability conditions for derivational equations of tangents and three such conditions for normals of the submanifold, as the corresponding Gauss-Codazzi equations too.

Split-Quaternionic Representation of the Moving Frame for Timelike Surfaces in 3-Dimensional Minkowski Spacetime

The moving frame and associated Gauss-Codazzi equations for surfaces in Minkowski three space are introduced. A split-quaternionic representation is used to identify the Gauss-Weingarten equations with a Lax pair representation. This Lax pair representaion is calculated and given explicitly.

On an invariant on isometric immersions into spaces of constant sectional curvature

In the present paper, we give an invariant on isometric immersions into spaces of constant sectional curvature. This invariant is a direct consequence of the Gauss equation and the Codazzi equation of isometric immersions. We apply this invariant on some examples. Further, we apply it to codimension 1 local isometric immersions of 2-step nilpotent Lie groups with arbitrary leftinvariant Riemannian metric into spaces of constant nonpositive sectional curvature. We also consider the more general class, namely, three-dimensional Lie groups G with nontrivial center and with arbitrary left-invariant metric. We show that if the metric of G is not symmetric, then there are no local isometric immersions of G into Q c 4.

Holomorphic curvature of complex Finsler submanifolds

Let M be a complex n-dimensional manifold endowed with a strongly pseudoconvex complex Finsler metric F. Let \( \mathcal{M} \) be a complex m-dimensional submanifold of M, which is endowed with the induced complex Finsler metric \( \mathcal{F} \). Let D be the complex Rund connection associated to (M, F). We prove that (a) the holomorphic curvature of the induced complex linear connection ∇ on (\( \mathcal{M},\mathcal{F} \)) and the holomorphic curvature of the intrinsic complex Rund connection ∇* on (\( \mathcal{M},\mathcal{F} \)) coincide; (b) the holomorphic curvature of ∇* does not exceed the holomorphic curvature of D; (c) (\( \mathcal{M},\mathcal{F} \)) is totally geodesic in (M, F) if and only if a suitable contraction of the second fundamental form B(·, ·) of (\( \mathcal{M},\mathcal{F} \)) vanishes, i.e., B(χ, ι) = 0. Our proofs are mainly based on the Gauss, Codazzi and Ricci equations for (\( \mathcal{M},\mathcal{F} \)).