Codazzi Equations Research Articles

Large-Strain Surface Modeling Using Plasticity.

Modeling arbitrarily large deformations of surfaces smoothly embedded in three-dimensional space is challenging. We give a new method to represent surfaces undergoing large spatially varying rotations and strains, based on differential geometry, and surface first and second fundamental forms. Methods that penalize the difference between the current shape and the rest shape produce sharp spikes under large strains, and variational methods produce wiggles, whereas our method naturally supports large strains and rotations without any special treatment. For stable and smooth results, we demonstrate that the deformed surface has to locally satisfy compatibility conditions (Gauss-Codazzi equations) on the first and second fundamental forms. We then give a method to locally modify the surface first and second fundamental forms in a compatible way. We use those fundamental forms to define surface plastic deformations, and finally recover output surface vertex positions by minimizing the surface elastic energy under the plastic deformations. We demonstrate that our method makes it possible to smoothly deform triangle meshes to large spatially varying strains and rotations, while meeting user constraints.

Fractional Sobolev isometric immersions of planar domains

We discuss $C^1$ regularity and developability of isometric immersions of flat domains into $\mathbb R^3$ enjoying a local fractional Sobolev $W^{1+s, \frac2s}$ regularity for $2/3 \le s< 1 $, generalizing the known results on Sobolev and Holder regimes. Ingredients of the proof include analysis of the weak Codazzi-Mainardi equations of the isometric immersions and study of $W^{2,\frac2s}$ planar deformations with symmetric Jacobian derivative and vanishing distributional Jacobian determinant. On the way, we also show that the distributional Jacobian determinant, conceived as an operator defined on the Jacobian matrix, behaves like determinant of gradient matrices under products by scalar functions.

The geometry of C1,α flat isometric immersions

We show that any isometric immersion of a flat plane domain into ${\mathbb {R}}^3$ is developable provided it enjoys the little Hölder regularity $c^{1,2/3}$ . In particular, isometric immersions of local $C^{1,\alpha }$ regularity with $\alpha &gt;2/3$ belong to this class. The proof is based on the existence of a weak notion of second fundamental form for such immersions, the analysis of the Gauss–Codazzi–Mainardi equations in this weak setting, and a parallel result on the very weak solutions to the degenerate Monge–Ampère equation analysed in [M. Lewicka and M. R. Pakzad. Anal. PDE 10 (2017), 695–727.].

On Tangent Bundles of Submanifolds of a Riemannian Manifold Endowed with a Quarter-Symmetric Non-metric Connection

The object of this article is to study a quarter-symmetric non-metric connection in the tangent bundle and induced metric and connection on submanifold of co-dimension 2 and hypersurface concerning the quarter-symmetric non-metric connection in the tangent bundle. The Weingarten equations concerning the quarter-symmetric non-metric connection on a submanifold of co-dimension 2 and the hypersurface in the tangent bundle are obtained. Finally, authors deduce the Riemannian curvature tensor and Gauss and Codazzi equations on a submanifold of co-dimension 2 and hypersurface of the Riemannian manifold concerning the quarter-symmetric non-metric connection in the tangent bundle.

Semi-slant submanifolds in a locally conformal Kaehler space form

In this paper, we consider semi-slant submanifolds in a locally conformal Kaehler manifold and a locally conformal Kaehler space form. We give inequalities for the length of the second fundamental form and the mean curvature and, using Gauss and Codazzi equations, we get many useful results in locally conformal Kaehler space form.

Asymptotics for minimizers of a Donaldson functional and mean curvature 1-immersions of surfaces into hyperbolic 3-manifolds

It has been shown in [17] that, for given |c|<1, the moduli space of constant mean curvature (CMC) c-immersions of a closed orientable surface of genus g≥2 into a hyperbolic 3-manifold can be parametrized by elements of the tangent bundle of the corresponding Teichmüller space. This is attained by showing the unique solvability of the Gauss-Codazzi equations governing (CMC) c-immersions. The corresponding unique solution is identified as the global minimum (and only critical point) of the Donaldson functional Dt (introduced in [11]) given in (1.11) with t=1−c2.When |c|≥1 (i.e. t≤0), so far nothing is known about the existence of analogous (CMC) c-immersions. Indeed, for t≤0 the functional Dt may no longer be bounded from below and evident non-existence situations do occur.Already the case |c|=1 (i.e. t=0) appears rather involved and actually (CMC) 1-immersions can be attained only as “limits” of (CMC) c-immersions for |c|⟶1− (see Theorem 2). To handle this situation, here we analyze the asymptotic behavior of minimizers of Dt as t⟶0+.We use an accurate asymptotic analysis (see [35]) to describe possible blow-up phenomena. In this way, we can relate the existence of (CMC) 1-immersions to the Kodaira map defined in (1.9). As a consequence, we obtain the first existence and uniqueness result about (CMC) 1-immersions of surfaces of genus g=2 into hyperbolic 3-manifolds.

A general inequality for warped product $CR$-submanifolds of Kähler manifolds

In this paper, warped product CRCR-submanifolds in Kahler manifolds and warped product contact CRCR-submanifolds in Sasakian, Kenmotsu and cosymplectic manifolds, are shown to possess a geometric property; namely DTDT-minimal. Taking benefit from this property, an optimal general inequality is established by means of the Gauss equation, we leave cosyplectic because it is an easy structure. Moreover, a rich geometry appears when the necessity and sufficiency are proved and discussed in the equality case. Applying this general inequality, the inequalities obtained by Munteanu are derived as particular cases. Up to now, the method used by Chen and Munteanu can not extended for general ambient manifolds, this is because many limitations in using Codazzi equation. Hence, Our method depends on the Gauss equation. The inequality is constructed to involve an intrinsic invariant (scalar curvature) controlled by an extrinsic one (the second fundamental form), which provides an answer for the well-know Chen's research problem (Problem 1.1???). As further research directions, we have addressed a couple of open problems arose naturally during this work and depending on its results.

From the Nash–Kuiper theorem of isometric embeddings to the Euler equations for steady fluid motions: Analogues, examples, and extensions

Direct linkages between regular or irregular isometric embeddings of surfaces and steady compressible or incompressible fluid dynamics are investigated in this paper. For a surface (M, g) isometrically embedded in R3, we construct a mapping that sends the second fundamental form of the embedding to the density, velocity, and pressure of steady fluid flows on (M, g). From a Partial Differential Equations perspective, this mapping sends solutions to the Gauss–Codazzi equations to the steady Euler equations. Several families of special solutions of physical or geometrical significance are studied in detail, including the Chaplygin gas on standard and flat tori as well as the irregular isometric embeddings of the flat torus. We also discuss tentative extensions to multiple dimensions.

Spectral Parameter as a Group Parameter

A large class of integrable non-linear partial differential equations is characterized by the existence of the associated linear problem (in the case of two independent variables, known as a Lax pair) containing the so-called spectral parameter. In this paper, we present and discuss the conjecture that the spectral parameter can be interpreted as the parameter of some one-parameter groups of transformation, provided that it cannot be removed by any gauge transformation. If a non-parametric linear problem for a non-linear system is known (e.g., the Gauss–Weingarten equations as a linear problem for the Gauss–Codazzi equations in the geometry of submanifolds), then, by comparing both symmetry groups, we can find or indicate the integrable cases. We consider both conventional Lie point symmetries and the so-called extended Lie point symmetries, which are necessary in some cases. This paper is intended to be a review, but some novel results are presented as well.

The origins of the fundamental theorem of surface theory

The Mainardi-Codazzi equations (MCE) and the fundamental theorem of surface theory (FT) are regarded as crucial achievements in the development of surface theory. The paper offers an analysis of three papers by Bour, Codazzi and Bonnet, submitted on the occasion of the Grand Prix des Mathématiques (1859), in which the MCE and the FT were systematically employed to deal with applicability problems. Our analysis provides a new insight into the historical process leading to a recognition of the relevance of the MCE and the FT and helps explaining why previous contributions on the subject could go unnoticed for years.

Gauss and Codazzi equations in a subspace of a Kähler space

Gauss and Codazzi equations in a subspace of a Kähler space

Generalisation of Gauss--Codazzi equations for Berwald's curvature tensor in a hypersurface of a Finsler space

Generalisation of Gauss--Codazzi equations for Berwald's curvature tensor in a hypersurface of a Finsler space

The Flag Curvature of a Submanifold of a Randers–Minkowski Space in Terms of Zermelo Data

The main result of this paper is an expression of the flag curvature of a submanifold of a Randers–Minkowski space ({mathscr {V}},F) in terms of invariants related to its Zermelo data (h, W). More precisely, these invariants are the sectional curvature and the second fundamental form of the positive definite scalar product h and some projections of the wind W. This expression allows for a promising characterization of submanifolds with scalar flag curvature in terms of Riemannian quantities, which, when a hypersurface is considered, seems quite approachable. As a consequence, we prove that any h-flat hypersurface S has scalar F-flag curvature and the metric of its Zermelo data is conformally flat. As a tool for making the computation, we previously reobtain the Gauss–Codazzi equations of a pseudo-Finsler submanifold using anisotropic calculus.

Coordinate-free compatibility conditions for deformations of material surfaces

The study of material surfaces uses notions from classical differential geometry, such as the covariant gradient, the mean and Gaussian curvatures, and the Peterson–Mainardi–Codazzi and Gauss equations. These notions are traditionally introduced relative to local surface coordinates and involve Christoffel symbols. We proceed instead without recourse to coordinates using direct notation. After developing the formula for the covariant gradient relative to a surface metric, we derive versions of the Peterson–Mainardi–Codazzi and Gauss equations and Gauss’ Theorema Egregium relevant to a deformed material surface. We then apply our framework to kinematically constrained material surfaces. For material surfaces that can sustain only deformations that preserve either angles or lengths, we obtain explicit representations for the covariant gradient relative to the surface metric in terms of the surface gradient. We show also that a deformation of a material surface that preserves angles and areas must be length preserving and vice versa. Finally, we present an alternative derivation of the Peterson–Mainardi–Codazzi and Gauss equations for a deformed material surface subject to the provision that the surface metric derives from the metric for the ambient Euclidean space within which the surface is embedded. An Appendix involving coordinates is included to ease comparisons between our approach to covariant differentiation and associated derivations of the Peterson–Mainardi–Codazzi and Gauss equations and standard coordinate-based approaches.

Submanifolds in Koszul–Vinberg Geometry

A Koszul–Vinberg manifold is a manifold M endowed with a pair \((\nabla ,h)\) where \(\nabla \) is a flat connection and h is a symmetric bivector field satisfying a generalized Codazzi equation. The geometry of such manifolds could be seen as a type of bridge between Poisson geometry and pseudo-Riemannian geometry, as has been highlighted in our previous article [Contravariant Pseudo-Hessian manifolds and their associated Poisson structures. Differential Geometry and its Applications (2020)]. Our objective here will be to pursue our study by focusing in this setting on submanifolds by taking into account some developments in the theory of Poisson submanifolds.

Geometric Mechanics on Warped Product Semi-Slant Submanifold of Generalized Complex Space Forms

In this study, we develop a general inequality for warped product semi-slant submanifolds of type M n = N T n 1 × f N ϑ n 2 in a nearly Kaehler manifold and generalized complex space forms using the Gauss equation instead of the Codazzi equation. There are several applications that can be developed from this. It is also described how to classify warped product semi-slant submanifolds that satisfy the equality cases of inequalities (determined using boundary conditions). Several results for connected, compact warped product semi-slant submanifolds of nearly Kaehler manifolds are obtained, and they are derived in the context of the Hamiltonian, Dirichlet energy function, gradient Ricci curvature, and nonzero eigenvalue of the Laplacian of the warping functions.

Evolving to non-round Weingarten spheres: integer linear Hopf flows

In the 1950’s Hopf gave examples of non-round convex 2-spheres in Euclidean 3-space with rotational symmetry that satisfy a linear relationship between their principal curvatures. In this paper, we investigate conditions under which evolving a smooth convex rotationally symmetric sphere by a linear combination of its radii of curvature yields a Hopf sphere. When the coefficients of the flow have certain integer values, the fate of an initial sphere is entirely determined by the local geometry of its isolated umbilic points. A variety of behaviours is uncovered: convergence to round spheres and non-round Hopf spheres, as well as divergence to infinity. The critical quantity is the rate of vanishing of the astigmatism—the difference of the radii of curvature—at the isolated umbilic points. It is proven that the size of this quantity versus the coefficient in the flow function determines the fate of the evolution. The geometric setting for the equation is Radius of Curvature space, viewed as a pair of hyperbolic/AdS half-planes joined along their boundary, the umbilic horizon. A rotationally symmetric sphere determines a parameterized curve in this plane with end-points on the umbilic horizon. The slope of the curve at the umbilic horizon is linked by the Codazzi–Mainardi equations to the rate of vanishing of astigmatism, and for generic initial conditions can be used to determine the outcome of the flow. The slope can jump during the flow, and a number of examples are given: instant jumps of the initial slope, as well as umbilic circles that contract to points in finite time and ‘pop’ the slope. Finally, we present soliton-like solutions: curves that evolve under linear flows by mutual hyperbolic/AdS isometries (dilation and translation) of Radius of Curvature space. A forthcoming paper will apply these geometric ideas to non-linear curvature flows.

A General Inequality for CR-Warped Products in Generalized Sasakian Space Form and Its Applications

In the present paper, by considering the Gauss equation in place of the Codazzi equation, we derive new optimal inequality for the second fundamental form of CR-warped product submanifolds into a generalized Sasakian space form. Moreover, the inequality generalizes some inequalities for various ambient space forms.

Computing the Riemannian curvature of image patch and single-cell RNA sequencing data manifolds using extrinsic differential geometry

Most high-dimensional datasets are thought to be inherently low-dimensional-that is, data points are constrained to lie on a low-dimensional manifold embedded in a high-dimensional ambient space. Here, we study the viability of two approaches from differential geometry to estimate the Riemannian curvature of these low-dimensional manifolds. The intrinsic approach relates curvature to the Laplace-Beltrami operator using the heat-trace expansion and is agnostic to how a manifold is embedded in a high-dimensional space. The extrinsic approach relates the ambient coordinates of a manifold's embedding to its curvature using the Second Fundamental Form and the Gauss-Codazzi equation. We found that the intrinsic approach fails to accurately estimate the curvature of even a two-dimensional constant-curvature manifold, whereas the extrinsic approach was able to handle more complex toy models, even when confounded by practical constraints like small sample sizes and measurement noise. To test the applicability of the extrinsic approach to real-world data, we computed the curvature of a well-studied manifold of image patches and recapitulated its topological classification as a Klein bottle. Lastly, we applied the extrinsic approach to study single-cell transcriptomic sequencing (scRNAseq) datasets of blood, gastrulation, and brain cells to quantify the Riemannian curvature of scRNAseq manifolds.

Stability analysis of Single-Brane with Gauss-Bonnet Term in a Bulk

We study single-brane 4 + n dimensions which is embedded in bulk 5 + n dimensions with scalar field and Gauss Bonnet terms in bulk. The brane field equation is obtained by performing a bulk field projection by using the Gauss-Codazzi equation. The Einstein brane field equation is formed into the standard Einstein field equation in the Theory of general relativity with additional terms Gauss Bonnet and extra terms. Furthermore cosmological application obtained by reviewing brane’s spacetime is homogeneous and isotropic. FRW metric is taken with two scale factors that are internal dimensions and external dimensions which have a relationship $b(t)=a(t)^\\gamma$. A dynamic analysis is performed to determine the stability of this model by taking the case of the absence of extra terms. Finally, we get a stable solution for this model occurs when the extra dimension n=2.