Higher-order Curvatures Research Articles

Total torsion of three-dimensional lines of curvature

A curve gamma in a Riemannian manifold M is three-dimensional if its torsion (signed second curvature function) is well-defined and all higher-order curvatures vanish identically. In particular, when gamma lies on an oriented hypersurface S of M, we say that gamma is well positioned if the curve’s principal normal, its torsion vector, and the surface normal are everywhere coplanar. Suppose that gamma is three-dimensional and closed. We show that if gamma is a well-positioned line of curvature of S, then its total torsion is an integer multiple of 2pi ; and that, conversely, if the total torsion of gamma is an integer multiple of 2pi , then there exists an oriented hypersurface of M in which gamma is a well-positioned line of curvature. Moreover, under the same assumptions, we prove that the total torsion of gamma vanishes when S is convex. This extends the classical total torsion theorem for spherical curves.

Boundary behaviour of the span metric and its higher-order curvatures

In this note, we use scaling principle to study the boundary behaviour of the span metric and its higher-order curvatures on finitely connected Jordan planar domains. A localization of this metric near boundary points of finitely connected Jordan domains is also obtained. Further, we obtain boundary sharp estimates for the span metric near \( C^2 \)-smooth boundary points on such domains.

Some properties of Bertrand curves in Lorentzian n-space 𝕃n

In this paper, Bertrand curves in [Formula: see text]-dimensional Lorentz space [Formula: see text] are defined and some of their properties are determined. Various relationships and characterizations are found between higher order curvatures and their derivatives for Bertrand curve pair. In addition, some relationships are obtained between these curves and general helix, harmonic curvature.

Strong (r, s)-stability in the de Sitter space

In this paper, we establish the notion of strong (r, s)-stability concerning closed space-like hypersurfaces immersed with higher-order mean curvatures linearly related in the de Sitter space . In this setting, we prove that totally umbilical round spheres of are strongly (r, s)-stable. Afterwards, we obtain sufficient geometric conditions that guarantee that a closed strongly (r, s)-stable space-like hypersurface in must be a totally umbilical round sphere.

A note on Weingarten hypersurfaces in the warped product manifold

In this paper, we consider the closed embedded hypersurface Σ in the warped product manifold [Formula: see text] equipped with the metric g = dr2 + λ(r)2 gN. We give some characterizations of slice {r} × N by the condition that Σ has constant weighted higher-order mean curvatures (λ′)αpk, or constant weighted higher-order mean curvature ratio (λ′)αpk/p1, which generalize Brendle's [Constant mean curvature surfaces in warped product manifolds, Publ. Math. Inst. Hautes Études Sci. 117 (2013) 247–269] and Brendle–Eichmair's [Isoperimetric and Weingarten surfaces in the Schwarzschild manifold, J. Differential Geom. 94(3) (2013) 387–407] results. In particular, we show that the assumption convex of Brendle–Eichmair's result [Isoperimetric and Weingarten surfaces in the Schwarzschild manifold, J. Differential Geom. 94(3) (2013) 387–407] is unnecessary. Here pk is the kth normalized mean curvature of the hypersurface Σ. As a special case, we also give some characterizations of geodesic spheres in ℝn, ℍn and [Formula: see text], which generalize the classical Alexandrov-type results.

STABILITY OF SPACELIKE HYPERSURFACES WITH HIGHER-ORDER MEAN CURVATURES LINEARLY RELATED IN CONFORMALLY STATIONARY SPACETIMES

In this paper, we establish the notion of (r, s)-stability concerning spacelike hypersurfaces with higher-order mean curvatures linearly related in conformally stationary spacetimes of constant sectional curvature. In this setting, we characterize (r, s)-stable closed spacelike hypersurfaces through the analysis of the first eigenvalue of an operator naturally attached to the higher-order mean curvatures. Moreover, we obtain sufficient conditions which assure the (r, s)-stability of complete spacelike hypersurfaces immersed in the de Sitter space.

Local Estimates for Elliptic Equations Arising in Conformal Geometry

In this paper we consider Yamabe type problem for higher order curvatures on manifolds with totally geodesic boundaries. We prove local gradient and second derivative estimates for solutions to the fully nonlinear elliptic equations associated with the problems.

The Intermediate Case of the Yamabe Problem for Higher Order Curvatures

In this paper, we prove the solvability, together with the compactness of the solution set, for the -Yamabe problem on compact Riemannian manifolds of arbitrary even dimension n > 2. These results had previously been obtained by Chang, Gursky, and Yang for the case n = 4 and by Li and Li for locally conformally flat manifolds in all even dimensions. Our proof also applies to more generally prescribed symmetric functions of the Ricci curvatures.

On the Curvatures of Complete Spacelike Hypersurfaces in de Sitter Space

We establish some estimates for the higher-order mean curvatures, the scalar curvature and the Ricci curvature of a complete spacelike hypersurface in de Sitter space which is contained in certain unbounded regions of the ambient space. Our results will be an application of a generalized maximum principle due to Omori.

Three-dimensional nonlinear analysis of elastic arches

This paper presents a curved finite element model for the three-dimensional nonlinear analysis of elastic arches. The model includes higher-order curvatures which make the order of the bending strains consistent with that of the membrane strains, and the same low-order polynomials are used for all the displacements. No approximations are made for the twist rotations, and the nonlinear strains and the incremental stiffness matrix include the exact coupling due to the twist rotations. All significant terms of strains are retained. Without these terms, superimposed rigid body motions may lead to overstiff solutions. The model also includes warping of the cross-section and the Wagner effect. Application of the model to several numerical examples and comparison with the existing experimental and analytical results demonstrate that the model is much more effective and efficient than other methods in terms of accuracy, the number of elements needed for convergence, and the ability to pass maximum load points. The model is also used to investigate the elastic flexural-torsional buckling and the elastic postbuckling behaviour of arches.