Biharmonic surfaces in Euclidean space $mathbb{E}^3$ are firstly studied from a differential geometric point of view by Bang-Yen Chen, who showed that the only biharmonic surfaces are minimal ones. A surface $x : M^2rightarrowmathbb{E}^{3}$ is called biharmonic if $Delta^2x=0$, where $Delta$ is the Laplace operator of $M^2$. We study the $L_k$-biharmonic spacelike hypersurfaces in the $4$-dimen...

The purpose of this paper is to summarize results on various aspects of sets with positive reach, which are up to now not available in such a compact form. After recalling briefly the results before 1959, sets with positive reach and their associated curvature measures are introduced. We develop an integral and current representation of these curvature measures and show how the current represen...

Abstract We present a simple discrete formula for the elastic energy of a bilayer. The formula is convenient for rapidly computing equilibrium configurations of actuated bilayers of general initial shapes. We use maps of principal curvatures and minimum-curvature direction fields to analyze configurations. We find good agreement between the computations and an approximate analytical solution fo...

In this paper we modify the maximum principal of (Galloway, 2000) for totally geodesic null hypersurfaces by proving a geometric maximum principle which obeys mean curvature inequalities of a family of totally umbilical null hypersurfaces of a spacetime manifold (Theorem 6). As a physical interpretation we show that, in particular, for a prescribed class of spacetimes the geometric inequality o...

The purpose of this paper is to summarize results on various aspects of sets with positive reach, which are up to now not available in such a compact form. After recalling briefly the results before 1959, sets with positive reach and their associated curvature measures are introduced. We develop an integral and current representation of these curvature measures and show how the current represen...

In this work, a novel method for determining the principal directions (maxima) of the diffusion orientation distribution function (ODF) is proposed. We represent the ODF as a symmetric high-order Cartesian tensor restricted to the unit sphere and show that the extrema of the ODF are solutions to a system of polynomial equations whose coefficients are polynomial functions of the tensor elements....

In this paper, we characterize and classify the isoparametric hypersurfaces with constant principal curvatures in product spaces Qc12×Qc22, where Qci2 is a space form sectional curvature ci, for ci∈{−1,0,1} c1≠c2.

In this paper we present a novel method to estimate curvature of iso grey-level surfaces in grey-value images. Our method succeeds where isophote curvature fails. There is neither a segmentation of the surface needed nor a parametric model assumed. Our estimator works on the orientation (normal vector) field of the surface. This orientation field and a description of local structure is obtained...

The motion equations for a Lagrangian L(k1), depending on the curvature k1 of the particle worldline, embedded in a space–time of constant curvature, are considered and reformulated in terms of the principal curvatures. It is shown that for arbitrary Lagrangian function L(k1) the general solution of the motion equations can be obtained by integrals. By analogy with the flat space–time case, the...

introduction: complete knowledge of root canal curvature is a critical factor in successful dental treatment .the aim of this study was to investigate the direction, radius and degree of curvature of maxillary anterior teeth and the relation between radius and degree of curvature in babol city. methods &materials;: this study was performed on 242 anterior teeth radiographs were taken by periapi...