On the Semicontinuity of Curvature Integrals

Let H_j(K, ·) be the j – th elementary symmetric function of the principal curvatures of a convex body K in Euclidean d – space. We show that the functionals ∫_bd f(H_j(K, x)) dℋ︁^d—1(x) depend upper semicontinuously on K, if f : [0, ∞) is concave, lim_t→0f(t) = 0, and lim_t→∞f(t)/t = 0. An analogous statement holds for integrals of elementary symmetric functions of the principal radii of curvature.     

Homogeneous hypersurfaces in complex hyperbolic spaces

We study the geometry of homogeneous hypersurfaces and their focal sets in complex hyperbolic spaces. In particular, we provide a characterization of the focal set in terms of its second fundamental form and determine the principal curvatures of the homogeneous hypersurfaces together with their multiplicities.      

Curvature Computations of 2-Manifolds in $\mathbb{R}^k$

In this paper, we provide simple and explicit formulas for computing Riemannian curvatures, mean curvature vectors, principal curvatures and principal directions for a 2-dimensional Riemannian manifold embedded in IR^k with k ≥ 3.     

反常霍尔效应理论的研究进展

文章介绍了在铁磁性材料中反常霍尔效应的发现及其机制研究的历史；阐述了反常霍尔效应理论研究最近取得的重大进展，即倒空间中布洛赫态的贝里曲率（规范场）特性决定了霍尔电导率；同时指出，建立系统地解释反常霍尔效应机制的理论仍然是一个挑战性的任务．     

On Vertices, focal curvatures and differential geometry of space curves

The focal curve of an immersed smooth curve γ : θ ↦ γ (θ), in Euclidean space ℝ^m+1, consists of the centres of its osculating hyperspheres. This curve may be parametrised in terms of the Frenet frame of γ (t, n₁, . . . , n_m), as C_γ (θ) = (γ +c₁n₁+ c₂n₂ + • • • + c_mn_m)(θ), where the coefficients c₁, . . . , c_m-1 are smooth functions that we call the focal curvatures of γ . We discovered a remarkable formula relating the Euclidean curvatures κ_i , i = 1, . . . ,m, of γ with its focal curvatures. We show that the focal curvatures satisfy a system of Frenet equations (not vectorial, but scalar!). We use the properties of the focal curvatures in order to give, for ℓ = 1, . . . ,m, necessary and sufficient conditions for the radius of the osculating ℓ-dimensional sphere to be critical. We also give necessary and sufficient conditions for a point of γ to be a vertex. Finally, we show explicitly the relations of the Frenet frame and the Euclidean curvatures of γ with the Frenet frame and the Euclidean curvatures of its focal curve C_γ.     

Distinguishing between evidence and its explanations in the steering of atomic clocks

Quantum theory reflects within itself a separation of evidence from explanations. This separation leads to a known proof that: (1) no wave function can be determined uniquely by evidence, and (2) any chosen wave function requires a guess reaching beyond logic to things unforeseeable. Chosen wave functions are encoded into computer-mediated feedback essential to atomic clocks, including clocks that step computers through their phases of computation and clocks in space vehicles that supply evidence of signal propagation explained by hypotheses of spacetimes with metric tensor fields.     

Asymmetric Polymer Particles with Anisotropic Curvatures by Annealing Polystyrene Microspheres on Poly(vinyl alcohol) Films

Anisotropic polymer particles such as Janus particles have attracted significant attention in recent years because of their unique properties and unusual self‐assembly behavior. Most anisotropic polymer particles synthesized so far, however, only have different chemical regions compartmentalized on the particles. It remains a great challenge to fabricate anisotropic polymer particles with different shapes within a single particle. A novel approach is developed to prepare anisotropic polymer particles that contain two hemispheres with different curvatures by annealing polystyrene microspheres on poly(vinyl alcohol) films. During the annealing process, the polymer microspheres gradually sink into the polymer films and transform to asymmetric polymer particles, driven by the surface and interfacial tensions of the polymers. Selective removal techniques are also used to confirm the morphologies of the asymmetric particles.

     

An extension of Takahashi theorem for the linearized operators of the higher order mean curvatures

We study hypersurfaces in Euclidean space whose position vector x satisfies the condition L _k x = Ax + b, where L _k is the linearized operator of the (k + 1)th mean curvature of the hypersurface for a fixed , is a constant matrix and is a constant vector. For every k, we prove that the only hypersurfaces satisfying that condition are hypersurfaces with zero (k + 1)th mean curvature and open pieces of round hyperspheres and generalized right spherical cylinders of the form , with . This extends a previous classification for hypersurfaces in satisfying , where is the Laplacian operator of the hypersurface, given independently by Hasanis and Vlachos [J. Austral. Math. Soc. Ser. A 53, 377–384 (1991) and Chen and Petrovic [Bull. Austral. Math. Soc. 44, 117–129 (1991)].      

The Embedding of Schwarzschild in Braneworld

The braneworlds models were inspired partly by Kaluza-Klein’s theory, where both the gravitational and the gauge fields are obtained from the geometry of a higher dimensional space. The positive aspects of these models consist in perspectives of modifications it could bring in to particle physics, such as: unification in a TeV scale, quantum gravity in this scale and deviation of Newton’s law for small distances. One of the principles of these models is to suppose that all space-times can be embedded in a bulk of higher dimension. The main result in these notes is a theorem showing a mathematical inconsistency of the Randall-Sundrum braneworld model, namely that the Schwarzschild space-time cannot be embedded locally and isometrically in a five dimensional bulk with constant curvature (for example AdS-5). From the point of view of semi-Riemannian geometry this last result represents a serious restriction to the Randall-Sundrum’s braneworld model.     

OPTICAL SPACE OF THE KERR SOLUTIONS

In this paper, we discuss the optical geometry of the Kerr solutions, and show the embedding diagram at the equator θ=π/2 for the optical space of the Kerr solution with a/M=1. It is noticed that at the equator θ=π/2, the Gaussian curvature K for the ordinary space is negative, and the \stackrel{\sim}{K} for the optical space is positive.