Laguerre geometry of surfaces with plane lines of curvature

We study surfaces with plane lines of curvature in the framework of Laguerre geometry and provide explicit representation formulae for these surfaces in terms of a potential function. As an application, we explicitly integrate allL- minimal surfaces with plane curvature lines. Partially supported by MURST 40.     

Some Theorems for Hypersurface of Randers Spaces

In this paper, we consider the hypersurfaces of Randers space with constant flag curvature. (1) Let (Mⁿ⁺¹, F) be a Randers-Minkowski space. If (Mⁿ, F) is a hypersurface of (Mⁿ⁺¹, F) with constant flag curvature K=1, then we can prove that M is Riemannian. (2) Let (Mⁿ⁺¹, F) be a Randers space with constant flag curvature. Assume (M, F) is a compact hypersurface of (Mⁿ⁺¹, F) with constant mean curvature|H|. Then a pinching theorem is established, which generalizes the result of[Proc. Amer. Math. Soc., 120, 1223-1229 (1994)] from the Riemannian case to the Randers space.     

Can a Circulating Light Beam Produce a Time Machine

In a recent paper, Mallett found a solution of the Einstein equations in which closed timelike curves (CTC’s) are present in the empty space outside an infinitely long cylinder of light moving in circular paths around an axis. Here we show that, for physically realistic energy densities, the CTC’s occur at distances from the axis greater than the radius of the visible universe by an immense factor. We then show that Mallett’s solution has a curvature singularity on the axis, even in the case where the intensity of the light vanishes. Thus it is not the solution one would get by starting with Minkowski space and establishing a cylinder of light.     

Solvent Relaxation in Phospholipid Bilayers: Principles and Recent Applications

Although there exist a number of methods, such as NMR, X-ray, e.g., which explore the hydration of phospholipid bilayers, the solvent relaxation (SR) method has the advantage of simple instrumentation, easy data treatment and possibility of measuring fully hydrated samples. The main information gained from SR by the analysis of recorded “time-resolved emission spectra” (TRES) is micro-viscosity and micro-polarity of the dye microenvironment. Based on these parameters, one can draw conclusions about water structure in the bilayer. In this review, we focus on physical background of this method, on all the procedures that are needed in order to obtain relevant parameters, and on the requirements on the fluorescence dyes. Furthermore, a few recent applications (the effect of curvature, binding of antibacterial peptides and phase transition) illustrating the versatility of this method are mentioned. Moreover, limitations and potential problems are discussed.     

Stable constant mean curvature hypersurfaces in ℝ n+1 and ℍ n+1(−1)

In this paper, we show that all complete stable hypersurfaces in ⁿ⁺¹(or ⁿ⁺¹ (-1)) (n = 3, 4, 5) with constant mean curvature H > 0 (or H > 1, respectively) and finite L ² norm of traceless second fundamental form are compact geodesic spheres. Keywords: stable hypersurface, constant mean curvature, isometric immersion, Bernstein theorem.*Supported by PolyU grant G-T575.**Partially supported by CNPq of Brazil.     

D = 5, N = 2 Geometric Higher Curvature Supergravity in the Second-Order Canonical Theory

The supersymmetric extension of the five-dimensional Chern–Simons gravity is studied from the Hamiltonian point of view. This model containing the Gauss–Bonnet term quadratic in the Riemann curvature is the gauge theory of the supergroup SU(2,2/1). In the first order, the theory has a polynomial structure, but the second-order leads to a nonpolynomial structure for both the Hamiltonian and the supersymmetry transformation rules of the fields. The second-order theory has the advantage that the apparent gauge degrees of freedom are unambiguously removed leaving only the physical ones. This important feature is analyzed by constructing the second-order Hamiltonian theory. The gauge invariances of the model and the generator of time evolution are found.     

Invariant surfaces of a three-dimensional manifold with constant Gauss curvature

We give a reduction procedure to determine (locally) the surfaces with constant Gauss curvature in a three-dimensional manifold which are invariant under the action of a one-parameter subgroup of the isometry group of the ambient space. We apply this procedure to describe the invariant surfaces with constant Gauss curvature in $H^2\times R$ and in $H_3$.     

Behavior of Distant Maximal Geodesics in Finitely Connected Complete Two-Dimensional Riemannian Manifolds II

We study the behavior of maximal geodesics in a finitely connected complete two-dimensional Riemannian manifold M admitting curvature at infinity. In the case where M is homeomorphic to ² the Cohn–Vossen theorem states that the total curvature of M, say c(M), is 2. We already studied the case c(M)<2 in our previous paper. So we study the behavior of geodesics in M with total curvature 2 in this paper. Next we consider the case where M has nonempty boundary. In order to know the behavior of distant geodesics in M with boundary, it is useful to investigate the 'visual image' of the boundary of M. The latter half of this paper will be spent to study the asymptotic behavior of the visual image of a subset of M with located point tending to infinity.     

Geometrical and Topological Properties of Real Polynomial Fibres

We establish a Gauss—Bonnet type formula for a smooth fibre of a nonproper real polynomial of ⁿ . For this we need to study topological properties of a generic hyperplane section of this fibre.     

A complete conformal metric of preassigned negative Gaussian curvature for a punctured hyperbolic Riemann surface

Leth be a complete metric of Gaussian curvature K₀ on a punctured Riemann surface of genusg ≥ 1 (or the sphere with at least three punctures). Given a smooth negative functionK withK =K ₀ in neighbourhoods of the punctures we prove that there exists a metric conformal toh which attains this function as its Gaussian curvature for the punctured Riemann surface. We do so by minimizing an appropriate functional using elementary analysis.