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We first define a complex angle between two oriented spacelike planes in 4-dimensional Minkowski space, and then study the constant angle surfaces in that space, i.e. the oriented spacelike surfaces whose tangent planes form a constant complex angle with respect to a fixed spacelike plane. This notion is the natural Lorentzian analogue of the notion of constant angle surfaces in 4-dimensional Euclidean space. We prove that these surfaces have vanishing Gauss and normal curvatures, obtain representation formulas for the constant angle surfaces with regular Gauss maps and construct constant angle surfaces using PDE's methods. We then describe their invariants of second order and show that a surface with regular Gauss map and constant angle ψ ≠ 0 [ π ∕ 2 ] is never complete. We finally study the special cases of surfaces with constant angle π ∕ 2 [ π ] , with real or pure imaginary constant angle and describe the constant angle surfaces in hyperspheres and lightcones. [ABSTRACT FROM AUTHOR]

It is well known that the space of oriented lines of Euclidean space has a natural symplectic structure. Moreover, given an immersed, oriented hypersurface S the set of oriented lines that cross S orthogonally is a Lagrangian submanifold. Conversely, if S ¯ an n-dimensional family of oriented lines is Lagrangian, there exists, locally, a 1-parameter family of immersed, oriented, parallel hypersurfaces S t whose tangent spaces cross orthogonally the lines of S ¯. The purpose of this paper is to generalize these facts to higher dimension: to any point x of a submanifold S of R m of dimension n and co-dimension k = m - n , we may associate the affine k-space normal to S at x. Conversely, given an n-dimensional family S ¯ of affine k-spaces of R m , we provide certain conditions granting the local existence of a family of n-dimensional submanifolds S which cross orthogonally the affine k-spaces of S ¯ . We also define a curvature tensor for a general family of affine spaces of R m which generalizes the curvature of a submanifold, and, in the case of a 2-dimensional family of 2-planes in R 4 , show that it satisfies a generalized Gauss–Bonnet formula. [ABSTRACT FROM AUTHOR]

In this paper we give a spinorial representation of submanifolds of any dimension and codimension into Lie groups equipped with left invariant metrics. As applications, we get a spinorial proof of the Fundamental Theorem for submanifolds into Lie groups, we recover previously known representations of submanifolds in R n and in the 3-dimensional Lie groups S 3 and E ( κ , τ ) , and we get a new spinorial representation for surfaces in the 3-dimensional semi-direct products: this achieves the spinorial representations of surfaces in the 3-dimensional homogeneous spaces. We finally indicate how to recover a Weierstrass-type representation for CMC-surfaces in 3-dimensional metric Lie groups recently given by Meeks, Mira, Perez and Ros. [ABSTRACT FROM AUTHOR]

We prove that an isometric immersion of a simply connected Lorentzian surface in R 2 , 2 is equivalent to a normalised spinor field solution of a Dirac equation on the surface. Using the quaternions and the Lorentz numbers, we also obtain an explicit representation formula of the immersion in terms of the spinor field. We then apply the representation formula in R 2 , 2 to give a new spinor representation formula for Lorentzian surfaces in 3-dimensional Minkowski space. Finally, we apply the representation formula to the local description of the flat Lorentzian surfaces with flat normal bundle and regular Gauss map in R 2 , 2 , and show that these surfaces locally depend on four real functions of one real variable, or on one holomorphic function together with two real functions of one real variable, depending on the sign of a natural invariant. [ABSTRACT FROM AUTHOR]

We prove that an isometric immersion of a simply connected Riemannian surface in four-dimensional Minkowski space, with given normal bundle and given mean curvature vector , is equivalent to a normalized spinor field solution of a Dirac equation on the surface. Using the immersion of the Minkowski space into the complex quaternions, we also obtain a representation of the immersion in terms of the spinor field. We then use these results to describe the flat spacelike surfaces with flat normal bundle and regular Gauss map in four-dimensional Minkowski space, and also the flat surfaces in three-dimensional hyperbolic space, giving spinorial proofs of results by J.A. Gálvez, A. Martínez and F. Milán. [ABSTRACT FROM AUTHOR]

An essay is presented on plagiarism. It discusses the theory that a comparable situation holds for writers and works of the future, which practice on past and present literature a kind of spread out radiation. It explains the particular perception that allows readers, in examining texts with enough care, to grasp the new aesthetic territories where the readers are headed. It implies that the entirety of the literary history conception must be altered, once anticipatory plagiarism is accepted.

Abstract: We first describe the numerical invariants and the curvature hyperbola attached to the second fundamental form of a timelike surface in four-dimensional Minkowski space: beside the four natural invariants, a new invariant appears at some special points of the surface, which are said to be quasi-umbilic; at such quasi-umbilic points, the curvature hyperbola degenerates to a line with one point removed. We then study the asymptotic lines on a timelike surface, and characterize the quasi-umbilic points of the surface as the points where the asymptotic directions degenerate to a double lightlike line. We also give an interpretation of the new invariant at a quasi-umbilic point, using the Gauss map of the surface. We finally describe the timelike surfaces which are quasi-umbilic at every point. [Copyright &y& Elsevier]

Abstract: We describe some numerical invariants and their relation with the curvature ellipse attached to the second fundamental form of a surface immersed in . We provide a geometric interpretation of them. Moreover, we describe the field of asymptotic directions in terms of these invariants and study this field in a neighborhood of an isolated inflection point of imaginary type. [Copyright &y& Elsevier]

Abstract: We prove the existence and uniqueness in of entire spacelike hypersurfaces contained in the future of the origin O and asymptotic to the light-cone, with scalar curvature prescribed at their generic point M as a negative function of the unit vector pointing in the direction of , divided by the square of the norm of (a dilation invariant problem). The solutions are seeked as graphs over the future unit-hyperboloid emanating from O (the hyperbolic space); radial upper and lower solutions are constructed which, relying on a previous result in the Cartesian setting, imply their existence. [Copyright &y& Elsevier]

We prove existence and stability of smooth entire strictly convex spacelike hypersurfaces of prescribed Gauß curvature in Minkowski space. The proof is based on barrier constructions and local a priori estimates. [ABSTRACT FROM AUTHOR]

Investigates whether it is possible to apply literature to psychoanalysis as an approach to literary text. Definition of applied literature; Discussion on hermeneutic criticism; Instruction of psychoanalysis through literature.