Ruled surfaces with lightlike ruling in 3-Minkowski space

In this paper we discuss ruled surfaces with lightlike ruling in 3-Minkowski space and give some characterizations and examples of so called B-scroll surfaces.     

Pitch functions of ruled surfaces and B-scrolls in Minkowski 3-space

In this paper, we define pitch function for any non developable ruled surfaces and use this notion to give a new characterization of B-scrolls in Minkowski 3-space.     

Hypersurfaces in lightlike cone

In this paper, we are concerned with hypersurfaces in (n+1)$(n+1)$ dimensional lightlike cone. We will give the fundamental theories and some properties of such hypersurfaces and characterize some of them. Finally, using the moving frame method, we calculate the first and the second variation formulas of the area integral for the hypersurface in lightlike cone.     

Schlesinger transforms for the discrete painlevé IV equation

We derive an auto-Bäcklund transformation for the discrete Painlevé IV equation and use it in order to derive Schlesinger transformations for the same equation as well as particular solutions in perfect analogy to the continuous case.     

A pinching theorem for the normal scalar curvature of invariant submanifolds

We prove some inequalities relating intrinsic and extrinsic curvature invariants for invariant submanifolds of Kaehlerian and Sasakian space forms. When we restrict to invariant submanifolds of odd-dimensional unit spheres or invariant submanifolds of complex Euclidean space, one of the inequalities gives a positive answer to a conjecture, proposed in [P.J. De Smet, F. Dillen, L. Verstraelen, L. Vrancken, A pointwise inequality in submanifold theory, Arch. Math. (Brno) 35 (1999) 115–128].     

The theorem of Schur in the Minkowski plane

We prove a Lorentzian analogue of the theorem of Schur for spacelike (or timelike) curves in the Minkowski plane.     

On leafwise conformal diffeomorphisms

For every diffeomorphism φ:M→N$\phi :M\to N$ between 3-dimensional Riemannian manifolds M$M$ and N$N$, there are locally two 2-dimensional distributions _D±$D_{\pm }$ such that φ$\phi$ is conformal on both of them. We state necessary and sufficient conditions for a distribution to be one of _D±$D_{\pm }$. These are algebraic conditions expressed in terms of the self-adjoint and positive definite operator induced from _φ∗${\phi }_{\ast }$. We investigate the integrability condition of _D+$D_{+}$ and _D−$D_{-}$. We also show that it is possible to choose coordinate systems in which leafwise conformal diffeomorphism is holomorphic on leaves.     

On time-like Willmore surfaces in Minkowski space

We study time-like surfaces in Minkowski space, which are critical points of the Willmore energy. Transforming the fourth order Willmore equation into a quasi-linear, second order hyperbolic system, we prove existence, uniqueness and symmetry properties of such surfaces, subject to geometric initial conditions.     

On Nurowski’s conformal structure associated to a generic rank two distribution in dimension five

For a generic distribution of rank two on a manifold M$M$ of dimension five, we introduce the notion of a generalized contact form. To such a form we associate a generalized Reeb field and a partial connection. From these data, we explicitly construct a pseudo-Riemannian metric on M$M$ of split signature. We prove that a change of the generalized contact form only leads to a conformal rescaling of this metric, so the corresponding conformal class is intrinsic to the distribution.     

The classification of 4-dimensional non-degenerate affine hypersurfaces with parallel cubic form

In this paper, we complete the classification of 4-dimensional non-degenerate affine hypersurfaces with parallel cubic form with respect to the Levi-Civita connection of the affine Berwald–Blaschke metric.     

Coupling solutions of BGG-equations in conformal spin geometry

BGG-equations are geometric overdetermined systems of partial differential equations (PDEs) on parabolic geometries. Normal solutions of BGG-equations are particularly interesting, and we give a simple formula for the necessary and sufficient additional integrability conditions on a solution. We then discuss a procedure for coupling known solutions of BGG-equations to produce new ones. Employing a suitable calculus for conformal spin structures, this yields explicit coupling formulas and conditions between almost Einstein scales, conformal Killing forms, and twistor spinors. Finally, we discuss a class of generic twistor spinors that provides an invariant decomposition of conformal Killing fields.     

Geometric realizations,curvature decompositions,and Weyl manifolds

We show that any Weyl curvature model can be geometrically realized by a Weyl manifold.     

Spacelike submanifolds of codimension two in de Sitter space

We investigate the differential geometry of spacelike submanifolds of codimension two in de Sitter space and classify the singularities of lightlike hypersurfaces and lightcone Gauss maps in de Sitter 4-space.     

Some examples of the behaviour of conformal geodesics

With the aid of concrete examples, we consider the question of whether, in the presence of conformal curvature, a conformal geodesic can become trapped in smaller and smaller sets, or phrased informally: Are spirals possible? We do not arrive at a definitive answer, but we are able to find situations where this behaviour is ruled out, including a reduction of the conformal-geodesic equations to quadratures in a specific non-conformally flat metric.     

Generalized complex structures on Kodaira surfaces

We compute the deformations in the sense of generalized complex structures of the standard classical complex structure on a primary Kodaira surface and we prove that the obtained family of deformations is a smooth locally complete family depending on four complex parameters. This family is the same as the extended deformations (in the sense of Kontsevich and Barannikov) in degree two, obtained by Poon using differential Gerstenhaber algebras.     

On the twistor-spinors

Two interesting conformal invariants which are constant on the manifold are given for twistor-spinors on a spin manifold following the notion of a twistor-spinor associated to a twisted spin bundle. For a twisted spin bundle corresponding to a flat Hermitian vector bundle, the associated twistor-spinors admit the same conformal invariants.An analysis is made of the twistor-spinors given by , where f is a complex-valued function. There is only one case where is not a Killing spinor. An example is given of a compact spin manifold for which the situation is realized.     

Definite affine spheres and loop groups

We obtain the Weierstrass-type representation and the dressing transformation for definite affine spheres in this paper. As an application of the Weierstrass-type representation, we construct the entire family of finite-Symes type affine spheres.     

Constant angle surfaces in product spaces

We classify all the surfaces in M²(_c1)×M²(_c2)$M^2\left(c_1\right)\times M^2\left(c_2\right)$ for which the tangent space _TpM²$T_p{M}^2$ makes constant angles with _Tp(M²(_c1)×{_p2})$T_p(M^2\left(c_1\right)\times \left\{p_2\right\})$ (or equivalently with _Tp({_p1}×M²(_c2))$T_p(\left\{p_1\right\}\times M^2\left(c_2\right))$ for every point p=(_p1,_p2)$p=(p_1,p_2)$ of M²$M^2$. Here M²(_c1)$M^2\left(c_1\right)$ and M²(_c2)$M^2\left(c_2\right)$ are 2$2$-dimensional space forms, not both flat. As a corollary we give a classification of all the totally geodesic surfaces in M²(_c1)×M²(_c2)$M^2\left(c_1\right)\times M^2\left(c_2\right)$.