On the Minkowski Formula for Hypersurfaces in Complex Space Forms

In this paper, we discuss various Minkowski-type formulas for real hypersurfaces in complex space forms. In particular, we investigate the formulas suggested by the natural splitting of the tangent space. In this direction, our main result concerns a new kind of 2nd Minkowski formula.

Horizontal Newton operators and high-order Minkowski formula

In this paper, we study the horizontal Newton transformations, which are nonlinear operators related to the natural splitting of the second fundamental form for hypersurfaces in a complex space form. These operators allow to prove the classical Minkowski formulas in the case of real space forms: unlike the real case, the horizontal ones are not divergence-free. Here, we consider the highest order of nonlinearity and we will show how a Minkowski-type formula can be obtained in this case.

Elastic strips with null and pseudo-null directrix

The solution of the variational problem which gives elastic strips in Minkowski [Formula: see text]-space is separately obtained for elastic strips with null and pseudo-null directrix. First, critical points of the modified Sadowsky functional (which depends on the modified torsion) for elastic strips with null directrix are characterized by three Euler–Lagrange equations. A connection is established between elastic curves on the two-dimensional null cone and elastic strips with null directrix. Then conservation laws of elastic strips with null directrix in Minkowski 3-space are given. Second, equilibrium equations for elastic strips with pseudo-null directrix are determined and solved. It is also shown the tangent and the binormal of a critical curve of the Sadowsky functional correspond to a null elastic curve in de Sitter 2-space and a spacelike elastic curve in the two-dimensional null cone, respectively. Finally, two conservation laws for elastic strips with pseudo-null directrix are derived.

General helices with spacelike slope axis in Minkowski 3-space

In the present paper, we consider general helix with spacelike slope axis for all possible types of curves in Minkowski [Formula: see text]-space. We give the conditions under which the curves in Minkowski [Formula: see text]-space have spacelike slope axis. In addition, we find the parametric equations of the curves. Also, we give the related examples and their graphics.

Foliations between crooked planes in 3-dimensional Minkowski space

We show that any two disjoint crooked planes in [Formula: see text] are leaves of a crooked foliation. This answers a question asked by Charette and Kim [V. Charette and Y. Kim, Foliations of Minkowski [Formula: see text] spacetime by crooked planes, Int. J. Math. 25(9) (2014) 1450088.].

Lorentzian Darboux images of curves on spacelike surfaces in Lorentz–Minkowski 3-space

For a regular curve on a spacelike surface in Lorentz–Minkowski [Formula: see text]-space, we have a moving frame along the curve which is called a Lorentzian Darboux frame. We introduce five special vector fields along the curve associated to the Lorentzian Darboux frame and investigate their singularities.

Spacelike hypersurfaces in de Sitter space with constant higher-order mean curvature

The authors apply the generalized Minkowski formula to set up a spherical theorem. It is shown that a compact connected hypersurface with positive constant higher-order mean curvatureHrfor some fixedr,1≤r≤n, immersed in the de Sitter spaceS1n+1must be a sphere.

A FIELD THEORETICAL MODEL IN NONCOMMUTATIVE MINKOWSKI SUPERSPACE

In this talk we present a field theoretical model constructed in Minkowski [Formula: see text] superspace with a deformed supercoordinate algebra. Our study is motivated in part by recent results from super-string theory, which show that in a particular scenario in Euclidean superspace the spinor coordinates θ do not anticommute. Field theoretical consequences of this deformation were studied in a number of articles. We present a way to extend the discussion to Minkowski space, by assuming non-vanishing anticommutators for both θ, and [Formula: see text]. We give a consistent supercoordinate algebra, and a star product that is real and preserves the (anti)chirality of a product of (anti)chiral superfields. We also give the Wess-Zumino Lagrangian [Formula: see text] that gains only Lorentz-invariant corrections due to non(anti)commutativity within our model. The Lagrangian in Minkowski superspace is also always manifestly Hermitian.