The regularized expectation value of the stress-energy tensor for a massless bosonic or fermionic field in 1+1 dimensions is calculated explicitly for the instantaneous vacuum relative to any Cauchy surface (here a spacelike curve) in terms of the length L of the curve (if closed), the local extrinsic curvature K of the curve, its derivative K ′ with respect to proper distance x along the curve...

The regularized expectation value of the stress-energy tensor for a massless bosonic or fermionic field in 1+1 dimensions is calculated explicitly for the instantaneous vacuum relative to any Cauchy surface (here a spacelike curve) in terms of the length L of the curve (if closed), the local extrinsic curvature K of the curve, its derivative K ′ with respect to proper distance x along the curve...

In this paper we introduce generalized Darboux frame of a spacelike curve $\alpha$ lying on lightlike surface in Minkowski space $\mathbb{E}_{1}^{3}$. We prove that has two such frames and obtain frame's equations. find the relations between curvature functions $k_g$, $k_n$, $\tau_g$ with respect to its $\tilde{k}_{g}$, $\tilde{k}_{n}$, $\tilde{\tau}_{g}$ frames. show exist along straight line ...

Bézier curves are a type of used in computer aided design and related fields. The can be defined with the help De Casteljau algorithm, which is one most basic elements curve surface design, Bernstein polynomials, facilitate theoretical developments. A rational may evaluated by applying de algorithm to both numerator denominator finally dividing through. suitable control points corresponding sca...

Spacelike Willmore surfaces in 4-dimensional Lorentzian space forms, a topic in Lorentzian conformal geometry which parallels the theory of Willmore surfaces in S, are studied in this paper. We define two kinds of transforms for such a surface, which produce the so-called left/right polar surfaces and the adjoint surfaces. These new surfaces are again conformal Willmore surfaces. For them holds...

We obtain a gradient estimate for the Gauss maps from complete spacelike constant mean curvature hypersurfaces in Minkowski space into the hyperbolic space. As applications, we prove a Bernstein theorem which says that if the image of the Gauss map is bounded from one side, then the spacelike constant mean curvature hypersurface must be linear. This result extends the previous theorems obtained...

In this paper we solve the Plateau problem for spacelike surfaces with constant mean curvature in Lorentz-Minkowski three-space L and spanning two circular (axially symmetric) contours in parallel planes. We prove that rotational symmetric surfaces are the only compact spacelike surfaces in L of constant mean curvature bounded by two concentric circles in parallel planes. As conclusion, we char...

In this paper, first we give a notion for linear Weingarten spacelike hypersurfaces with P + aH = b in a locally symmetric Lorentz space Ln+1 1 . Furthermore, we study complete or compact linear Weingarten spacelike hypersurfaces in locally symmetric Lorentz spaces Ln+1 1 satisfying some curvature conditions. By modifying Cheng-Yau’s operator given in [7], we introduce a modified operator L and...

We show that if one takes into account Penrose’s conformal technique in the Einstein-Cartan gravity, it is possible to obtain a relation between Hubble’s expansion and Birch’s rotation of the Universe at spacelike infinity. This relation also leads to the final reduction of the gravitational coupling to a dimensionless quantity of order unity at spacelike infinity. Initial-value equations at sp...

Pseudo-Riemannian manifolds of balanced signature which are both spacelike and timelike Jordan Osserman nilpotent of order 2 and of order 3 have been constructed previously. In this short note, we shall construct pseudo-Riemannian manifolds of signature (2s, s) for any s ≥ 2 which are spacelike Jordan Osserman nilpotent of order 3 but which are not timelike Jordan Osserman. Our example and tech...