In this paper, we investigate the chord properties of the non-null W-curves in Lorentz-Minkowski space. We give the general equation form for W-curves in (2n+1)-dimension. We define some special curves and give the relations between these curves and isoparametric surfaces. Finally we obtain the geodesics of the pseudospherical cylinder and pseudohyperbolic cylinder in 4-dimensional space.

In this paper, we consider the Mannheim curve and the slant helix together. We called this curve as a Mannheim slant helix shortly. First we calculate the (first) curvature ( ), and the curvature of the tangent indicatrix of the Mannheim curve, in terms of the arc-lenght parameter of the curve. Also, we proved that if the Mannheim curve is also slant helix, i.e. if it is Mannheim slant helix, then the partner curve is general helix. Moreover, we show the striction curve of the ruled surface such that the base curve is Mannheim curve, and the rulings are the normal vector field of the Mannheim curve, is the Mannheim partner curve. Finally, we show the ruled surface such that the base curve is Mannheim curve, and the rulings are the normal vector field of the Mannheim curve is non-developable while the torsion of the Mannheim partner curve ( ) ≠ ±∞ for all s.

In the present study, we find the parametric equations of non-null W-curves through the semi skewsymmetric matrix in three dimensional Lorentz-Minkowski space. Our technic provides more simple but efficient method for find the parametric equations of these curves in comparison to previous studies in mentioned space. Finally, we give some pictures of W-curves in polynomial form.

In this paper, we examine the image of geodesic curves of Riemann 2-manifolds under the isometric immersions in three dimensional Euclidean space. We show that the curvature of these curves is equal to the normal curvature of the manifold in the direction of tangent vector field of the geodesics. Moreover, we prove that if the parameter curves of the manifold are the line of curvature, then the geodesic torsion of geodesics is equal to the torsion of the image curve.

In this study, we determine the isoparametric surfaces and we give the Gauss map of these surfaces by semi symmetric matrix, in Lorentz space. Also we define any chord property and we show that the surfaces which have the chord property corresponds to isoparametric surfaces. Moreover, we consider the chord property locally and we give some examples in the Euclidean space.

In this paper, we characterize the de Sitter space by means of spacelike and timelike curves that fully lies on it. For this purpose, we consider the tangential part of the second derivative of the unit speed curve on the hypersurface, and obtain the vector equations of the geodesics. We find the geodesics as hyperbolas, ellipses, and helices. Moreover, we give an example of null curve with constant curvature in 4−dimensional Minkowski space and we illustrate the geodesics of S 1 1 (r) × R.