Akademik Veri Yönetim Sistemi
3. Azerbaycan Türkiye Matematik Sempozyumu, Trabzon, Türkiye, 31 Ağustos - 04 Eylül 1993, ss.67-68
By L^3 we denote the space R^3 endowed with the inner product of index 1 and call it Lorentzian 3-space. In L^3 every tangent space of a surface and be considered as a subspace of L^3 in a canonical way. Thus if a surface in L^3 has the tangent spaces of index 1 then we call the surface Lorentzian. In addition, a curve in a Lorentzian surface called time-like, space-like or null whether its velocity vector is.
In the Riemannian case, it is well known that all the curves pass through a point, say p, and have common and non asymptotic tangents at the point p have their curvature centers on a unique sphere and also have their curvature circles on another unique sphere. This fact known as the Meusnier's Theorem. The essential part of this work devoted to give an analog of this fact in L^3.
Let M be a Lorentzian surface in L^3 and peM, XpeM. We assume that XpeTpM is not an asymptotic on M then,
i) The locus of the curvature centers of all the non-null section curves determined by Xp with space-like second Frenet vectors is a pseudosphere,
ii) The locus of the fourth vertex point of the parallelogram which constructed with one diagonal (CCi) and there vertices P, C, Cj is a pseudosphere where Ci and C are the curvature centers of an arbitrary section curve and the normal section curve determined by Xp, respectively.
iii) All curvature circles of all the non-null section curves determined Xp with space-like second Frenet vectors lie on a pseudosphere centered at the point C, the curvature center of the normal section curve determined by Xp.
iv) All the special translated curvature circles of all non-null section curves, determined Xp, with time-like second Frenet vectors lie on a pseudosphere or a pseudohyperbolic space and the center of the pseudosphere or the hyperbolic space is the fourth vertex point of the parallelogram which is determined by the vertex P, C and Ci and one diagonal the line segment (CCj).