In this paper, by considering a Frenet curve lying on an oriented hypersurface, we extend the Darboux frame field into Euclidean 4-space E 4 . Depending on the linear independency of the curvature vector with the hypersurface's normal, we obtain two cases for this extension. For each case, we obtain some geometrical meanings of new invariants along the curve on the hypersurface. We also give the relationships between the Frenet frame curvatures and Darboux frame curvatures in E 4 . Finally, we compute the expressions of the new invariants of a Frenet curve lying on an implicit hypersurface.

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The taxicab metric, of course, was known before the taûcab geometry was introduced in 1975. Since then, the studies have shown that the taxicab geometry is better model in urban world. The defînition of inner-product and norm in taxicab geometry are given in [1]. In this paper, we will discuss some properties of taxicab norm and the isometries of taxicab geometry.

We planned this paper into two main sections. In the first section, we gjve an analog for the Lorentzian case of some characterizations given in [2 ]. There is no difference between the characterizations in both cases of inunersions with (pointwise) 2-planar normal sections of Riemannian and Lorentzian manifolds into R™ and R,"*, respectively, but the ptoofs.In the second part of paper, we deal with the Tlıeorem. 3.2 given in [1 ] and show that there must be some extra hypothesis to get the characterizations given as Theorems 2.1 and 2.2 in the present paper.

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