On The Normalized Cartan Frame Of Null Curves And The Extended B-Scrolls In Minkowski 3-Space

In this paper, The parameter s with x3=1 is called the normalized parameter. We call frame F={X, Y, Z} with the Frenet equations: a normalized Cartan Frame and (a, F) a normalized Cartan Framed null curve. k=||α"|| is the curvature of (α, F). Furthermore, under the choice of normalized parameter s and the curvature function k(s), we obtained the parametrization of the Cartan framed null curve (α,F) in R13. Under the normalized Cartan Framed null curve (α, F), we defined the B-scroll:M:χ(s, t)= α(s)+tY(s), s∈I, t∈J, and calculated the first fundamental form I, the second fundamental form Ⅱ, the Gaussian curvature K=-k2/4, the mean curvature H= -k/2 of the B-scrolls M. Moreover, we construct the extended B-scroll ME:χ(s,t)=α(s)+tL(s), s∈I, t∈J, which is obtained by a null generator L(s) moving with normalized Cartan frame of a null curve a(s) in Minkowski 3-space R3. We also calculated the first fundamental form Ⅰ, the second fundamental formⅡ, the Gaussian curvature K, the mean curvature H of the extended B-scrolls, and the Gaussian curvature K and the mean curvature H satisfy K=-H2. And we also obtained the parameter of the extended B-scrolls with prescribed mean curvature H. Last, we given some examples of the extended B-scrolls and drew the graphs.