ON THE GEOMETRY OF CLOSED TIMELIKE
RULED SURFACES IN DUAL LORENTZIAN SPACE
Abstract. In this paper, a dual timelike curve $c(t)$ which is a Lorentzian spherical indicatrix of a timelike closed ruled surface $\left({\overrightarrow{V}}^*_1(t)\right)$ with a real parameter $t$, two frames $D^*({\overrightarrow{V}}^*_1,{\overrightarrow{V}}^*_2,{\overrightarrow{V}}^*_3)$ related to the timelike closed ruled surface $\left({\overrightarrow{V}}^*_1(t)\right)$ and $D({\overrightarrow{V}}_1,{\overrightarrow{V}}_2,{\overrightarrow{V}}_3)$ which moves respect to $D^*$ and related to a timelike closed ruled surface drawn by a timelike vector ${\overrightarrow{V}}_1(t)$, and a timelike vector $\overrightarrow{V}$ which is fixed in the frame $D$ are considered; and dual integral invariants of the timelike closed ruled surfaces which correspond to the dual timelike closed curves drawn by the vectors $\overrightarrow{V},\ {\overrightarrow{V}}_1$ and ${\overrightarrow{V}}^*_1$ are studied; and it is found same relations among the dual integral invariants of the timelike closed ruled surfaces which correspond to the dual timelike closed curves drawn by the timelike vectors $\overrightarrow{V},\ {\overrightarrow{V}}_1$ and ${\overrightarrow{V}}^*_1$. In addition, these results are carried to the Lorentzian line space ${\mathbb {R}}^3_1$ and give some theorems by means of Study's mapping.
AMS Subject Classification: 53A17, 53B30


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DOI: 10.12732/ijam.v29i1.2

Volume: 29
Issue: 1
Year: 2016