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According to E. Study map in Minkowski space, we give in this present paper, a one-to-one correspondence between "the curves on tangent bundles of Lorentzian (or de Sitter) unit sphere TS21 and hyperbolic unit sphere TH2" and "the space-like or time-like ruled surface in E31 ". In fact, we can consider each curve on TS21 or TH2 as a ruled surface in E31 . Moreover, we study the relationships between the developability conditions of these corresponding ruled surfaces in E31 and their base and striction curves, and we show that if the curves on TS21 or TH2 are involute-evolute couples, then the corresponding ruled surfaces are developable.

Tangent bundle of pseudo-sphere and ruled surfaces in Minkowski 3-space