The E. Study Maps of Circles on Dual Hyperbolic and Lorentzian Unit Spheres <i>H</i>²₀ and <i>S</i>²₁

On pseudohyperbolic space motions

“…The geodetic curvatures of the moving and the fixed polodes are 17) respectively, whereQ andQ −Ŵ are the elements of the geodetic curvatures of the polodes.…”

Tangent Bundle of Pseudo-sphere and Ruled Surfaces in Minkowski 3-space

“…We extend the E. Study correspondence map to the tangent bundles of the Lorentzian (or de Sitter) unit sphere S 2 1 and hyperbolic unit sphere H 2 as: There exists a one-to-one correspondence between any smooth curve Γ on T S 2 1 and a space-like or time-like ruled surface in E 3 1 , and a one-to-one correspondence between any smooth curve Γ on T H 2 and a time-like ruled surface in E 3 1 , respectively, see Propositions 3 and 3. We devote our study to the developability conditions of these ruled surfaces (with different causal characters) depending to their corresponding smooth curves on T S 2 1 and T H 2 to be Legendre or involute-evolute curve couples (see [4], [11], [12]). The paper is presented as follows: In section 2, we give some notions and definitions of "the tangent bundles of Lorentzian (or de Sitter) unit sphereS 2 1 and hyperbolic unit sphere H 2 ", "ruled surfaces" and "Legendre curves".…”

Evolutes of Hyperbolic Dual Spherical Curve in Dual Lorentzian 3-Space

“…Detailed descriptions are found in the books [11,12]. Let F : (D × D r , (S 0 , X 0 )) → D be a dual smooth function, and F(S) = F X0 , F X0 (S) = F (S, X 0 ).…”

“…Let the (k − 1)-jet of the partial derivative ∂F ∂Xi at S 0 be j The bifurcation dual set B F of F is the set [11,12]:…”

“…More details on the necessary basic concepts about the dual elements and the one-to-one correspondence between ruled surfaces and one-parameter dual spherical motions can be found in [7][8][9][10]. If we take the Minkowski 3-space E [12]. Then a differentiable curve on H 2 + corresponds to a timelike ruled surface at E 3 1 .…”