LetKbe the restricted contact Lie algebras K(n, 1) over an algebraically closed field F of characteristic p >3. We prove that each skew-symmetric biderivation of K is inner and show that commuting maps on K are scalar multiplication maps. Moreover, it is showed that the commuting automorphisms and dertivations of K are proved to be the identity mappings and zero mappings, respectively. Meanwhile, the commutative post-Lie algebra structures on K are given.

In this paper, we focus on a developable surface tangent to a timelike surface along a curve in Minkowski 3-space, which is called the osculating developable surface of the timelike surface along the curve. The ruling of the osculating developable surface is parallel to the osculating Darboux vector field. The main goal of this paper is to classify the singularities of the osculating developable surface. To this end, two new invariants of curves are defined to characterize these singularities. Meanwhile, we also research the singular properties of osculating developable surfaces near their lightlike points. Moreover, we give a relation between osculating Darboux vector fields and normal vector fields of timelike surfaces along curves from the viewpoint of Legendrian dualities. Finally, some examples with symmetrical structures are presented to illustrate the main results.

In the theory of cosmology, de Sitter space is the symmetrical model of accelerated expansions of the universe. It is derived from the solution of the Einstein field equation, which has a positive cosmological constant. In this paper, we define the evolutes and focal surfaces of timelike Sabban curves in de Sitter space. We find that de Sitter focal surfaces can be regarded as caustics and de Sitter evolutes corresponding to the locus of the polar vectors of osculating de Sitter subspaces. By using singularity theory, we classify the singularities of the de Sitter focal surfaces and de Sitter evolutes and show that there is a close relationship between a new geometric invariant and the types of singularities. Moreover, the Legendrian dual relationships between the hyperbolic tangent indicatrix of timelike Sabban curves and the focal surfaces are given. Finally, we provide an example to illustrate our main results.