A Type Of Weingarten Surfaces With Prescribed Principal Curvatures

In this thesis, we mainly discuss a new type of Weingarten surfaces. First we introduce the method of the construction of this new type of Weingarten surfaces. Then based on [1], we obtain a differential relation on the principal curvatures of this type of surfaces, as well as the existence of this type of Weingarten surface with prescribed two smooth functions satisfied the differential relation as the principal curvatures (see the theorems below).Theorem4.1 Let C : r(s) = (x(s),0,z(s)) be a plane curve parametrized by arc length s, x(s) > 0, s ∈ [0, L]. A = A(t) be a smooth curve in 50(3), t E [0, α]. Then f(t, s) and g(s), as the principal curvatures of surface 5 : X(t, s) = r(s)A(t), satisfyingTheorem4.2 Let a(t), c(t) be two smooth functions denned on [0, α], satisfying a(t)≠ 0, a(0) = -1, c(0) = 0;f(t, s),g(s) be two smooth functions, satisfyingand f(t, s) ≠ g(s), for all s E [0, L],t E [0, α]. In addition, we supposethen there is a surface in R³ with principal curvatures f(t,s) and g(s), and s is the arc length of its generating curve.For the more, we give two examples of this new type of Weingarten surfaces. In example 1, A₁(t) construct surface S₁. And as a curve in 50(3), the relative curvature of it k = 0 and the relative torsion r = 1/2. The surface of revolution in [1] is a special situation when λ = 0,a(t) ≡ - 1. In example 2, A₂(t) construct surface S₁. And as a curve in 50(3), the relative curvature of it k = 1 and the relative torsion Ï„ = 1/2. This is a really new Weingarten surface different from the surface of revolution. So our conclusion is the generalization of [1].