On Cyclic Surfaces With Restricted Gaussian Curvature

In this thesis, the geometry properties of cyclic surfaces in Eucilidean 3-space is mainly studied. Let n = n(s) be the smooth unit vector field of each plane including s-circle.Then cyclic surface is parametrized bywhere a=n(s),b= n'(s),c=n(s)∧n'(s), p - p(s) and r = r(s) denote respectively the radius and center of each s-circle, and s is the arc parameter of n(s), t is the radian of each s-circle.Let T = a,N = b cos t + c sin t,B = -b sin t + c cos t,choose moving frame {X;T,N,B}. Then use this moving frame to compute the coefficients of the first fundamental form E, F, G and the coefficients of the second fundamental form L, M, N. Let us denote by [,, ] the mixed product in R³ and put W = EG - F²,then the Gauss curvature H can write aswhere K₁ = [X_s,X_t,X_ss][X_s,X_t,X_tt] - [X_s,X_t,X_st]².It is intended to study the cyclic surfaces that satisfy (?)K/(?)t = 0. It is easy to check that K_1tW - 2K₁W-t = 0 is equivalent to (?)K/(?)t = 0. So it is sufficient to study the cyclic surfaces that satisfy K_1tW - 2K₁W_t = 0. Expand K_1tW - 2K₁W_t into Fourier expansion about t. ThenAll coefficients E_i,F_i are smooth function on s. So K_1tW - 2K₁W_t = 0 holds if and only if E₀ = E_i = F_i = 0, i =1,2,3,4.The main results of this paper are:Proposition 1 Suppose that the non-sphere cyclic surface S satisfies the type (?)K/(?)t = 0 , then the planes containing the circles of the foliation are parallel.Proposition 2 Suppose that cyclic surface S is foliated by pieces of circles lying in parallel planes,and (?) = 0 , then(1)If K≠0,then S is a surface of revolution.(2)If K≠0,then S can be parameterized as S is a quadratic cone or a elliptic cylinder.