三维双曲空间中平行曲面族的两个定理

TWO THEOREMS FOR THE FAMILIES OF PARALLEL SURFACES IN IH^3

设 M 是三维双曲空间 H^3中的光滑曲面,M 的两个主曲率为λ_1和λ_2.设{M_t}是 M的平行曲面族(-ε<t<ε),M_t 的两个主曲率为λ_1(t)和λ_2(t).本文得到两个结果定理1 M 有常主曲率的充要条件是 sum from i=1 to 2 λ_i^k(t)只是 t 的函数(k=1,2).定理2 设λ_1λ_2≠0,且λ_1(λ_1~2+1)(λ_2~4+1)≠λ_2(λ_2~2+1)(λ_1~4+1),则 M 具有常主曲率的充要条件是每个 M_(?)有常值的 Gauss 曲率.

Let M be a c~∞ surface isometrically immersed in H^3.λ_1 and λ_2 are twoprincipal curvatures of M.Let{M_t} be parallel surfaces of M.(-ε<t<ε).λ_1(t)and λ_2(t)are two principal curvatures of M_t.We obtainTheorem 1 M has constant principal curvatures if and only if (?) λ_i^k(t)is a constant for fixed t.(k=1,2).Theorem 2 Let λ_1λ_2≠0 and λ_1(λ_1~2+1)(λ_2~4+1)≠λ_2(λ_2~2+1)(λ_1~4+1),Then Mhas constant principal curvatures if and only if each M_t has constant Gaus-sian curvature.