三維黎曼空間中曲面的平均曲率及第二基本形式

<正> §1.T.Y.Thomas在三維歐氏空間的曲面論中將一般曲面的第二基本形式的係數用平均曲率及測度張量代數地表示出來.作者利用了高維歐氏空間超曲面的變形理論在高維歐氏空間的可變形超曲面上也得到同樣的結果,因此得知:對於歐氏空間的可變形超曲面一般地可以由測度張量及平均曲率完全予以確定.在該文的證明中曾經推廣T.Y.Thomas的結果到三維常曲率空間的曲面論.本文的目的是關

DETERMINATION OF THE SECOND FUNDAMENTAL FORM OF A SURFACE V_2 IN A RIEMANNIAN SPACE V_3 BY ITS MEAN CURVATURE

T. Y. Thomas has shown that in general the mean curvature of a surface yields an algebraic determination of its second fundamental form b_(ii)dx~idx~i and has given the explicit ex pression of b_(ii) in terms of the metric tensor g_(ii), the mean curvature H and their derivatives.In this paper we consider the same problem for the surface V_2 in 3 dimensional Riemannian space V_3.At first, we consider the isometric correspondence which preserves the mean curvature between pairs of surfaces in V_3. By solving a system of exterior differential equations we find that the degree of freedom of such pairs of surfaces is 4 functions of single argmefit. Consequently, in general V_2 is determined by the mean curvature and the metric tensor.In order to get the expression of b_(ii), we choose such a coordinate by which the equation of V_2 is y~3=0 and the normal of the surface is ξ~α(0,0,1) then the equations of Peterson-Gauss Codazzi becomes R_(1212)=(b_(11)b_(22)-b_(12)~2+R_(1212)b_(ij,k)-b_(ik,i)=R_(iεik) (i, i, k = 1, 2)From the condition of inttgrability of b_(ij,k) and using the similar but much more complicated calculation as T. Y. Thomas we get A 2 b_(12) + B (b_(11)-b_(22)) = C, A (b_(22)-b_(11)) + B (2 b_(12)) = D. where A,B,C,D stand for the expression in (17), (18), (19), (20). Here also we notice that when B=C=D=0 it corresponds to the work of T. Y. Thomas of V_2 E_3.We discuss the problem at any arbitrary paint P and choose the system of coordinate to be orthogonal at that point.Let △=4b_(12)~2+(b_(11)-b_(22))~2=4(H~2-K). If △ = 0, we get the expression of b_(ii) in the following form b_(ii)= H g_(ii). Under the supposition of △≠0, we get a b_(12)~2 + β (b_(11)-b_(22)) = γ where α,β, γ, stand for the expression in (24).When α≠0, b_(ii) are expressed by (26),When α=0, there are three cases(i) βγ≠0(ii) γ = 0 β≠ 0(iii) β= 0, then γ= 0.In the case (i) and (ii), we can also express b_(ii) by the mean curvature and the coefficients of first fundamental form and their derivatives. In the case (iii), we have H~2-K=const. generally. All the surfaces which admit continuous deformation which preserves mean curvature are included in this case.We also obtain the equations which must be satisfied by the mean curvature.