By Pfeffer, Riemannian
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First let us assume that T3 = 0 and T 2 ^ 0. Then there exist vectors X\,X-i, X3 and such that K(K(Xi,X2),X3) = dv. Therefore T3 = 0 implies that K(dv,X) = 0 for any vector field X. By the assumption, there exist also vector fields Xi and X 2 such that K(Xi,X2) = 9u> + xdv. Since T3 = 0, we deduce that 0 = K{K(XUX2), K(XUX2)) = K(dw,dw). Also, 0 = h(K(du,dv),8u) = h(K(du,du),dv) 0 = h(K(dw,dv),8u) = h{K(du,dw),dv) 0 = h{K(dw, dw), du) = h(K(du, dw), dw). 3) we have r = h(K(du,du),dw) = h(K(du,dw),du) = t(l + h(u)wf.
Magid and P. Ryan, Flat affine spheres, Geometriae Dedicata 33 (1990), 277-288. , Affine 3-spheres with constant affine curvature, Trans. Amer. Math. Soc. 330 (1992), 887-901. K. Nomizu and T. Sasaki, Affine Differential Geometry, Cambridge University press, Cambridge, 1994. J. Radon, Zur Affingeometrie der Regelflachen, Leipziger Berichte 70 (1918), 147-155. U. Simon, Local classification of two-dimensional affine spheres with constant curvature metric, Differential Geom. Appl. 1 (1991), 123-132.
Hence M is a flat improper affine sphere. 4 of [DV2] that M is affine equivalent with the hypersurface y = ju) 2 + zx + i z 3 . 2: VK does not vanish identically, but (VK)(X 3 , X\, Xi) = 0. 3. So in this case, a2 = 0. 3) that \ = h(R(XuX3)X3,X2) = h(Vx, (-C3X2) - V x , ( - a 3 X 2 ) - V-a,x2-o3x3X3,X2) = 0. 37 Thus A = 0 and M is a flat improper afiine sphere. 6) = VxA-aiX2) = -X2(ai)X2. 7) = ( - X 1 ( 6 3 ) + X 2 (a 3 )-63C3)X 2 . 8) X2(7) = -63, * s ( 7 ) = -C3. 8) exists. We now consider the following change of frame: Y1=Xi-\12X2+1X3, Y2=X2, Y3 = X3-yX2.