By Janos Kollar
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Extra info for Complex Algebraic Geometry (Ias Park City Mathematics Series, V. 3) PCMS 3
18, as was first shown by Nomizu . 19. Let f W M n ! M space form. Then for each principal curvature , the leaves of the principal foliation Q nC1 and totally geodesic in M n . 18, and it is useful in the study of Dupin hypersurfaces. 9 that if a principal curvature has constant multiplicity m 1, then is constant along each leaf of T in the domain U of f if and only if f itself is constant along each leaf of T in U. In that case, the curvature sphere map K is also constant along each leaf of T in U.
6. M; 0 Riemannian manifolds with g . X; Y/ for all X; Y tangent to Q at x. Let M be an oriented hypersurface in M, Q and let be a smooth principal M curvature function of constant multiplicity m on M. 4 Focal Submanifolds 21 x τ (x) P Rk Fig. 1 Stereographic projection D e h. grad h; // is a smooth principal curvature function of multiplicity m on respective principal distributions of and coincide on M. 7 (Stereographic projection and inversions in spheres). 6 to the case of hypersurfaces in real space forms by considering the conformal transformation given by stereographic projection from Sk or H k into Rk , for any positive integer k.
X/, where ranges over the principal curvatures of M at x that are not equal to . x/, for ¤ . x/. x/ for each ¤ for all x 2 . This occurs precisely when Á D 0 for all Á 2 T ? x/ for all x 2 . Since is assumed constant along , this happens precisely when assumes a critical value along . 18, as was first shown by Nomizu . 19. Let f W M n ! M space form. Then for each principal curvature , the leaves of the principal foliation Q nC1 and totally geodesic in M n . 18, and it is useful in the study of Dupin hypersurfaces.
Complex Algebraic Geometry (Ias Park City Mathematics Series, V. 3) PCMS 3 by Janos Kollar