Read e-book online Global Differential Geometry and Global Analysis: PDF

By D. Ferus, U. Simon, U. Pinkall, B. Wegner

ISBN-10: 0387547282

ISBN-13: 9780387547282

All papers showing during this quantity are unique study articles and feature now not been released somewhere else. They meet the necessities which are valuable for e-book in a quality basic magazine. E.Belchev, S.Hineva: at the minimum hypersurfaces of a in the community symmetric manifold. -N.Blasic, N.Bokan, P.Gilkey: The spectral geometry of the Laplacian and the conformal Laplacian for manifolds with boundary. -J.Bolton, W.M.Oxbury, L.Vrancken, L.M. Woodward: minimum immersions of RP2 into CPn. -W.Cieslak, A. Miernowski, W.Mozgawa: Isoptics of a strictly convex curve. -F.Dillen, L.Vrancken: Generalized Cayley surfaces. -A.Ferrandez, O.J.Garay, P.Lucas: On a definite category of conformally flat Euclidean hypersurfaces. -P.Gauduchon: Self-dual manifolds with non-negative Ricci operator. -B.Hajduk: at the obstruction team toexistence of Riemannian metrics of optimistic scalar curvature. -U.Hammenstaedt: Compact manifolds with 1/4-pinched adverse curvature. -J.Jost, Xiaowei Peng: The geometry of moduli areas of reliable vector bundles over Riemannian surfaces. - O.Kowalski, F.Tricerri: A canonical connection for in the neighborhood homogeneous Riemannian manifolds. -M.Kozlowski: a few fallacious affine spheres in A3. -R.Kusner: A greatest precept at infinity and the topology of whole embedded surfaces with consistent suggest curvature. -Anmin Li: Affine completeness and Euclidean completeness. -U.Lumiste: On submanifolds with parallel larger order basic shape in Euclidean areas. -A.Martinez, F.Milan: Convex affine surfaces with consistent affine suggest curvature. -M.Min-Oo, E.A.Ruh, P.Tondeur: Transversal curvature and tautness for Riemannian foliations. -S.Montiel, A.Ros: Schroedinger operators linked to a holomorphic map. -D.Motreanu: commonplace lifestyles of Morse capabilities on countless dimensional Riemannian manifolds and purposes. -B.Opozda: a few extensions of Radon's theorem.

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Extra resources for Global Differential Geometry and Global Analysis: Proceedings, Berlin 1991

Example text

First, we assume that 111 is not empty. Then 111 is a non empty open part of M and in each point p of Ill, we know that (Vh)p = 0. The classical Pick-Berwald theorem then implies that 111 is an open part of a nondegenerate ellipsoid or hyperboloid. Thus detS is a constant different from zero on 111" The continuity of detS then implies that fit = M. Finally, we may assume that S = 0 on the whole of M. Thus by Proposition 2, we can suppose that M is given by the equation z -- P(x,y), where P is a polynomial of degree at most k + 1, and that the canonical affine normal vector field is given by (0,0,1).

N be the Frenet frame of its directrix curve ~o" through ~p2 if and only if (i) n = 2m, (ii) ~ = ~m, (iii) ~o(z) Theorem 5 where z is a stereographic complex coordinate. - ~n(-I/~) be a linearly full minimal L e t %b : S 2 --+ CP n, n - 2m, immersion which factors through ~P~, ~o is k-point ramified for k < 2. and suppose that the directrix curve Let z be a stereographic complex coordinate such that if k ~ 0 then 4o has a higher order singularity at z - O. Then ~o is given by g/o(Z) = /kp zk' + ' " "+ kp ep (t) p-O where e_o .

Remark The condition of factoring through RP 2 is crucial here. result is not true in general. [l]). This, together with Theorem 2, is a first step in dealing with the space of minimal immersions of S ~ into Sn. Finally, combining Theorems 2 and 6, we have the following partial generalisation of Theorem 5. Theorem 7 Let # : S ~ --+ CP n be a linearly full minimal immersion which factors through Rp2. Then up to holomorphic isometries of CP n there are only finitely many minimal immersions of S 2 into CP n which factor through ~p2 and have the same singularity type as ~.

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Global Differential Geometry and Global Analysis: Proceedings, Berlin 1991 by D. Ferus, U. Simon, U. Pinkall, B. Wegner

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