By F Defever; J M Morvan; et al (eds.)
This quantity offers a scientific and unified method of the research, id and optimum keep watch over of continuous-time dynamical structures through orthogonal polynomials (such as Legendre, Laguerre, Hermite, Tchebycheff, Jacobi and Gegenbauer) and through orthogonal capabilities reminiscent of sine-cosine, block-pulse, and Walsh. This publication concentrates at the program of orthogonal polynomials in platforms and keep an eye on and goals to set up the prevalence of orthogonal polynomials over different orthogonal features
Read Online or Download Geometry and topology of submanifolds, IX : dedicated to Prof. Radu Rosca on the ocasion of his 90th birthday, Valenciennes, France, 26-27 March, Lyon, France, 17-18 May, Leuven, Belgium, 19-20 September, 1997 PDF
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Additional resources for Geometry and topology of submanifolds, IX : dedicated to Prof. Radu Rosca on the ocasion of his 90th birthday, Valenciennes, France, 26-27 March, Lyon, France, 17-18 May, Leuven, Belgium, 19-20 September, 1997
Example text
Hicks, Linear perturbations of connections, Mich. Math. J. 12 (1965), 389-397. [KOB-N] S. Kobayashi and K. Nomizu, Foundations of Differential Geometry I, II, Interscience Publis hers (1962, 1969). [KOW] O. Kowalski, A classification of Riemannian 3-manifolds with constant principal Ricci curva tures p\ = pi ^ p3, Nagoya Math. J. 132 (1993), 1-36. [L-S-V-W] H. L. Liu, U. Simon, L. Verstraelen and C. P. Wang, The third fundamental hypersurfaces in nonflat space forms, J. of Geometry, to appear.
Corollary. Let M 2 (7) be a closed, orientable surface of genus 7 > 1. p(n) = 1. Proof. Consider the principal curvatures fcj with eigenvectors Ei of S. 1) IC(Ei, Ej) = 1 + hkj. 3) there is a point p € M such that 0 = (detS)(p) = *i(p)fc2(p)fc3(p), thus at least one of the principal curvatures vanishes at p € M. 1) then gives the assertion. 4. Four-dimensional manifolds. 1. Lemma. Let x,x" : M 4 -> S 5 (l) be a polar pair. (i) -Ufa. = - £ « is a polarization invariant; 44 (ii) at p € M , K" = 1 if and only if K = 1.
2, since A = 0 if and only if / " = 0 and since / ' = b = const is the unique solution of (18) for the given initial conditions. 1 we can assume ||T||2 ^ 0. From 0 =/= const = A = nf" we get /'(t) = n\t + c. Because of K = 0 and (8) this surface would be regular. The linear / ' leads to a contradiction in (18). ii \\C\\2 = const is equivalent to r' = 0 or w = 3 / " = const, which in turn is equivalent to K = 0. 3 Singular Tchebychev surfaces Singularity leads to the second order ODE r = \f3-vf (19) for /'.