Download Geometry and topology of submanifolds, IX : dedicated to by F Defever; J M Morvan; et al (eds.) PDF

By F Defever; J M Morvan; et al (eds.)

This quantity offers a scientific and unified method of the research, id and optimum keep an eye on of continuous-time dynamical structures through orthogonal polynomials (such as Legendre, Laguerre, Hermite, Tchebycheff, Jacobi and Gegenbauer) and through orthogonal features reminiscent of sine-cosine, block-pulse, and Walsh. This booklet concentrates at the program of orthogonal polynomials in structures and keep an eye on and goals to set up the prevalence of orthogonal polynomials over different orthogonal capabilities

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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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Extra 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

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I). 4). 3. Lemma. Let x, x* be polar. , p* of (M, /*) are constant. 4. Corollary. Let x, x* be a polar pair and n = 3. Then the following statements are equivalent: (i) (M, I) is curvature homogeneous and H\ = constant; 42 (ii) (M, I*) is curvature homogeneous and H* = constant. 5. Corollary. Let x,x* : M -t S " + 1 ( l ) be polar hypersurfaces of dimension n > 3. Then the following statements are equivalent; (i) x is umbilical; (ii) x" is umbilical; (iii) (M, I) has constant sectional curvature; (iv) (M, f ) has constant sectional curvature.

There is no classification so far, but there are many examples satisfying these conditions, and we give isoparametric and non-isoparametric examples. For a better understanding, recall the following properties of the Veronese surface V2 C §4(1). l. The Veronese surface in S 4 (l). (i) V2 has constant Gauss curvature K, of the metric: K = f; (ii) V 2 is minimal in S4; (iii) let £ be an arbitrary local normal section and S(£) its Weingarten operator with associated bilinear symmetric Weingarten form 5(f).

U„-l) where 7(s) = (7i(s),72(s),73(s)) (See [4] for the case 7 is a plane curve) . Let {'Y(s) = Vi,V2,v3} be the oriented Frenet frame along 7 and Ki(s) and K2(s) be the Frenet curvatures of 7. ,0), where 1 occurs at the (j + 2) -nd place. Now we obtain that V x , xs = 0, VXlsxu = V ^ xs = 0 and V x „ xu = 0; and also that i fc JVi(s) = h{x„x,) h{xt,xu) = KI(S)I^(S), = h ( a v , x u ) = 0. Form these and Frenet formulas N2(s) = (Vx. h)(x„x,) = «i(s)i;2 + KI(S)K 2 (S)V 3 , 26 and all other derivatives of h are zero.

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