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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**

**Sample text**

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.