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# 4 - Mathematical Communications 6(2001 39-45 39...

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Mathematical Communications 6 (2001), 39-45 39 Hyperspherical curves in n -dimensional k -isotropic space I k n ˇ Zeljka Milin ˇ Sipuˇ s Abstract . In this paper we give the characterization of hyperspher- ical curves in n -dimensional k -isotropic space I k n . Key words: n -dimensional k -isotropic space, osculating hypersphere, hyperspherical curve AMS subject classifications: 53A35 Received July 15, 2000 Accepted March 10, 2001 1. Introduction The n -dimensional k -isotropic space I k n is introduced in [5] where it is defined as a pair ( A, V ) where A is a real n -dimensional aﬃne space and V its corresponding vector space decomposed in a direct sum of subspaces V = U 1 U 2 , dim U 1 = n k , dim U 2 = k . The space U 1 is endowed with a scalar product · : U 1 × U 1 R which is extended on the whole V by x · y = π 1 ( x ) · π 1 ( y ) , where π 1 : V U 1 denotes the canonical projection. In such a way a semi-definite scalar product on V is defined. In this paper we describe the osculating hyperspheres of an admissible curve in the space I k n . The theory of curves in I k n is developed in [2]. Furthermore, we study the conditions under which an admissible curve is hyperspherical. 2. Osculating hypersphere in I k n As it is shown in [5], a hypersphere in I k n is defined in aﬃne coordinates by the equation n k i =1 x 2 i + 2 n i =1 α i x i + α 0 = 0 , (1) Department of Mathematics, University of Zagreb, Bijeniˇ cka c. 30, HR- 10 000 Zagreb, Croa- tia, e-mail: [email protected]

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40 ˇ Z. Milin ˇ Sipuˇ s where α 1 , . . . , α n R . We distinguish the following types of hyperspheres in I k n ([5], Theorem 7.1). If ( α n k +1 , . . . , α n ) = (0 , . . . , 0), then by an isotropic motion we obtain the normal form of a parabolic hypersphere of type l , l ∈ { 0 , . . . , k 1 } , n k i =1 x 2 i + α n l x n l = 0 , Its radius is defined by α n - l 2 , α n l = 0. If α n k +1 = · · · = α n = 0, then by an isotropic motion we get a cylindrical hypersphere n k i =1 x 2 i = r 2 . Its radius is defined by r . Definition 1. Let c : I I k n be a regular C r -curve and f ( x ) = 0 a regular C r -hypersurface, r 1 . A point P 0 ( t 0 ) is a point of contact of r th order of the curve c and the hypersurface f if the function F ( t ) = f ( c ( t )) satisfies F ( t 0 ) = . . . = F ( r ) ( t 0 ) = 0 , F ( r +1) ( t 0 ) = 0 . Definition 2. Let c be an admissible C r -curve in I k n , r n , P 0 a point of c . A hypersphere which has contact of n th order with the curve c in P 0 is called the osculating hypersphere. We can write equation (1) of a hypersphere in the following form. Let P 0 = ( p 1 , . . . , p n ) be a point of hypersphere (1). Then n k i =1 p 2 i + 2 n i =1 α i p i + α 0 = 0 . (2) Subtracting (2) from (1) we get n k i =1 ( x 2 i p 2 i ) + 2 n i =1 α i ( x i p i ) = 0 , which can be written as n k i =1 ( x i p i ) 2 + 2 n k
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4 - Mathematical Communications 6(2001 39-45 39...

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