Certain Conditions on the Ricci Tensor of Real Hypersurfaces in Quaternionic Projective Spaces

The purpose of this paper is to classify real hypersurfaces of quaternionic projective spaces whose Ricci tensor satisfy a pair of conditions on the maximal quaternionic distribution D? = Span fU1; U2; U3g. x0. Introduction Throughout this paper let us denote by M a connected real hypersurface in a quaternionic projective space QP, m=3, endowed with the metric g of constant quaternionic sectional curvature 4. Let N be a unit local normal vector eld onM and Ui = JiN , i = 1; 2; 3, where fJigi=1;2;3 is a local basis of the quaternionic structure of QP, [3]. Then Ui = JiN , i = 1; 2; 3 are tangent to M . Recently, the rst author [7] has studied and classi ed real hypersurfaces such that the Ricci tensor satis es (1:1) (rXS)Y = c X3 i=1 g( iX;Y )Ui + fi(Y ) iX for any X, Y tangent to M , c being a nonzero constant where r is the covariant derivative onM and iX denotes the tangential component of JiX and fi(X) = g(X;Ui). Also in the same paper [7] real hypersurfaces of QP , m=3 such that iS = S i, i = 1; 2; 3 are classi ed. Now let us de ne a distribution D by D(x) = X 2 TxM : X?Ui(x), i = 1; 2; 3 , x 2M , of a real hypersurface M in QP, which is orthogonal to the structure vector elds fU1; U2; U3g and invariant with respect to structure tensors f 1; 2; 3g. So the distribution D is said to be the maximal quaternionic distribution on M , and D = Span fU1; U2; U3g is its orthogonal complement in TM . Equation (1.1) restricted to vector elds on D gives us (1:2) (rXS)Y = c X3 i=1 g( iX;Y )Ui This work was supported by the grant from Korea Research Foundation, Korea, 1999, KRF-99-015-DI0009. 0236{5294/1/$ 5.00 c 2001 Akad emiai Kiad o, Budapest 344 J. DE DIOS P EREZ and YOUNG JIN SUH for any X;Y 2 D, c being a nonzero constant. The purpose of the present paper is to classify real hypersurfaces of QP, m=2 such that (1.2) is satis ed for any X;Y 2 D and (1:3) (S i iS)X = i(X)Ui; i = 1; 2; 3 for any X 2 D, i being a 1-form on M , i = 1; 2; 3, is also satis ed. Our result is Theorem. Let M be a real hypersurface of QP, m=2, satisfying (1:2) and (1:3). Then M is congruent to an open subset of either i) a geodesic hypersphere, or ii) a tube of radius r, 0 < r < 2 , over a totally geodesic QP , k 2 f1; : : : ;m 2g such that cot r = (2k + 1)=(2m 2k 1). x2. Preliminaries Let X be a tangent vector eld to M . We write JiX = iX + fi(X)N , i = 1;2;3, where iX is the tangent component of JiX and fi(X) = g(X;Ui), i = 1; 2; 3. As J i = id, i = 1; 2; 3, where id denotes the identity endomorphism on TQP, we get (2:1) 2 i X = X + fi(X)Ui; fi( iX) = 0; iUi = 0; i = 1; 2; 3 for any X tangent to M . As JiJj = JjJi = Jk, where (i; j; k) is a cyclic permutation of (1; 2; 3) we obtain (2:2) iX = j kX fk(X)Uj = k jX + fj(X)Uk