ON THE RlCCl TENSOR OF REAL HYPERSURFACES OF QUATERNIONIC PROJECTIVE SPACE

We study some conditions on the Ricci tensor of real hypersurfaces of quaternionic projective space obtaining among other results an improvement of the main theorem in [9].

QP', see [2].Then U, d,(, 1, 2, 3, are tangent to M. Let S be the Ricci tensor of M. In [6] we studied pseudo-Einstein real hypersurfaces of Qpm.These are real hypersurfaces satisfying qX aX + bL,=IB(.X U,)U, (1.1) for any X tangent to M, where a and b are constant.If m _> 3 we obtained that M is pseudo-Einstein if it is an open subset of either a geodesic hypersphere or of a tube of radius r over Qpk, 0 < k < rn 1, 0 < r < n/2 and co2r (4k + 2)/(4m 4k 2).
As a corollary we also obtained that the unique Einstein real hypersurfaces of Qp,n, rn >_ 2, are open subsets ofgeodesic hyperspheres of QP' of radius r such that co9r 1/2m.
The purpose of the present paper is to study several conditions on the Ricci tensor of M. Concretely in 3 we prove the following result if X is tangent to M we shall write J,X==X+fi(X),i= 1,2,3, where ,I,,X denotes the tangent component of J/X and f,(X) g(X, U,).Then THEOREM 1.Let M be a real hypersurface of QP', m _> 3, such that b,S SI,,, Then M is an open subset of a tube of radius r, 0 < r < YI/2, over QPk, k E {0, , m 1}.
This theorem generalizes results obtained by Pak in [7].In [9] we studied real hypersurfaces of Qpm with harmonic curvature for which U,, 1, 2, 3, are eigenvectors of the Weingarten endomorphism of M with the same principal curvature.A real for any X, Y tangent to M, where , denotes the covariant differentiation of M In 4 we shall improve the result of [9] showing that the condition about principality of U,,i 1,2,3, is unnecessary Concretely we obtain THEOREM 2. A real hypersurface of QP', m _> 2, has harmonic curvature it" and only if it is

Einstein
As a consequence we can classify Ricci-parallel real hypersurfaces of Qpm, that is, real hypersurfaces such that q7 x S 0 for any X tangent to M. We get COROLLARY 3. The unique Ricci-parallel real hypersurfaces of Qpm, m _> 2, are open subsets of geodesic hyperspheres of radius r, 0 < r < 7r/2, such that corer 1/2m.
From this result we introduce in 5 a condition that generalize Ricci-parallel real hypersurfaces We shall say that a real hypersurface of Qpm is pseudo Ricci- for any X, Y tangent to M, c being a nonnull constant We obtain THEOREM 4. M is a pseudo Ricci-parallel real hypersurface of Qpm, m _> 2, if and only if it is "an open subset of a geodetic hypersphere.
As M has harmonic curvature for any X, Y tangent to M we get V x SY 27 rSX S([X, Y]) (4.1) Then for any X, Y, Z tangent to M we obtain R(Z, X)SY , z V xSY-x 7 z SY r7 [z,xl SY S(R(Z,X)Y) + V z( rS)X + V zS)( V x Y) <7 X( V yS)Z where R denotes the curvature tensor of M. From (4.2), (1.2) and the first identity of Bianchi we get g(R(X, Y)SZ) 0 (4.3) for any X, Y, Z tangent to M, where cr denotes the cyclic sum.The result now/bllows from the main theorem of [8].