A characterization of pseudo-Einstein real hypersurfaces in a quaternionic projective space

Abstract

Let HPn be a quaternionic projective space, n = 3, with metric G of constant quaternionic sectional curvature 4, and let M be a connected real hypersurface of HPn. Let ξ be a unit local normal vector field on M and {I, J,K} a local basis of the quaternionic structure of HPn (cf. [4]). Then U1 = −Iξ, U2 = −Jξ, U3 = −Kξ are unit vector fields tangent to M . We call them structure vectors. Now we put fi(X) = g(X, Ui), for arbitrary X ∈ TM , i = 1, 2, 3, where TM is the tangent bundle of M and g denotes the Riemannian metric induced from the metric G. We denote D and D⊥ the subbundles of TM generated by vectors perpendicular to structure vectors, and structure vectors, respectively. There are many theorems from the point of view of the second fundamental tensor A of M (cf. [1], [8] and [9]). It is known that if M satisfies g(AD,D⊥) = 0 then there is a local basis of quaternionic structure such that structure vectors are principal vectors. Berndt classified the real hypersurfaces which satisfy this condition (cf. [1]). On the other hand we know some results on real hypersurfaces of HPn in terms of the Ricci tensor S of M (cf. [3] and [8]). If the Ricci tensor satisfies that SX = aX + b ∑3 i=1 fi(X)Ui for some smooth functions a and b on M , then M is called a pseudo-Einstein real hypersurface of HPn. This notion comes from the problem for the real hypersurfaces in complex projective space CPn. Kon studied it under the assumption that they have constant coefficients (cf. [5]) and Cecil and Ryan gave a complete classification (cf [2]). In [8] Martinez and Perez studied pseudoEinstein real hypersurfaces of HPn, n = 3, under the condition that a and b are constant. Using Berndt’s classification we show that we do not need the assumption. The main purpose of this paper is to provide a characterization of pseudo-Einstein real hypersurface in HPn by using an estimate of the length of the Ricci tensor S, which is a quaternionic version of a result of Kimura and Maeda (cf. [5]).