Abstract. In the present article we study a special class of surfaces in the four-dimensional Euclidean space, which are one-parameter systems of meridians of the standard rotational hypersurface. They are called meridian surfaces. We show that a meridian surface has a harmonic Gauss map if and only if it is part of a plane. Further, we give necessary and sufficient conditions for a meridian surface to have pointwise 1-type Gauss map and find all meridian surfaces with pointwise 1-type Gauss map.

We give necessary and sufficient conditions for warped product manifolds (M, g) , of dimension ⩾ 4 , with 1 -dimensional base, and in particular, for generalized Robertson-Walker spacetimes, to satisfy some generalized Einstein metric condition. Namely, the difference tensor R•C −C •R , formed from the curvature tensor R and the Weyl conformal curvature tensor C , is expressed by the Tachibana tensor Q(S, R) formed from the Ricci tensor S and R . We also construct suitable examples of such manifolds. They are quasi-Einstein, i.e. at every point of M rank (S − α g) ⩽ 1 , for some α ∈ R , or non-quasi-Einstein.

In the present study we consider generalized rotation surfaces imbedded in an Euclidean space of four dimensions. We also give some special examples of these surfaces in E 4 . Further, the curvature properties of these surfaces are investigated. We give necessary and sufficient conditions for generalized rotation surfaces to become pseudo-umbilical. We also show that every general rotation surface is Chen surface in E 4 . Finally we give some examples of generalized rotation surfaces in E 4 .Mathematics Subject Classification (2010). Primary 53C40; Secondary 53C42.

Abstract. Tensor product immersions of a given Riemannian manifold was initiated by B.-Y. Chen. In the present article we study the tensor product surfaces of two Euclidean plane curves. We show that a tensor product surface M of a plane circle c 1 centered at origin with an Euclidean planar curve c 2 has harmonic Gauss map if and only if M is a part of a plane. Further, we give necessary and sufficient conditions for a tensor product surface M of a plane circle c 1 centered at origin with an Euclidean planar curve c 2 to have pointwise 1-type Gauss map.

0. Introduction. Let M be a smooth m-dimensional submanifold in (m + d)-dimensional Euclidean space U m+d . For x e M and a non-zero vector X in T X M, we define theIn a neighbourhood of x, the intersection M DE(x,X) is a regular curve y : ( -e , e)-»M. We suppose the parameter f e (-e, e) is a multiple of the arc-length such that y(0) = x and y(0) = X. Each choice of X e T(M) yields a different curve which is called the normal section of M at JC in the direction of X, where X s T X (M) (Section 3).For such a normal section we can write Submanifolds with pointwise 3-planar normal sections have been studied by S. J. Li in the case when M is isotropic [6] and also in the case when M is spherical [7].In this paper we consider product submanifolds M = M X X M 2 with P3-PNS and we show that this implies strong conditions on Mj and M 2 .

In the present study we consider curves and surfaces of AW(k) ( k =1, 2 or 3 ) type. We also give related examples of curves and surfaces satisfying AW(k) type conditions.

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