In this paper, we study biconservative hypersurfaces of index 2 in E 5 2 . We give the complete classification of biconservative hypersurfaces with diagonalizable shape operator at exactly three distinct principal curvatures. We also give an explicit example of biconservative hypersurfaces with four distinct principal curvatures.

An almost Golden Riemannian structure (ϕ, g) on a manifold is given by a tensor field ϕ of type (1,1) satisfying the Golden section relation ϕ 2 = ϕ + 1, and a pure Riemannian metric g, i.e., a metric satisfying g(ϕX, Y ) = g(X, ϕY ). We study connections adapted to such a structure, finding two of them, the first canonical and the well adapted, which measure the integrability of ϕ and the integrability of the G-structure corresponding to (ϕ, g).

In this paper, we study biconservative hypersurfaces in Sn and Hn. Further, we obtain complete explicit classifications of biconservative hypersurfaces in 4‐dimensional Riemannian space forms with exactly three distinct principal curvatures.

In this paper we study biconservative submanifolds in S n × R and H n × R with parallel mean curvature vector field and co-dimension 2. We obtain some necessary and sufficient conditions for such submanifolds to be conservative. In particular, we obtain a complete classification of 3-dimensional biconservative submanifolds in S 4 ×R and H 4 ×R with nonzero parallel mean curvature vector field. We also get some results for biharmonic submanifolds in S n × R and H n × R.MSC 2010: Primary: 53A10; Secondary: 53C40, 53C42