We give some examples of slant submanifolds of cosymplectic manifolds. Also, we study some special slant submanifolds, called austere submanifolds, and establish a relation between minimal and anti-invariant submanifolds which is based on properties of the second fundamental form. Moreover, we give an example to illustrate our result.

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In this paper, we study biharmonic hypersurfaces in E 5 . We prove that every biharmonic hypersurface in Euclidean space E 5 must be minimal.Mathematics Subject Classification. 53D12, 53C40, 53C42.

In this paper, we introduce the notion of a slant lightlike submanifold of an indefinite Cosymplectic manifold. We provide a nontrivial example and obtain necessary and sufficient conditions for the existence of a slant lightlike submanifold. Also, we give an example of a minimal slant lightlike submanifold of R 9 2 and prove some characterization Theorems. Mathematics Subject Classification (2010). Primary 53C40; Secondary 53C15, 53C50, 53D15.

The following Chen's bi-harmonic conjecture made in 1991 is wellknown and stays open: The only bi-harmonic submanifolds of Euclidean spaces are the minimal ones. In this paper, we prove that the bi-harmonic conjecture is true for biharmonic hypersurfaces with three distinct principal curvatures of a Euclidean space of arbitrary dimension.