• Honam Mathematical Journal
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• v.36 no.4
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• pp.723-730
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• 2014

In the present paper, we introduce a type of Riemannian manifolds (namely, $W_4$-birecurrent manifold) and study the several properties of such a manifold on which some geometric conditions are imposed.

• Bulletin of the Korean Mathematical Society
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• v.26 no.2
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• pp.159-164
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• 1989

Let M be a hypersurface of dimension n(.geq.2) in an (n+1)-dimensional real space form over bar M(c) with constant curvature c and H the second fundamental tensor of M. M is said to be birecurrent if here exists a covariant tensor field .alpha. of order 2 such that .del.$^{2}$H=H .alpha., where .del. is the connection of M. Also, M is said to be recurrent if there exists a 1-form .betha. such that .del.H=H .betha.. Matsuyama [2] recently proved that a recurrent hypersurface M in a real space form is locally symmetric and a complete irreducible birecurrent hypersurface M in a real space form is recurrent. The main purpose of this paper is to characterize the birecurrent or recurrent hypersurface M of a Riemannian manifold with constant curvature c and to prove that M is classified as a cylinder, $M^{n}$ (c) or ( $c_{1}$)* $M^{n-r}$ ( $c_{2}$) where 1/ $c_{1}$+1/ $c_{2}$=1/c.

• Journal of the Korean Mathematical Society
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• v.32 no.2
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• pp.301-309
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• 1995

Manifolds with recurrent connections have been studied by many authors, such as Chung, Datta, E.M.Patterson, M.Prvanovitch, Singal, and TAkano, etc (refer to [2] and [3]). Examples of such manifolds are those of recurrent curvature, Ricci-recurrent manifolds, and birecurrent manifolds.