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A characterization of the Riemann extension in terms of harmonicity

Cornelia-Livia Bejan, Şemsi Eken (2017)

Czechoslovak Mathematical Journal

If ( M , ∇ ) is a manifold with a symmetric linear connection, then T * M can be endowed with the natural Riemann extension g ¯ (O. Kowalski and M. Sekizawa (2011), M. Sekizawa (1987)). Here we continue to study the harmonicity with respect to g ¯ initiated by C. L. Bejan and O. Kowalski (2015). More precisely, we first construct a canonical almost para-complex structure 𝒫 on ( T * M , g ¯ ) and prove that 𝒫 is harmonic (in the sense of E. García-Río, L. Vanhecke and M. E. Vázquez-Abal (1997)) if and only if g ¯ reduces to the...

A characterization of totally η -umbilical real hypersurfaces and ruled real hypersurfaces of a complex space form

Mayuko Kon (2008)

Czechoslovak Mathematical Journal

We give a characterization of totally η -umbilical real hypersurfaces and ruled real hypersurfaces of a complex space form in terms of totally umbilical condition for the holomorphic distribution on real hypersurfaces. We prove that if the shape operator A of a real hypersurface M of a complex space form M n ( c ) , c ≠ 0 , n ≥ 3 , satisfies g ( A X , Y ) = a g ( X , Y ) for any X , Y ∈ T 0 ( x ) , a being a function, where T 0 is the holomorphic distribution on M , then M is a totally η -umbilical real hypersurface or locally congruent to a ruled real hypersurface....

A class of 3-dimensional almost Kenmotsu manifolds with harmonic curvature tensors

Yaning Wang (2016)

Open Mathematics

Let M3 be a three-dimensional almost Kenmotsu manifold satisfying ▽ξh = 0. In this paper, we prove that the curvature tensor of M3 is harmonic if and only if M3 is locally isometric to either the hyperbolic space ℍ3(-1) or the Riemannian product ℍ2(−4) × ℝ. This generalizes a recent result obtained by [Wang Y., Three-dimensional locally symmetric almost Kenmotsu manifolds, Ann. Polon. Math., 2016, 116, 79-86] and [Cho J.T., Local symmetry on almost Kenmotsu three-manifolds, Hokkaido Math. J., 2016,...

A class of metrics on tangent bundles of pseudo-Riemannian manifolds

H. M. Dida, A. Ikemakhen (2011)

Archivum Mathematicum

We provide the tangent bundle T M of pseudo-Riemannian manifold ( M , g ) with the Sasaki metric g s and the neutral metric g n . First we show that the holonomy group H s of ( T M , g s ) contains the one of ( M , g ) . What allows us to show that if ( T M , g s ) is indecomposable reducible, then the basis manifold ( M , g ) is also indecomposable-reducible. We determine completely the holonomy group of ( T M , g n ) according to the one of ( M , g ) . Secondly we found conditions on the base manifold under which ( T M , g s ) ( respectively ( T M , g n ) ) is Kählerian, locally symmetric or Einstein...

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