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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...

A classification of locally homogeneous connections on 2-dimensional manifolds via group-theoretical approach

Oldřich Kowalski, Barbara Opozda, Zdeněk Vlášek (2004)

Open Mathematics

The aim of this paper is to classify (lócally) all torsion-less locally homogeneous affine connections on two-dimensional manifolds from a group-theoretical point of view. For this purpose, we are using the classification of all non-equivalent transitive Lie algebras of vector fields in ℝ2 according to P.J. Olver [7].

A curvature identity on a 6-dimensional Riemannian manifold and its applications

Yunhee Euh, Jeong Hyeong Park, Kouei Sekigawa (2017)

Czechoslovak Mathematical Journal

We derive a curvature identity that holds on any 6-dimensional Riemannian manifold, from the Chern-Gauss-Bonnet theorem for a 6-dimensional closed Riemannian manifold. Moreover, some applications of the curvature identity are given. We also define a generalization of harmonic manifolds to study the Lichnerowicz conjecture for a harmonic manifold “a harmonic manifold is locally symmetric” and provide another proof of the Lichnerowicz conjecture refined by Ledger for the 4-dimensional case under a...

A framed f-structure on the tangent bundle of a Finsler manifold

Esmaeil Peyghan, Chunping Zhong (2012)

Annales Polonici Mathematici

Let (M,F) be a Finsler manifold, that is, M is a smooth manifold endowed with a Finsler metric F. In this paper, we introduce on the slit tangent bundle T M ˜ a Riemannian metric G̃ which is naturally induced by F, and a family of framed f-structures which are parameterized by a real parameter c≠ 0. We prove that (i) the parameterized framed f-structure reduces to an almost contact structure on IM; (ii) the almost contact structure on IM is a Sasakian structure iff (M,F) is of constant flag curvature...

A local characterization of affine holomorphic immersions with an anti-complex and ∇-parallel shape operator

Maria Robaszewska (2002)

Annales Polonici Mathematici

We study the complex hypersurfaces f : M ( n ) n + 1 which together with their transversal bundles have the property that around any point of M there exists a local section of the transversal bundle inducing a ∇-parallel anti-complex shape operator S. We give a class of examples of such hypersurfaces with an arbitrary rank of S from 1 to [n/2] and show that every such hypersurface with positive type number and S ≠ 0 is locally of this kind, modulo an affine isomorphism of n + 1 .

A lossless reduction of geodesics on supermanifolds to non-graded differential geometry

Stéphane Garnier, Matthias Kalus (2014)

Archivum Mathematicum

Let = ( M , 𝒪 ) be a smooth supermanifold with connection and Batchelor model 𝒪 Γ Λ E * . From ( , ) we construct a connection on the total space of the vector bundle E M . This reduction of is well-defined independently of the isomorphism 𝒪 Γ Λ E * . It erases information, but however it turns out that the natural identification of supercurves in (as maps from 1 | 1 to ) with curves in E restricts to a 1 to 1 correspondence on geodesics. This bijection is induced by a natural identification of initial conditions for geodesics...

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