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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 ) → ℂ 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 ℂ n + 1 .

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

Stéphane Garnier, Matthias Kalus (2014)

Archivum Mathematicum

Let ℳ = ( M , 𝒪 ℳ ) be a smooth supermanifold with connection ∇ and Batchelor model 𝒪 ℳ ≅ Γ Λ E * . From ( ℳ , ∇ ) 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 ℳ ≅ Γ Λ E * . It erases information, but however it turns out that the natural identification of supercurves in ℳ (as maps from ℝ 1 | 1 to ℳ ) 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...

A new characterization of r -stable hypersurfaces in space forms

H. F. de Lima, M. A. Velásquez (2011)

Archivum Mathematicum

In this paper we study the r -stability of closed hypersurfaces with constant r -th mean curvature in Riemannian manifolds of constant sectional curvature. In this setting, we obtain a characterization of the r -stable ones through of the analysis of the first eigenvalue of an operator naturally attached to the r -th mean curvature.

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