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( h , Φ ) -entropy differential metric

María Luisa Menéndez, Domingo Morales, Leandro Pardo, Miquel Salicrú (1997)

Applications of Mathematics

Burbea and Rao (1982a, 1982b) gave some general methods for constructing quadratic differential metrics on probability spaces. Using these methods, they obtained the Fisher information metric as a particular case. In this paper we apply the method based on entropy measures to obtain a Riemannian metric based on ( h , Φ ) -entropy measures (Salicrú et al., 1993). The geodesic distances based on that information metric have been computed for a number of parametric families of distributions. The use of geodesic...

A characterization of n-dimensional hypersurfaces in R n + 1 with commuting curvature operators

Yulian T. Tsankov (2005)

Banach Center Publications

Let Mⁿ be a hypersurface in R n + 1 . We prove that two classical Jacobi curvature operators J x and J y commute on Mⁿ, n > 2, for all orthonormal pairs (x,y) and for all points p ∈ M if and only if Mⁿ is a space of constant sectional curvature. Also we consider all hypersurfaces with n ≥ 4 satisfying the commutation relation ( K x , y K z , u ) ( u ) = ( K z , u K x , y ) ( u ) , where K x , y ( u ) = R ( x , y , u ) , for all orthonormal tangent vectors x,y,z,w and for all points p ∈ M.

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 second-order identity for the Riemann tensor and applications

Carlo Alberto Mantica, Luca Guido Molinari (2011)

Colloquium Mathematicae

A second-order differential identity for the Riemann tensor is obtained on a manifold with a symmetric connection. Several old and some new differential identities for the Riemann and Ricci tensors are derived from it. Applications to manifolds with recurrent or symmetric structures are discussed. The new structure of K-recurrency naturally emerges from an invariance property of an old identity due to Lovelock.

Algebraic theory of affine curvature tensors

Novica Blažić, Peter Gilkey, S. Nikčević, Udo Simon (2006)

Archivum Mathematicum

We use curvature decompositions to construct generating sets for the space of algebraic curvature tensors and for the space of tensors with the same symmetries as those of a torsion free, Ricci symmetric connection; the latter naturally appear in relative hypersurface theory.

Anti-invariant Riemannian submersions from almost Hermitian manifolds

Bayram Ṣahin (2010)

Open Mathematics

We introduce anti-invariant Riemannian submersions from almost Hermitian manifolds onto Riemannian manifolds. We give an example, investigate the geometry of foliations which are arisen from the definition of a Riemannian submersion and check the harmonicity of such submersions. We also find necessary and sufficient conditions for a Langrangian Riemannian submersion, a special anti-invariant Riemannian submersion, to be totally geodesic. Moreover, we obtain decomposition theorems for the total manifold...

Basic equations of G -almost geodesic mappings of the second type, which have the property of reciprocity

Mića S. Stanković, Milan L. Zlatanović, Nenad O. Vesić (2015)

Czechoslovak Mathematical Journal

We study G -almost geodesic mappings of the second type θ π 2 ( e ) , θ = 1 , 2 between non-symmetric affine connection spaces. These mappings are a generalization of the second type almost geodesic mappings defined by N. S. Sinyukov (1979). We investigate a special type of these mappings in this paper. We also consider e -structures that generate mappings of type θ π 2 ( e ) , θ = 1 , 2 . For a mapping θ π 2 ( e , F ) , θ = 1 , 2 , we determine the basic equations which generate them.

Bochner's formula for harmonic maps from Finsler manifolds

Jintang Li (2008)

Colloquium Mathematicae

Let ϕ :(M,F)→ (N,h) be a harmonic map from a Finsler manifold to any Riemannian manifold. We establish Bochner's formula for the energy density of ϕ and maximum principle on Finsler manifolds, from which we deduce some properties of harmonic maps ϕ.

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