Affine Osserman Connections Which Are Ricci Flat but Not Flat

Diallo, Abdoul Salam; Hassirou, Mouhamadou; Katamb\'e, I.

doi:10.12732/ijpam.v91i3.3

Cited by 3 publications

(3 citation statements)

References 6 publications

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“…The connection-flatness PDEs system is non-tensorial, while curvature-flatness, Ricci-flatness, and scalar curvature-flatness PDEs systems are tensorial. Our ideas come also from the papers [11][12][13][14][15][16][17][18][19].…”

Section: Pdes In Differential Geometrymentioning

confidence: 99%

“…Let (M, g) be a Riemannian manifold. In this case the Ricci flatness was described in the papers [16,18,19,[23][24][25] as locally underlining the difference between an "Euclidean ball" and a "geodesic ball". Surprisingly, there are Ricci-flatness solutions that are not Riemann-flatness solutions, for example the Schwarzschild solution.…”

Section: Least Squares Lagrangian Density Attached To Ricci-flatnessmentioning

confidence: 99%

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Least Squares Approximation of Flatness on Riemannian Manifolds

et al. 2020

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The purpose of this paper is fourfold: (i) to introduce and study the Euler–Lagrange prolongations of flatness PDEs solutions (best approximation of flatness) via associated least squares Lagrangian densities and integral functionals on Riemannian manifolds; (ii) to analyze some decomposable multivariate dynamics represented by Euler–Lagrange PDEs of least squares Lagrangians generated by flatness PDEs and Riemannian metrics; (iii) to give examples of explicit flat extremals and non-flat approximations; (iv) to find some relations between geometric least squares Lagrangian densities.

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Section: Pdes In Differential Geometrymentioning

confidence: 99%

Section: Least Squares Lagrangian Density Attached To Ricci-flatnessmentioning

confidence: 99%

Least Squares Approximation of Flatness on Riemannian Manifolds

et al. 2020

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“…Ricci flatness was described in [8], [10], [20], [11], [24], [25], [13] underlining locally the difference between an "Euclidean ball" and a "geodesic ball".…”

Section: Ricci-flat Manifoldsmentioning

confidence: 99%

Flatness produced by some geometric PDEs

Hirica,

Udriste,

Pripoae

et al. 2019

Preprint

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This paper has several goals. The first idea is to study the geometric PDEs of connection-flatness, curvature-flatness, Ricci-flatness, scalar curvature-flatness in a modern and rigorous way. Although the idea is not new, our main Theorems about flatness introduce a different point of view in Differential Geometry. The second idea is to introduce and study the Euler-Lagrange prolongations of PDEs-flatness solutions via associated least squares Lagrangian densities and functionals on Riemannian manifolds. All geometric PDEs turned into one of the most intensively developing branches of modern differential geometry.

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