Applications of Affine and Weyl Geometry

Garcı́a-Rı́o, Eduardo; Gilkey, Peter Β.; Nikčević, Stana; Vázquez-Lorenzo, Ramón

doi:10.1007/978-3-031-02405-4

Cited by 4 publications

(2 citation statements)

References 145 publications

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“…This notion clearly is invariant under rescaling and there are many examples. One has, for example, the following result of García-Río et al [13].…”

Section: Affine Osserman Manifoldsmentioning

confidence: 93%

“…S − (M, g)). The situation is rather different here as the Jacobi operator is no longer diagonalizable and can have nontrivial Jordan normal form as shown by García-Río et al [13]; in the algebraic context, the Jordan normal form can be arbitrarily complicated [17]. One says (M, g) is nilpotent if J (x) is nilpotent for any tangent vector x; this does not imply (M, g) is flat in the pseudo-Riemannian setting.…”

Section: Osserman Geometry In the Pseudo-riemannian Geometrymentioning

confidence: 95%

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Affine projective Osserman structures

Gilkey¹,

Nikčević²

2013

Class. Quantum Grav.

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By considering the projectivized spectrum of the Jacobi operator, we introduce the concept of projective Osserman manifold in both the affine and in the pseudo-Riemannian settings. If M is an affine projective Osserman manifold, then the modified Riemannian extension metric on the cotangent bundle is both spacelike and timelike projective Osserman. Since any rank 1 symmetric space is affine projective Osserman, this provides additional information concerning the cotangent bundle of a rank 1 Riemannian symmetric space with the modified Riemannian extension metric. We construct other examples of affine projective Osserman manifolds where the Ricci tensor is not symmetric and thus the connection in question is not the Levi-Civita connection of any metric. If the dimension is odd, we use methods of algebraic topology to show the Jacobi operator of an affine projective Osserman manifold has only one non-zero eigenvalue and that eigenvalue is real.1.2. Osserman geometry in the pseudo-Riemannian geometry. Suppose that M = (M, g) is a pseudo-Riemannian manifold of signature (p, q) for p > 0 and q > 0. The pseudo-sphere bundles are defined by setting:One says that (M, g) is spacelike (resp. timelike) Osserman if the eigenvalues of J are constant on S ± (M, g). The situation is rather different here as the Jacobi 2000 Mathematics Subject Classification. 53C50, 53C44. Key words and phrases. affine Osserman, affine projective Osserman, spacelike projective Osserman, timelike projective Osserman. 1 x 1 := cos θe 1 + sin θe 3 ,x 2 := − sin θe 1 + cos θe 3 , x 3 := Jx 1 = cos θe 2 + sin θe 4 , x 4 := Jx 2 = − sin θe 2 + cos θe 4 .Let ⋆ be a coefficient which we do not need to specify. We then haveJ (x 1 )x 3 = (λ 0 + 3λ 1 )x 3 + ⋆e 1 = ⋆x 1 + ⋆x 2 + (λ 0 + 3λ 1 )x 3 , J (x 1 )x 4 = λ 0 x 4 + ⋆e 1 = ⋆x 1 + ⋆x 2 + λ 0 x 4 .The matrix of J (x 1 ) on this 4-dimensional subspace is therefore given by

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“…This notion clearly is invariant under rescaling and there are many examples. One has, for example, the following result of García-Río et al [13].…”

Section: Affine Osserman Manifoldsmentioning

confidence: 93%

Section: Osserman Geometry In the Pseudo-riemannian Geometrymentioning

confidence: 95%

Affine projective Osserman structures

Gilkey¹,

Nikčević²

2013

Class. Quantum Grav.

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Recurrent Lorentzian Weyl Spaces

Dikarev,

Galaev,

Schneider

2024

J Geom Anal

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Integrability and cosmological solutions in Einstein-æther-Weyl theory

Paliathanasis

León

2021

Eur. Phys. J. C

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We consider a Lorentz violating scalar field cosmological model given by the modified Einstein-æther theory defined in Weyl integrable geometry. The existence of exact and analytic solutions is investigated for the case of a spatially flat Friedmann–Lemaître–Robertson–Walker background space. We show that the theory admits cosmological solutions of special interests. In addition, we prove that the cosmological field equations admit the Lewis invariant as a second conservation law, which indicates the integrability of the field equations.

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Applications of Affine and Weyl Geometry

Cited by 4 publications

References 145 publications

Affine projective Osserman structures

Affine projective Osserman structures

Recurrent Lorentzian Weyl Spaces

Integrability and cosmological solutions in Einstein-æther-Weyl theory

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