Geometric Structures Modeled on Affine Hypersurfaces and Generalizations of the Einstein–Weyl and Affine Sphere Equations

Fox, Daniel J. F.

doi:10.1007/978-3-319-21284-5_3

Cited by 7 publications

(19 citation statements)

References 4 publications

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“…Proof. We first remark that iteration of the commutator formula (8) gives [δ, L i ] = −2iL i−1 d for i ≥ 1. Then δK = 0 implies 0 = δK 0 + i≥1 δL i K i = δK 0 − 2dK 1 + Lk 1 for some symmetric tensor k 1 .…”

Section: A Killing Tensor If and Only If The Complete Symmetrization mentioning

confidence: 99%

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Killing and conformal Killing tensors

Heil

Moroianu

Semmelmann

2016

Journal of Geometry and Physics

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Abstract. We introduce an appropriate formalism in order to study conformal Killing (symmetric) tensors on Riemannian manifolds. We reprove in a simple way some known results in the field and obtain several new results, like the classification of conformal Killing 2-tensors on Riemannian products of compact manifolds, Weitzenböck formulas leading to non-existence results, and construct various examples of manifolds with conformal Killing tensors.

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Section: A Killing Tensor If and Only If The Complete Symmetrization mentioning

confidence: 99%

“…in connection with geometric inverse problems, integrable systems and Einstein-Weyl geometry, cf. [5], [8], [9], [13], [15], [23], [26].…”

Section: Introductionmentioning

confidence: 99%

Killing and conformal Killing tensors

Heil

Moroianu

Semmelmann

2016

Journal of Geometry and Physics

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“…By (2.33), W ∧ N = 0 if and only if Z = 0. This shows the equivalence of (6) and (7). Since Z p i p = U(F)μ i − N p μ p F i , it also shows that W ∧ N = 0 if and only if μ ∧ ρ = 0.…”

Section: Example 217mentioning

confidence: 57%

“…The approach described now has as a side benefit a somewhat more general construction, namely it attaches to nondegenerate immersed hypersurface in an

(n + 1)

‐dimensional manifold M equipped with an affine connection

\overset{\nabla}{true}

a distinguished transverse line field that transforms covariantly under the action of the group of affine automorphisms of

(() M, \hat{\nabla})

. This was described previously in section 4 of , but is hidden there in a still more general context requiring substantially more preparation, so it is recapitulated here.…”

Section: Equiaffine Geometry Of Level Setsmentioning

confidence: 99%

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Equiaffine geometry of level sets and ruled hypersurfaces with equiaffine mean curvature zero

Fox

2016

Mathematische Nachrichten

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A Schwarz lemma for Kähler affine metrics and the canonical potential of a proper convex cone

Fox

2013

Annali di Matematica

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This is an account of some aspects of the geometry of Kähler affine metrics based on considering them as smooth metric measure spaces and applying the comparison geometry of Bakry-Emery Ricci tensors. Such techniques yield a version for Kähler affine metrics of Yau's Schwarz lemma for volume forms. By a theorem of Cheng and Yau there is a canonical Kähler affine Einstein metric on a proper convex domain, and the Schwarz lemma gives a direct proof of its uniqueness up to homothety. The potential for this metric is a function canonically associated to the cone, characterized by the property that its level sets are hyperbolic affine spheres foliating the cone. It is shown that for an n-dimensional cone a rescaling of the canonical potential is an n-normal barrier function in the sense of interior point methods for conic programming. It is explained also how to construct from the canonical potential Monge-Ampère metrics of both Riemannian and Lorentzian signatures, and a mean curvature zero conical Lagrangian submanifold of the flat para-Kähler space.The ∞-Ricci tensor was introduced by A. Lichnerowicz in [46] where he used it to prove a generalization of the Cheeger-Gromoll splitting theorem.The idea related to these tensors relevant in what follows is that a lower bound on the (N + n)dimensional Bakry Emery Ricci tensor of the metric measure structure (k, φ) on the n-manifold M corresponds to a lower bound on the usual Ricci curvature of some metric associated to (k, φ) on some (n + N )-dimensional manifold fibering over M . In the example relevant here, N = n and the fibering manifold is the tube domain over an n-dimensional convex cone.One aim of this paper is to explain that it is profitable to regard a Riemannian Hessian metric g ij = ∇ i dF j as the mm-space determined by g ij in conjunction with the volume form H(F )Ψ = H(F ) 1/2 vol g , which is the pullback via the differential dF of the parallel volume form dual to Ψ on the dual affine space R n+1 * . More generally, a Kähler affine structure (g, ∇) is identified with the local mm-structure formed by g and the half Koszul one-form 1 2 H i .

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Geometric Structures Modeled on Affine Hypersurfaces and Generalizations of the Einstein–Weyl and Affine Sphere Equations

Cited by 7 publications

References 4 publications

Killing and conformal Killing tensors

Killing and conformal Killing tensors

Equiaffine geometry of level sets and ruled hypersurfaces with equiaffine mean curvature zero

A Schwarz lemma for Kähler affine metrics and the canonical potential of a proper convex cone

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