The Weitzenböck machine

Semmelmann, Uwe; Weingart, Gregor

doi:10.1112/s0010437x09004333

Cited by 24 publications

(39 citation statements)

References 6 publications

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“…∇ g τ 2 + 2R g (τ , τ ) dV g , see [12]. This is a special instance of an example considered in [35], namely the Weitzenböck formula for two forms in Λ 2 14 .…”

Section: Proofs Of Main Theoremsmentioning

confidence: 97%

Curvature decomposition of G2-manifolds

Cleyton¹,

Ivanov²

2008

Journal of Geometry and Physics

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a b s t r a c tExplicit formulas for the G 2 -components of the Riemannian curvature tensor on a manifold with a G 2 -structure are given in terms of Ricci contractions. We define a conformally invariant Ricci-type tensor that determines the 27-dimensional part of the Weyl tensor and show that its vanishing on compact G 2 -manifold with closed fundamental form forces the three-form to be parallel. A topological obstruction for the existence of a G 2 -structure with closed fundamental form is obtained in terms of the integral norms of the curvature components. We produce integral inequalities for closed G 2 -manifold and investigate limiting cases. We make a study of warped products and cohomogeneity-one G 2 -manifolds. As a consequence every Fernandez-Gray type of G 2 -structure whose scalar curvature vanishes may be realized such that the metric has holonomy contained in G 2 .

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“…∇ g τ 2 + 2R g (τ , τ ) dV g , see [12]. This is a special instance of an example considered in [35], namely the Weitzenböck formula for two forms in Λ 2 14 .…”

Section: Proofs Of Main Theoremsmentioning

confidence: 97%

Curvature decomposition of G2-manifolds

Cleyton¹,

Ivanov²

2008

Journal of Geometry and Physics

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“…[15]) of the classical selection rule, which is helpful to better visualize the classification of elliptic operators, that turns out to be strongly related to the selection rule.…”

Section: Remark 22mentioning

confidence: 98%

“…15) We recall that the condition λ i = λ i+1 , for 1 i m − 2, is equivalent to the fact that the weights {−ε i , ε i+1 } are not relevant for λ and λ m−1 = λ m = 0 is the only case when −ε m−1 is not relevant. It can be then checked by the branching rule (cf.…”

Section: Proofmentioning

confidence: 99%

A representation-theoretical proof of Bransonʼs classification of elliptic generalized gradients

Pilca

2011

Differential Geometry and its Applications

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The purpose of this paper is to present a new proof of Branson's classification (Branson, 1997 [3]), of minimal elliptic sums of generalized gradients. The advantage of this proof is that it is local, being mainly based on representation theory and on the relationship between ellipticity and refined Kato inequalities. This approach is promising for the classification of elliptic generalized gradients of G-structures, for other subgroups G of the special orthogonal group.

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“…where ∇ ab···c ≡ ∇ a ∇ b · · · ∇ c , K a p ···a 1 is a KT of order p. We remark that one needs (p + 1) prolongation variables to carry out the prolongation for KTs, while that for Killing-Yano tensor fields involves only two prolongation variables for any order (see [34,36] or Appendix D). This fact complicates the prolongation for the Killing equation (1).…”

Section: Prolonged Equationsmentioning

confidence: 99%

“…The integrability conditions for p ≥ 3 were discussed in [33]. Similar techniques have been applied to other hidden symmetries [34,35,37,36,38,39].…”

Section: Introductionmentioning

confidence: 99%

On integrability of the Killing equation

Houri

Tomoda

Yasui

2018

Class. Quantum Grav.

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Killing tensor fields have been thought of as describing hidden symmetry of space(-time) since they are in one-to-one correspondence with polynomial first integrals of geodesic equations. Since many problems in classical mechanics can be formulated as geodesic problems in curved spaces and spacetimes, solving the defining equation for Killing tensor fields (the Killing equation) is a powerful way to integrate equations of motion. Thus it has been desirable to formulate the integrability conditions of the Killing equation, which serve to determine the number of linearly independent solutions and also to restrict the possible forms of solutions tightly. In this paper, we show the prologation for the Killing equation in a manner that uses Young symmetrizers. Using the prolonged equations, we provide the integrability conditions explicitly.

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The Weitzenböck machine

Cited by 24 publications

References 6 publications

Curvature decomposition of G2-manifolds

Curvature decomposition of G2-manifolds

A representation-theoretical proof of Bransonʼs classification of elliptic generalized gradients

On integrability of the Killing equation

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