On second-order, divergence-free tensors

@article{Navarro2013OnSD,
  title={On second-order, divergence-free tensors},
  author={Jos'e Navarro},
  journal={Journal of Mathematical Physics},
  year={2013},
  volume={55},
  pages={062501}
}
  • J. Navarro
  • Published 18 June 2013
  • Mathematics, Physics
  • Journal of Mathematical Physics
The aim of this paper is to describe the vector spaces of those second-order tensors on a pseudo-Riemannian manifold (i.e., tensors whose local expressions only involve second derivatives of the metric) that are divergence-free. The main result establishes isomorphisms between these spaces and certain spaces of tensors (at a point) that are invariant under the action of an orthogonal group. This result is valid for tensors with an arbitrary number of indices and symmetries among them and, in… 
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References

SHOWING 1-10 OF 28 REFERENCES
A Class of Conserved Tensors in an Arbitrary Gravitational Field
Four-index tensors with all possible terms either ( a ) quadratic in the Riemann curvature tensor, or ( b ) linear in its second derivatives, with coefficients constructed from products of metric
Variational principles for natural divergence-free tensors in metric field theories
Abstract Let T a b = T b a = 0 be a system of differential equations for the components of a metric tensor on R m . Suppose that T a b transforms tensorially under the action of the diffeomorphism
On the naturalness of Einstein’s equation
We compute all 2-covariant tensors naturally constructed from a semiriemannian metric g which are divergence-free and have weight greater than 2. As a consequence, it follows a characterization of
Variational properties of the Gauss–Bonnet curvatures
The Gauss–Bonnet curvature of order 2k is a generalization to higher dimensions of the Gauss–Bonnet integrand in dimension 2k, as the scalar curvature generalizes the two dimensional Gauss–Bonnet
Lovelock's theorem revisited
Let (X,g) be an arbitrary pseudo-riemannian manifold. A celebrated result by Lovelock ([4], [5], [6]) gives an explicit description of all second-order natural (0,2)-tensors on X, that satisfy the
The Einstein Tensor and Its Generalizations
The Einstein tensorGij is symmetric, divergence free, and a concomitant of the metric tensorgab together with its first two derivatives. In this paper all tensors of valency two with these properties
On the structure of divergence‐free tensors
Contravariant rank two tensors which are divergence‐free on one index and which are constructed from the metric tensor, an auxiliary collection of arbitrary tensor fields, and the first and second
Gauge-Invariant Characterization of Yang–Mills–Higgs Equations
Abstract.Let C → M be the bundle of connections of a principal G-bundle P → M over a pseudo-Riemannian manifold (M,g) of signature (n+, n−) and let E → M be the associated bundle with P under a
The Riemann-Lovelock Curvature Tensor
In order to study the properties of Lovelock gravity theories in low dimensions, we define the kth-order Riemann–Lovelock tensor as a certain quantity having a total 4k-indices, which is kth order in
Universal curvature identities II
Abstract We show that any universal curvature identity which holds in the Riemannian setting extends naturally to the pseudo-Riemannian setting. Thus the Euh–Park–Sekigawa identity also holds for
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