Journal ArticleDOI
Third-Order Tensor Potentials for the Riemann and Weyl Tensors. II: Singular Solutions
Franco Bampi,Giacomo Caviglia +1 more
TLDR
The analysis of the admissibility of a potential representation for the Riemann tensor is continued in this article, where it is shown that there never exist ordinary solutions in a four-dimensional manifold and the existence of singular solutions is established without requiring any integrability condition.Abstract:
The analysis of the admissibility of a potential representation for the Riemann tensor is here continued. As in the preceding paper, the starting point is to regard the relationship between the Riemann tensor and its possible potential as a system of partial differential equations determining the unknown potential. The first result, strengthening a previous conclusion, is that there never exist ordinary solutions. Surprisingly enough, in a four-dimensional Riemannian manifold the existence of singular solutions is established without requiring any integrability condition. Possible applications and generalizations are also suggested.read more
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On effective constraints for the Riemann–Lanczos system of equations
TL;DR: In this paper, the authors give a simple direct derivation of a constraint equation, confirm explicitly that known exact solutions of the Riemann-Lanczos problem satisfy it, and argue that the Bampi and Caviglia conclusion must therefore be flawed.
Journal ArticleDOI
Exterior differential systems, Janet–Riquier theory and the Riemann–Lanczos problems in two, three, and four dimensions
TL;DR: In this paper, the Riemann-Lanczos problem in two, three, and four dimensions is discussed using the theory of exterior differential systems and Janet-Riquier theory.
Posted Content
The Riemann-Lanczos Problem as an Exterior Differential System with Examples in 4 and 5 Dimensions
TL;DR: In this article, the Riemann-Lanczos problem is shown to be in involution in both 4 and 5 dimensions, but not in 5 dimensions. But it is known from the work of Bampi and Caviglia that it is always in involuction in 4 dimensions, and hence it is possible to find singular solutions for the Goedel, Kasner and Debever-Hubaut spacetimes.
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Some potentials for the curvature tensor on three-dimensional manifolds
TL;DR: In this paper, a second order potential for the Ricci tensor is introduced and it is shown that elliptic equations can be obtained by relaxing those symmetry requirements in at least two different ways, leading to global existence of potentials under natural conditions.
Journal ArticleDOI
Prolongation methods and Cartan characters for the three-dimensional Riemann–Lanczos problem
TL;DR: A prolongation of the three-dimensional Riemann–Lanczos problem in the same way as Bampi and Caviglia did for four dimensions is suggested and becomes involutive with Cartan characters (17,8,2) or (20,10,3) if no cyclic conditions are imposed.
References
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Book
Les systèmes différentiels extérieurs et leurs applications géométriques
Elie Cartan,M. E. Cartan +1 more
Journal ArticleDOI
The splitting of the riemann tensor
TL;DR: In this article, the selfdual part of R/sub ijkm/ is analyzed and reduced to a new tensor of third order H/sub Ijk/ of essentially 16 components.
Journal ArticleDOI
Third-order tensor potentials for the Riemann and Weyl tensors
Franco Bampi,Giacomo Caviglia +1 more
TL;DR: In this paper, the representations of the Riemann and the Weyl tensors through covariant derivatives of third-order potentials are examined in detail, and the possibility of introducing gauges on the potentials is reexamined in connection with the previous result.