Hostname: page-component-848d4c4894-r5zm4 Total loading time: 0 Render date: 2024-07-02T20:43:25.007Z Has data issue: false hasContentIssue false

A theorem on polynomial lorentz structures

Published online by Cambridge University Press:  18 May 2009

Krzysztof Deszyński
Affiliation:
Jagellonian University, Institute of Mathematics, 00-059 KrakóW, Ul Reymonta 4, Poland
Rights & Permissions [Opens in a new window]

Extract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Let M be a differentiable manifold of dimension m. A tensor field f of type (1, 1) on M is called a polynomial structure on M if it satisfies the equation:

where a1, a2, …, an are real numbers and I denotes the identity tensor of type (1, 1).

We shall suppose that for any xM

is the minimal polynomial of the endomorphism fx: TxMTxM.

We shall call the triple (M, f, g) a polynomial Lorentz structure if f is a polynomial structure on M, g is a symmetric and nondegenerate tensor field of type (0, 2) of signature

such that g (fX, fY) = g(X, Y) for any vector fields X, Y tangent to M. The tensor field g is a (generalized) Lorentz metric.

Type
Research Article
Copyright
Copyright © Glasgow Mathematical Journal Trust 1986

References

REFERENCES

1.Bureš, J. and Vanžura, J., Metric polynomial structures, Kodai Math. Sent. Rep. 27 (1976), 345352.Google Scholar
2.Deszyński, K., Notes on polynomial structures equipped with a Lorentzian metric, to appear.Google Scholar
3.Kobayashi, E. T., A remark on the Nijenhuis tensor, Pacific J. Math. 12 (1962), 936977.Google Scholar
4.Lehman-Lejeune, J., Intégrabilité des G-structures définies par une 1-forme 0-déiormable à valeurs dans le fibre tangent, Ann. Inst. Fourier (Grenoble) 16 (1966), 329387.CrossRefGoogle Scholar
5.Opozda, B., A theorem on metric polynomial structures, Ann. Polon. Math. 41 (1983), 139147.CrossRefGoogle Scholar