Abstract
Let M be a smooth oriented connected n-dimensional manifold and let \({\mathfrak{M}}\) be the space of pseudo-Riemannian metrics on M of a given signature \({(n^+, n^-), n^{+} + n^- = n > 1}\). A system of n metric invariants is attached to each metric in \({\mathfrak{M}}\), called the Ricci invariants, and by using the geometric properties of such invariants, the following result is proved: The subset \({\mathfrak{O} \subset \mathfrak{M}}\) of metrics with no Killing vector fields other than the trivial one is open and dense with respect to the strong topology.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Beig R., Chruściel P.T., Schoen R.: KIDs are non–generic. Ann. Henri Poincaré 6(1), 155–194 (2005)
Bourbaki N.: Eléments de Mathématique. Hermann, Paris (1964)
Ebin D. G.: On the space of Riemannian metrics. Bull. Amer. Math. Soc. 74, 1001–1003 (1968)
D. G. Ebin, The manifold of Riemannian metrics, Global Analysis (Proc. Sympos. Pure Math., Volume XV, Berkeley, California, 1968), Amer. Math. Soc., Providence, R.I., 1970, pp. 11–40.
Y. Eliashberg, N. Mishachev, Introduction to the h–principle, Graduate Studies in Mathematics, 48, Amer. Math. Soc., Providence, RI, 2002.
A. Grothendieck, Topological vector spaces, translated from the French by Orlando Chaljub, Notes on Mathematics and its Applications, Gordon and Breach Science Publishers, New York–London–Paris, 1973.
Hamilton R. S.: A compactness property for solutions of the Ricci flow, Amer. J.Math. 117(3), 545–572 (1995)
S. Kobayashi, K. Nomizu, Foundations of Differential Geometry, John Wiley & Sons, Inc. (Interscience Division), New York, Volume I, 1963; Volume II, 1969.
Mather J.N.: Stability of C ∞ mappings. II. Infinitesimal stability implies stability. Ann. of Math. 89(2), 254–291 (1969)
Mather J.N.: Stability of C ∞ mappings. V. Transversality. Advances in Math. 4, 301–336 (1970)
Mounoud P.: Dynamical properties of the space of Lorentzian metrics, Comment. Math. Helv. 78(3), 463–485 (2003)
P. Mounoud, Metrics without isometries are generic, arXiv:1403.0182v1 [math.DG] (to be published in Monatshefte für Mathematik).
Muñoz Masqué J., Valdés Morales A.: The number of functionally independent invariants of a pseudo–Riemannian metric. J.Phys. A: Math.Gen. 27, 7843–7855 (1994)
B. O’Neill, Semi–Riemannian geometry. With applications to relativity, Pure and Applied Mathematics, 103, Academic Press, Inc., New York, 1983.
Nomuzi K.: On local and global existence of Killing vector fields. Ann. of Math. 72, 105–120 (1960)
Prüfer F., Tricerri F., Vanhecke L.: Curvature Invariants, Differential Operators and Local Homogeneity. Trans. Amer. Math. Soc. 348(11), 4643–4652 (1996)
Whitney H.: Elementary structure of real algebraic varieties. Ann. of Math. 66, 545–556 (1957)
Author information
Authors and Affiliations
Corresponding author
Additional information
We are indebted to the anonymous referee of this paper for some suggestions that have really improved several of its results. Supported by Ministerio de Ciencia e Innovación of Spain under #MTM2011–22528.
Rights and permissions
About this article
Cite this article
Castrillón López, M., Muñoz Masqué, J. & Rosado María, E. Killing Vector Fields of Generic Semi-Riemannian Metrics. Milan J. Math. 83, 47–54 (2015). https://doi.org/10.1007/s00032-014-0229-3
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00032-014-0229-3