The connected components of the zero set of any conformal vector field v, in a pseudo-Riemannian manifold (M, g) of arbitrary signature, are of two types, which may be called ‘essential’ and ‘nonessential’. The former consist of points at which v is essential, that is, cannot be turned into a Killing field by a local conformal change of the metric. In a component of the latter type, points at which v is nonessential form a relatively-open dense subset that is at the same time a totally umbilical submanifold of (M, g). An essential component is always a null totally geodesic submanifold of (M, g), and so is the set of those points in a nonessential component at which v is essential (unless this set, consisting precisely of all the singular points of the component, is empty). Both kinds of null totally geodesic submanifolds arising here carry a 1-form, defined up to multiplications by functions without zeros, which satisfies a projective version of the Killing equation. The conformal-equivalence type of the 2-jet of v is locally constant along the nonessential submanifold of a nonessential component, and along an essential component on which the distinguished 1-form is nonzero. The characteristic polynomial of the 1-jet of v is always locally constant along the zero set.
@article{bwmeta1.element.doi-10_2478_s11533-012-0049-z,
author = {Andrzej Derdzinski},
title = {Two-jets of conformal fields along their zero sets},
journal = {Open Mathematics},
volume = {10},
year = {2012},
pages = {1698-1709},
zbl = {1260.53048},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.doi-10_2478_s11533-012-0049-z}
}
Andrzej Derdzinski. Two-jets of conformal fields along their zero sets. Open Mathematics, Tome 10 (2012) pp. 1698-1709. http://gdmltest.u-ga.fr/item/bwmeta1.element.doi-10_2478_s11533-012-0049-z/
[1] Beig R., Conformal Killing vectors near a fixed point, Institut für Theoretische Physik, Universität Wien, 1992 (unpublished manuscript)
[2] Belgun F., Moroianu A., Ornea L., Essential points of conformal vector fields, J. Geom. Phys., 2011, 61(3), 589–593 http://dx.doi.org/10.1016/j.geomphys.2010.11.007 | Zbl 1220.53042
[3] Capocci M.S., Essential conformal vector fields, Classical Quantum Gravity, 1999, 16(3), 927–935 http://dx.doi.org/10.1088/0264-9381/16/3/021
[4] Derdzinski A., Zeros of conformal fields in any metric signature, Classical Quantum Gravity, 2011, 28(7), #075011 http://dx.doi.org/10.1088/0264-9381/28/7/075011
[5] Derdzinski A., Maschler G., A moduli curve for compact conformally-Einstein Kähler manifolds, Compos. Math., 2005, 141(4), 1029–1080 http://dx.doi.org/10.1112/S0010437X05001612 | Zbl 1087.53066
[6] Hall G.S., Symmetries and Curvature Structure in General Relativity, World Sci. Lecture Notes Phys., 46, World Scientific, River Edge, 2004 http://dx.doi.org/10.1142/1729
[7] Kobayashi S., Fixed points of isometries, Nagoya Math. J., 1958, 13, 63–68 | Zbl 0084.18202
[8] Lampe M., On conformal connections and infinitesimal conformal transformations, PhD thesis, Universität Leipzig, 2010
[9] Milnor J., Morse Theory, Ann. of Math. Stud., 51, Princeton University Press, Princeton, 1963
[10] Weyl H., Zur Infinitesimalgeometrie: Einordnung der projektiven und der konformen Auffassung, Nachrichten von der Königlichen Gesellschaft der Wissenschaften und der Georg-Augusts-Universität zu Göttingen, 1921, 99–112 | Zbl 48.0844.04