# Two-jets of conformal fields along their zero sets

Open Mathematics (2012)

• Volume: 10, Issue: 5, page 1698-1709
• ISSN: 2391-5455

top

## Abstract

top
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.

## How to cite

top

Andrzej Derdzinski. "Two-jets of conformal fields along their zero sets." Open Mathematics 10.5 (2012): 1698-1709. <http://eudml.org/doc/269186>.

@article{AndrzejDerdzinski2012,
abstract = {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.},
author = {Andrzej Derdzinski},
journal = {Open Mathematics},
keywords = {Conformal vector field; Fixed-point set; Two-jet; conformal vector field; fixed-points set; two-jet},
language = {eng},
number = {5},
pages = {1698-1709},
title = {Two-jets of conformal fields along their zero sets},
url = {http://eudml.org/doc/269186},
volume = {10},
year = {2012},
}

TY - JOUR
AU - Andrzej Derdzinski
TI - Two-jets of conformal fields along their zero sets
JO - Open Mathematics
PY - 2012
VL - 10
IS - 5
SP - 1698
EP - 1709
AB - 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.
LA - eng
KW - Conformal vector field; Fixed-point set; Two-jet; conformal vector field; fixed-points set; two-jet
UR - http://eudml.org/doc/269186
ER -

## References

top
1. [1] Beig R., Conformal Killing vectors near a fixed point, Institut für Theoretische Physik, Universität Wien, 1992 (unpublished manuscript)
2. [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 Zbl1220.53042
3. [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. [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. [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 Zbl1087.53066
6. [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. [7] Kobayashi S., Fixed points of isometries, Nagoya Math. J., 1958, 13, 63–68 Zbl0084.18202
8. [8] Lampe M., On conformal connections and infinitesimal conformal transformations, PhD thesis, Universität Leipzig, 2010
9. [9] Milnor J., Morse Theory, Ann. of Math. Stud., 51, Princeton University Press, Princeton, 1963
10. [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 Zbl48.0844.04

## NotesEmbed?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.