Two-jets of conformal fields along their zero sets

Andrzej Derdzinski

Open Mathematics (2012)

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

Abstract

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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

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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

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  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

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