# Two-jets of conformal fields along their zero sets

Open Mathematics (2012)

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

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topAndrzej 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 -

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