Killing vector

more precisely, Killing vector field or infinitesimal motion

The field of velocities of a (local) one-parameter group of motions on a Riemannian manifold . More precisely, a vector field on is called a Killing vector field if it satisfies the Killing equation

(*)

where is the Lie derivative along and is the Riemannian metric of . These fields were first systematically studied by W. Killing [1], who also introduced equation (*) for them. In a complete Riemannian manifold any Killing vector field is complete, that is, it is the field of velocities of a one-parameter group of motions. The set of all Killing vector fields on forms a Lie algebra of dimension not exceeding , where , and this dimension is equal to only for spaces of constant curvature. The set of all complete Killing vector fields forms a subalgebra of , which is the Lie algebra of the group of motions of . The Lie derivative along the direction of a Killing vector field annihilates not only the metric but also all fields that are canonically constructed in terms of the metric, for example, the Riemann curvature tensor, the Ricci operator, etc. This enables one to establish a connection between the properties of Killing vector fields and the curvature tensor. For example, at a point where all eigen values of the Ricci operator are distinct, a Killing vector field cannot vanish.

A Killing vector field , regarded as a function

on the cotangent bundle , is a first integral of the (Hamilton) geodesic flow on determined by the Riemannian metric. Analogously, a field of contravariant symmetric tensors on is called a Killing tensor field if the function

on (polynomial on the fibres) corresponding to it is a first integral of the geodesic flow. The equation determining a Killing tensor field is also called the Killing equation. The set of all Killing tensor fields, regarded as functions on , forms a (generally infinite-dimensional) Lie algebra with respect to the Poisson brackets defined by the standard symplectic structure on .

More generally, let be a geometric object of order on the manifold , that is, a -equivariant mapping of the manifold of -frames on into the space on which the group of -jets of diffeomorphisms of at zero (preserving the origin) acts. A vector field on is called an infinitesimal automorphism, or a Killing field of the object , if the corresponding (local) one-parameter group of transformations of induces a group of transformations of the manifold of frames preserving : . The equation determining a Killing field of the object is called a Lie–Killing equation and the operator corresponding to it is called the Lie operator [6].

References

[1]	W. Killing, "Ueber die Grundlagen der Geometrie" J. Reine Angew. Math. , 109 (1892) pp. 121–186
[2]	P.K. [P.K. Rashevskii] Rashewski, "Riemannsche Geometrie und Tensoranalyse" , Deutsch. Verlag Wissenschaft. (1959) (Translated from Russian)
[3]	L.P. Eisenhart, "Riemannian geometry" , Princeton Univ. Press (1949)
[4]	S. Helgason, "Differential geometry, Lie groups, and symmetric spaces" , Acad. Press (1978)
[5]	S. Kobayashi, "Transformation groups in differential geometry" , Springer (1972)
[6]	A. Kumpera, D. Spencer, "Lie equations" , 1. General theory , Princeton Univ. Press (1972)
[7]	S. Kobayashi, K. Nomizu, "Foundations of differential geometry" , 2 , Interscience (1969)
[8]	I.P. Egorov, "Motions in spaces of affine connection" , Motions, spaces, affine connections , Kazan' (1965) pp. 5–179 (In Russian)