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 $ M $. More precisely, a vector field $ X $ on $ M $ is called a Killing vector field if it satisfies the Killing equation

$$ \tag{* } L _ {X} g = 0 , $$

where $ L _ {X} $ is the Lie derivative along $ X $ and $ g $ is the Riemannian metric of $ M $. 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 $ i ( M) $ of all Killing vector fields on $ M $ forms a Lie algebra of dimension not exceeding $ n ( n+ 1 ) / 2 $, where $ n = \mathop{\rm dim} M $, and this dimension is equal to $ n ( n+ 1 ) /2 $ only for spaces of constant curvature. The set of all complete Killing vector fields forms a subalgebra of $ i ( M) $, which is the Lie algebra of the group of motions of $ M $. The Lie derivative along the direction of a Killing vector field annihilates not only the metric $ g $ 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 $ X $, regarded as a function

$$ X : T ^ {*} M \ni \alpha \rightarrow \alpha ( X) $$

on the cotangent bundle $ T ^ {*} M $, is a first integral of the (Hamilton) geodesic flow on $ T ^ {*} M $ determined by the Riemannian metric. Analogously, a field $ S $ of contravariant symmetric tensors on $ M $ is called a Killing tensor field if the function

$$ S : \alpha \rightarrow S ( \alpha \dots \alpha ) $$

on $ T ^ {*} M $( 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 $ T ^ {*} M $, forms a (generally infinite-dimensional) Lie algebra with respect to the Poisson brackets defined by the standard symplectic structure on $ T ^ {*} M $.

More generally, let $ Q : \mathop{\rm Rep} ^ {k} M \rightarrow W $ be a geometric object of order $ k $ on the manifold $ M $, that is, a $ \mathop{\rm GL} ^ {k} ( n) $- equivariant mapping of the manifold of $ k $- frames on $ M $ into the space $ W $ on which the group $ \mathop{\rm GL} ^ {k} ( n) $ of $ k $- jets of diffeomorphisms of $ \mathbf R ^ {n} $ at zero (preserving the origin) acts. A vector field $ X $ on $ M $ is called an infinitesimal automorphism, or a Killing field of the object $ Q $, if the corresponding (local) one-parameter group of transformations $ \phi _ {t} $ of $ M $ induces a group $ \phi _ {t} ^ {(} k) $ of transformations of the manifold of frames $ \mathop{\rm Rep} ^ {k} M $ preserving $ Q $: $ Q \circ \phi _ {t} ^ {(} k) = Q $. The equation determining a Killing field of the object $ Q $ 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)