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)