Spray

on a differentiable manifold $M$

A vector field $W$ on the tangent space $TM$ which, in terms of the local coordinates $(x^1,\dots,x^n,v^1,\dots,v^n)$ on $TM$ associated in a natural way with the local coordinates $(x^1,\dots,x^n)$ on $M$, has components $(v^1,\dots,v^n,f^1,\dots,f^n)$, where $f^i=f^i(x^1,\dots,x^n,v^1,\dots,v^n)$ are functions of class $C^1$ which are, for fixed $x^1,\dots,x^n$, positive homogeneous functions in $v^1,\dots,v^n$ of degree 2 (these properties of $W$ do not depend on the actual choice of the local coordinates). The system of differential equations determined by this vector field,

$$\frac{dx^i}{dt}=v^i,\quad\frac{dv^i}{dt}=f^i(x^1,\dots,x^n,v^1,\dots,v^n),\quad i=1,\dots,n,$$

is equivalent to the system of second-order differential equations

$$\frac{d^2x^i}{dt^2}=f^i\left(x^1,\dots,x^n,\frac{dx^1}{dt},\dots,\frac{dx^n}{dt}\right);$$

therefore a spray describes (and moreover in an invariant manner, that is, independent of the coordinate system) a system of such equations on $M$.

The most important case of a spray is when the $f^i$ are polynomials of the second degree in the $v^i$:

$$f^i=\sum\Gamma_{jk}^i(x^1,\dots,x^n)v^jv^k,\quad\Gamma_{jk}^i=\Gamma_{kj}^i.\label{*}\tag{*}$$

In this case the $\Gamma_{jk}^i$ give an affine connection on $M$ with zero torsion tensor. Conversely, for every affine connection the equations of the geodesic lines are given by a certain spray with $f^i$ of the form \eqref{*} (where when going from the connection to the spray, the $\Gamma_{jk}^i$ symmetrize with respect to the suffixes). If the field $W$ is of class $C^2$, then $f^i$ must have the form \eqref{*}. In the general case, however, $W$ may be smooth outside the zero section of the bundle $TM$, but need not be a field of class $C^2$ near this section. In such a situation one sometimes talks about a generalized spray, leaving the term "spray" only for the special case \eqref{*}. The differential equations for geodesics in Finsler geometry give rise to a generalized spray.

It is possible to give a definition of a spray in invariant terms, which is suitable also for Banach manifolds (see [1]).

References

[1]	S. Lang, "Introduction to differentiable manifolds" , Interscience (1967)

Comments

The spray of an affine connection is also called the geodesic spray of this connection.

References

[a1]	D. Gromoll, W. Klingenberg, W. Meyer, "Riemannsche Geometrie im Grossen" , Springer (1968)