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on a differentiable manifold

A vector field on the tangent space which, in terms of the local coordinates on associated in a natural way with the local coordinates on , has components , where are functions of class which are, for fixed , positive homogeneous functions in of degree 2 (these properties of do not depend on the actual choice of the local coordinates). The system of differential equations determined by this vector field,

is equivalent to the system of second-order differential equations

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

The most important case of a spray is when the are polynomials of the second degree in the :


In this case the give an affine connection on with zero torsion tensor. Conversely, for every affine connection the equations of the geodesic lines are given by a certain spray with of the form (*) (where when going from the connection to the spray, the symmetrize with respect to the suffixes). If the field is of class , then must have the form (*). In the general case, however, may be smooth outside the zero section of the bundle , but need not be a field of class near this section. In such a situation one sometimes talks about a generalized spray, leaving the term "spray" only for the special case (*). 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]).


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


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


[a1] D. Gromoll, W. Klingenberg, W. Meyer, "Riemannsche Geometrie im Grossen" , Springer (1968)
How to Cite This Entry:
Spray. Encyclopedia of Mathematics. URL:
This article was adapted from an original article by D.V. Anosov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article