Spray
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]).
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) |
Spray. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Spray&oldid=13449