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''on a differentiable manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086880/s0868801.png" />''
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''on a differentiable manifold $M$''
  
A vector field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086880/s0868802.png" /> on the tangent space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086880/s0868803.png" /> which, in terms of the local coordinates <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086880/s0868804.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086880/s0868805.png" /> associated in a natural way with the local coordinates <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086880/s0868806.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086880/s0868807.png" />, has components <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086880/s0868808.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086880/s0868809.png" /> are functions of class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086880/s08688010.png" /> which are, for fixed <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086880/s08688011.png" />, positive homogeneous functions in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086880/s08688012.png" /> of degree 2 (these properties of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086880/s08688013.png" /> do not depend on the actual choice of the local coordinates). The system of differential equations determined by this vector field,
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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,
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086880/s08688014.png" /></td> </tr></table>
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$$\frac{dx^i}{dt}=v^i,\quad\frac{dv^i}{dt}=f^i(x^1,\dots,x^n,v^1,\dots,v^n),\quad i=1,\dots,n,$$
  
 
is equivalent to the system of second-order differential equations
 
is equivalent to the system of second-order differential equations
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086880/s08688015.png" /></td> </tr></table>
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$$\frac{d^2x^i}{dt^2}=f^i\left(x^1,\dots,x^n,\frac{dx^1}{dt},\dots,\frac{dx^n}{dt}\right);$$
  
therefore a spray describes (and moreover in an invariant manner, that is, independent of the coordinate system) a system of such equations on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086880/s08688016.png" />.
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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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086880/s08688017.png" /> are polynomials of the second degree in the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086880/s08688018.png" />:
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The most important case of a spray is when the $f^i$ are polynomials of the second degree in the $v^i$:
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086880/s08688019.png" /></td> <td valign="top" style="width:5%;text-align:right;">(*)</td></tr></table>
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$$f^i=\sum\Gamma_{jk}^i(x^1,\dots,x^n)v^jv^k,\quad\Gamma_{jk}^i=\Gamma_{kj}^i.\tag{*}$$
  
In this case the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086880/s08688020.png" /> give an [[Affine connection|affine connection]] on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086880/s08688021.png" /> with zero torsion tensor. Conversely, for every affine connection the equations of the geodesic lines are given by a certain spray with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086880/s08688022.png" /> of the form (*) (where when going from the connection to the spray, the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086880/s08688023.png" /> symmetrize with respect to the suffixes). If the field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086880/s08688024.png" /> is of class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086880/s08688025.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086880/s08688026.png" /> must have the form (*). In the general case, however, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086880/s08688027.png" /> may be smooth outside the zero section of the bundle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086880/s08688028.png" />, but need not be a field of class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086880/s08688029.png" /> 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.
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In this case the $\Gamma_{jk}^i$ give an [[Affine connection|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 \ref{*} (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 \ref{*}. 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 \ref{*}. 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 [[#References|[1]]]).
 
It is possible to give a definition of a spray in invariant terms, which is suitable also for Banach manifolds (see [[#References|[1]]]).

Revision as of 11:27, 9 November 2014

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 $v^i$:

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

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 \ref{*} (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 \ref{*}. 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 \ref{*}. 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)
How to Cite This Entry:
Spray. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Spray&oldid=34402
This article was adapted from an original article by D.V. Anosov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article