One-parameter transformation group
flow
The action of the additive group of real numbers on a manifold
.
Thus, a one-parameter family of transformations of a manifold
is a one-parameter transformation group if the following conditions are satisfied:
![]() | (*) |
If the manifold is smooth, then the group is usually assumed to be smooth also, that is, the corresponding mapping
![]() |
is a differentiable mapping of differentiable manifolds.
A more general concept is that of a local one-parameter transformation group of a manifold . It is defined as a mapping
of some open submanifold
of the form
, where
,
for
, satisfying the conditions (*) for all
,
for which both sides of the equations are defined.
With each smooth local one-parameter transformation group of
one associates the vector field
![]() |
called the velocity field, or infinitesimal generator, of the group . Conversely, any smooth vector field
generates a local one-parameter transformation group
having velocity field
. In local coordinates
on
this one-parameter transformation group is given as the solution of the system of ordinary differential equations
![]() |
with the initial conditions , where
.
If the local one-parameter transformation group generated by the vector field can be extended to a global one, then the field
is called complete. On a compact manifold any vector field is complete, so that there is a one-to-one correspondence between one-parameter transformation groups and vector fields. This is not the case for non-compact manifolds, and the set of complete vector fields is not even closed under addition.
References
[1] | V.I. Arnol'd, "Ordinary differential equations" , M.I.T. (1973) (Translated from Russian) |
[2] | R. Palais, "A global formulation of the Lie theory of transformation groups" , Amer. Math. Soc. (1957) |
Comments
References
[a1] | G.R. Sell, "Topological dynamics and ordinary differential equations" , v. Nostrand-Reinhold (1971) |
One-parameter transformation group. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=One-parameter_transformation_group&oldid=48043