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One-parameter transformation group

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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)
How to Cite This Entry:
One-parameter transformation group. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=One-parameter_transformation_group&oldid=48043
This article was adapted from an original article by D.V. Alekseevskii (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article