One-parameter transformation group

flow

The action of the additive group of real numbers $ \mathbf R $ on a manifold $ M $.

Thus, a one-parameter family $ \{ {\phi _ {t} } : {t \in \mathbf R } \} $ of transformations of a manifold $ M $ is a one-parameter transformation group if the following conditions are satisfied:

$$ \tag{* } \phi _ {t + s } x = \ \phi _ {t} ( \phi _ {s} x),\ \ \phi _ {-t} x = \ \phi _ {t} ^ {-1} x,\ \ t, s \in \mathbf R ,\ \ x \in M. $$

If the manifold $ M $ is smooth, then the group is usually assumed to be smooth also, that is, the corresponding mapping

$$ \phi : \mathbf R \times M \rightarrow M,\ \ ( t, x) \rightarrow \phi _ {t} x , $$

is a differentiable mapping of differentiable manifolds.

A more general concept is that of a local one-parameter transformation group of a manifold $ M $. It is defined as a mapping $ \phi : U \rightarrow M $ of some open submanifold $ U \subset \mathbf R \times M $ of the form $ U = \cup _ {x \in M } ( \left ] \epsilon _ {-} ( x), \epsilon _ {+} ( x) \right [ , x) $, where $ \epsilon _ {+} ( x) > 0 $, $ \epsilon _ {-} ( x) < 0 $ for $ x \in M $, satisfying the conditions (*) for all $ t, s \in \mathbf R $, $ x \in M $ for which both sides of the equations are defined.

With each smooth local one-parameter transformation group $ \{ \phi _ {t} \} $ of $ M $ one associates the vector field

$$ M \ni x \rightarrow X _ {x} = \ \left . { \frac{d}{dt } } \phi _ {t} x \right | _ {t = 0 } , $$

called the velocity field, or infinitesimal generator, of the group $ \{ \phi _ {t} \} $. Conversely, any smooth vector field $ X $ generates a local one-parameter transformation group $ \phi _ {t} $ having velocity field $ X $. In local coordinates $ x ^ {i} $ on $ M $ this one-parameter transformation group is given as the solution of the system of ordinary differential equations

$$ \frac{d \phi ^ {i} ( t, x ^ {j} ) }{dt } = X ^ {i} ( \phi ^ {j} ( t, x ^ {k} ) ) $$

with the initial conditions $ \phi ^ {i} ( 0, x ^ {j} ) = x ^ {i} $, where $ X = \sum _ {i} X ^ {i} \partial / \partial x ^ {i} $.

If the local one-parameter transformation group generated by the vector field $ X $ can be extended to a global one, then the field $ X $ 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)

References

[a1]	G.R. Sell, "Topological dynamics and ordinary differential equations" , v. Nostrand-Reinhold (1971)