One-Parameter Groups
Let 
be a differentiable map. For each 
we can define the differentiable map 
by 
. We call the collection 
a “one-parameter group” of diffeomorphisms — since it has, obviously, a single parameter — so long as it satisfies the two conditions 
and 
. That is, 
is a homomorphism from the additive group of real numbers to the diffeomorphism group of 
. Indeed, each 
is a diffeomorphism of — a differentiable isomorphism of the manifold to itself — with inverse 
If we define a vector field 
by 
then 
is a flow for this vector field. Indeed, it’s a maximal flow, since it’s defined for all time at each point.
Conversely, if 
is the maximal flow of a vector field 
, then defines something like a one-parameter subgroup. Indeed, “flowing forward” by 
and the again by 
is the same as flowing forward by 
along each integral curve, and so wherever both sides of this equation are well-defined. But they might not, since even if both 
and 
are in 
for all 
the point 
might not be. But if every integral curve can be extended for all times, then we call the vector field “complete” and conclude that its maximal flow is a one-parameter group of diffeomorphisms.
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May 31, 2011 -
Posted by John Armstrong |
Differential Topology, Topology
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