The Unapologetic Mathematician

Mathematics for the interested outsider

One-Parameter Groups

Let \Phi:\mathbb{R}\times M\to M be a differentiable map. For each t\in\mathbb{R} we can define the differentiable map \Phi_t:M\to M by \Phi_t(p)=\Phi(t,p). We call the collection \left\{\Phi_t\right\}_{t\in\mathbb{R}} a “one-parameter group” of diffeomorphisms — since it has, obviously, a single parameter — so long as it satisfies the two conditions \Phi_0=1_M and \Phi_{t_1+t_2}=\Phi_{t_1}+\Phi_{t_2}. That is, t\mapsto\Phi_t is a homomorphism from the additive group of real numbers to the diffeomorphism group of M. Indeed, each \Phi_t is a diffeomorphism of M — a differentiable isomorphism of the manifold to itself — with inverse \Phi_{-t}

If we define a vector field X by X(p)=\Phi_*\frac{\partial}{\partial t}(0,p) then \Phi 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 \Phi:W\to M is the maximal flow of a vector field X\in\mathfrak{X}M, then \Phi defines something like a one-parameter subgroup. Indeed, “flowing forward” by t_1 and the again by t_2 is the same as flowing forward by t_1+t_2 along each integral curve, and so \Phi_{t_1+t_2}=\Phi_{t_1}+\Phi_{t_2} wherever both sides of this equation are well-defined. But they might not, since even if both (t_1,p) and (t_2,p) are in W for all p\in M the point (t_1+t_2,p) 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.

May 31, 2011 Posted by | Differential Topology, Topology | 2 Comments

   

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