## 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.