Vector flow

In mathematics, the vector flow refers to a set of closely related concepts of the flow determined by a vector field. These appear in a number of different contexts, including differential topology, Riemannian geometry and Lie group theory. These related concepts are explored in a spectrum of articles:

*exponential map
**matrix exponential
**exponential function
*infinitesimal generator (→ Lie group)
*integral curve (→ vector field)
*one-parameter subgroup
*flow (geometry)
**geodesic flow
**Hamiltonian flow
**Ricci flow
**Anosov flow
*injectivity radius (→ glossary)

Vector flow in differential topology

Relevant concepts: "(flow, infinitesimal generator, integral curve, complete vector field)"

Let "V" be a smooth vector field on a smooth manifold "M". There is a unique maximal flow "D" → "M" whose infinitesimal generator is "V". Here "D" ⊆ R × "M" is the flow domain. For each "p" ∈ "M" the map "D"_"p" → "M" is the unique maximal integral curve of "V" starting at "p".

A global flow is one whose flow domain is all of R × "M". Global flows define smooth actions of R on "M". A vector field is complete if it generates a global flow. Every vector field on a compact manifold is complete.

Vector flow in Riemannian geometry

Relevant concepts: "(geodesic, exponential map, injectivity radius)"

The exponential map:exp : "T"_"p""M" → "M"is defined as exp("X") = γ(1) where γ : "I" → "M" is the unique geodesic passing through "p" at 0 and whose tangent vector at 0 is "X". Here "I" is the maximal open interval of R for which the geodesic is defined.

Let "M" be a pseudo-Riemannian manifold (or any manifold with an affine connection) and let "p" be a point in "M". Then for every "V" in "T"_"p""M" there exists a unique geodesic γ : "I" → "M" for which γ(0) = "p" and $dotgamma(0)\; =\; V.$ Let "D"_"p" be the subset of "T"_"p""M" for which 1 lies in "I".

Vector flow in Lie group theory

Relevant concepts: "(exponential map, infinitesimal generator, one-parameter group)"

Every left-invariant vector field on a Lie group is complete. The integral curve starting at the identity is a one-parameter subgroup of "G". There are one-to-one correspondences:{one-parameter subgroups of "G"} ⇔ {left-invariant vector fields on "G"} ⇔ g = "T"_"e""G".

Let "G" be a Lie group and g its Lie algebra. The exponential map is a map exp : g → "G" given by exp("X") = γ(1) where γ is the integral curve starting at the identity in "G" generated by "X".
*The exponential map is smooth.
*For a fixed "X", the map "t" $mapsto$ exp("tX") is the one-parameter subgroup of "G" generated by "X".
*The exponential map restricts to a diffeomorphism from some neighborhood of 0 in g to a neighborhood of "e" in "G".
*The image of the exponential map always lies in the connected component of the identity in "G".