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 (Riemannian geometry)
- infinitesimal generator (→ Lie group)
- integral curve (→ vector field)
- one-parameter subgroup
- flow (geometry)
- 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 Dp → 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 smooth vector field on a compact manifold without boundary is complete.
Vector flow in Riemannian geometry
Relevant concepts: (geodesic, exponential map, injectivity radius)
The exponential map
- exp : TpM → 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 TpM there exists a unique geodesic γ : I → M for which γ(0) = p and Let Dp be the subset of TpM 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 = TeG.
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 ↦ 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.