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)
  • 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 without boundary is complete.

Vector flow in Riemannian geometry

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

The exponential map

exp : T_pM → 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_pM there exists a unique geodesic γ : I → M for which γ(0) = p and $\dot\gamma(0) = V.$ Let D_p be the subset of T_pM 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_eG.

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.