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:
Relevant concepts: '
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: '
The exponential map
is defined as exp = γ 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 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 = p and Let Dp be the subset of TpM for which 1 lies in I''.

Vector flow in Lie group theory

Relevant concepts: '
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
Let G be a Lie group and g its Lie algebra. The exponential map is a map exp : g → G given by exp = γ where γ is the integral curve starting at the identity in G generated by X.