# 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 *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 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 :
*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 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*↦ 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*.