# Vector flow

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Concepts in mathematics

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In [mathematics](/source/Mathematics), the **vector flow** refers to a set of closely related concepts of the [flow](/source/Flow_(mathematics)) determined by a [vector field](/source/Vector_field). These appear in a number of different contexts, including [differential topology](/source/Differential_topology), [Riemannian geometry](/source/Riemannian_geometry) and [Lie group](/source/Lie_group) theory.

## In differential topology

Let *V* be a smooth [vector field](/source/Vector_field) on a smooth [manifold](/source/Manifold) *M*. There is a unique maximal [flow](/source/Flow_(geometry)) *D* → *M* whose [infinitesimal generator](/source/Lie_group#The_Lie_algebra_associated_to_a_Lie_group) 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](/source/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](/source/Vector_field#Complete_vector_fields) if it generates a global flow. Every smooth vector field on a compact manifold without boundary is complete.

## In Riemannian geometry

In [Riemannian geometry](/source/Riemannian_geometry), a vector flow can be thought of as a solution to the system of [differential equations](/source/Differential_equations) induced by a vector field.[1] That is, if a (conservative) vector field is a [map](/source/Map_(mathematics)) to the [tangent space](/source/Tangent_space), it represents the [tangent vectors](/source/Tangent_vector) to some function at each point. Splitting the tangent vectors into directional derivatives, one can solve the resulting system of differential equations to find the function. In this sense, the function is the flow and both induces and is induced by the vector field.

From a point, the rate of change of the i-th component with respect to the parametrization of the flow (“how much the flow has acted”) is described by the i-th component of the field. That is, if one parametrizes with *L* ‘length along the path of the flow,’ as one proceeds along the flow by *dL* the first position component changes as described by the first component of the vector field at the point one starts from, and likewise for all other components.

The [exponential map](/source/Exponential_map_(Riemannian_geometry))

- exp : *T**p**M* → *M*

is defined as exp(*X*) = γ(1) where γ : *I* → *M* is the unique [geodesic](/source/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](/source/Pseudo-Riemannian_manifold) (or any manifold with an [affine connection](/source/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 γ ˙ ( 0 ) = V . {\displaystyle {\dot {\gamma }}(0)=V.} Let *D**p* be the subset of *T**p**M* for which 1 lies in *I*.

## In Lie group theory

Every [left-invariant](/source/Left_invariant) vector field on a [Lie group](/source/Lie_group) is complete. The [integral curve](/source/Integral_curve) starting at the identity is a [one-parameter subgroup](/source/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](/source/Lie_algebra). The [exponential map](/source/Exponential_map_(Lie_theory)) 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](/source/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*.

## See also

- [Gradient vector flow](/source/Gradient_vector_flow) – Computer vision framework

## References

1. **[^](#cite_ref-1)** Chen, Ricky T. Q.; Lipman, Yaron (2024-02-26). "Flow Matching on General Geometries". [arXiv](/source/ArXiv_(identifier)):[2302.03660](https://arxiv.org/abs/2302.03660) [[cs.LG](https://arxiv.org/archive/cs.LG)].

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Adapted from the Wikipedia article [Vector flow](https://en.wikipedia.org/wiki/Vector_flow) by Wikipedia contributors ([contributor history](https://en.wikipedia.org/wiki/Vector_flow?action=history)). Available under [Creative Commons Attribution-ShareAlike 4.0 International](https://creativecommons.org/licenses/by-sa/4.0/). Changes may have been made.