Vector flow

Inmathematics,thevector flowrefers to a set of closely related concepts of theflowdetermined by avector field.These appear in a number of different contexts, includingdifferential topology,Riemannian geometryandLie grouptheory. 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)

LetVbe a smooth vector field on a smooth manifoldM.There is a unique maximalflowD→Mwhoseinfinitesimal generatorisV.HereD⊆R×Mis theflow domain.For eachp∈Mthe mapD_p→Mis the unique maximalintegral curveofVstarting atp.

Aglobal flowis one whose flow domain is all ofR×M.Global flows define smooth actions ofRonM.A vector field iscompleteif 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)

A vector flow can be thought of as a solution to the system of differential equations induced by a vector field. That is, if a (conservative) vector field is a map to the tangent space, it represents the tangent vectors 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 withL‘length along the path of the flow,’ as one proceeds along the flow bydLthe 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.

Theexponential map

exp:T_pM→M

is defined as exp(X) = γ(1) where γ:I→Mis the unique geodesic passing throughpat 0 and whose tangent vector at 0 isX.HereIis the maximal open interval ofRfor which the geodesic is defined.

LetMbe a pseudo-Riemannian manifold (or any manifold with anaffine connection) and letpbe a point inM.Then for everyVinT_pMthere exists a unique geodesic γ:I→Mfor which γ(0) =pand${\dot {\gamma }}(0)=V.$LetD_pbe the subset ofT_pMfor which 1 lies inI.

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. Theintegral curvestarting at the identity is aone-parameter subgroupofG.There are one-to-one correspondences

{one-parameter subgroups ofG} ⇔ {left-invariant vector fields onG} ⇔g=T_eG.

LetGbe a Lie group andgits Lie algebra. Theexponential mapis a map exp:g→Ggiven by exp(X) = γ(1) where γ is the integral curve starting at the identity inGgenerated byX.

• The exponential map is smooth.
• For a fixedX,the mapt↦ exp(tX) is the one-parameter subgroup ofGgenerated byX.
• The exponential map restricts to a diffeomorphism from some neighborhood of 0 ingto a neighborhood ofeinG.
• The image of the exponential map always lies in the connected component of the identity inG.

See also

• Gradient vector flow– Computer vision framework