In mathematics

Mathematics

Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

, the vector flow refers to a set of closely related concepts of the flow

Flow (mathematics)

In mathematics, a flow formalizes the idea of the motion of particles in a fluid. Flows are ubiquitous in science, including engineering and physics. The notion of flow is basic to the study of ordinary differential equations. Informally, a flow may be viewed as a continuous motion of points over...

 determined by a vector field

Vector field

In vector calculus, a vector field is an assignmentof a vector to each point in a subset of Euclidean space. A vector field in the plane for instance can be visualized as an arrow, with a given magnitude and direction, attached to each point in the plane...

. These appear in a number of different contexts, including differential topology

Differential topology

In mathematics, differential topology is the field dealing with differentiable functions on differentiable manifolds. It is closely related to differential geometry and together they make up the geometric theory of differentiable manifolds.- Description :...

, Riemannian geometry

Riemannian geometry

Riemannian geometry is the branch of differential geometry that studies Riemannian manifolds, smooth manifolds with a Riemannian metric, i.e. with an inner product on the tangent space at each point which varies smoothly from point to point. This gives, in particular, local notions of angle, length...

 and Lie group

Lie group

In mathematics, a Lie group is a group which is also a differentiable manifold, with the property that the group operations are compatible with the smooth structure...

 theory. These related concepts are explored in a spectrum of articles:

• exponential map
  Exponential map
  In differential geometry, the exponential map is a generalization of the ordinary exponential function of mathematical analysis to all differentiable manifolds with an affine connection....
  
  
  • matrix exponential
    Matrix exponential
    In mathematics, the matrix exponential is a matrix function on square matrices analogous to the ordinary exponential function. Abstractly, the matrix exponential gives the connection between a matrix Lie algebra and the corresponding Lie group....
    
    
  • exponential function
    Exponential function
    In mathematics, the exponential function is the function ex, where e is the number such that the function ex is its own derivative. The exponential function is used to model a relationship in which a constant change in the independent variable gives the same proportional change In mathematics,...
    
    
• infinitesimal generator (→ Lie group)
• integral curve
  Integral curve
  In mathematics, an integral curve is a parametric curve that represents a specific solution to an ordinary differential equation or system of equations...
  
   (→ vector field)
• one-parameter subgroup
• flow (geometry)
  • geodesic flow
  • Hamiltonian flow
  • Ricci flow
    Ricci flow
    In differential geometry, the Ricci flow is an intrinsic geometric flow. It is a process that deforms the metric of a Riemannian manifold in a way formally analogous to the diffusion of heat, smoothing out irregularities in the metric....
    
    
  • 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

Integral curve

In mathematics, an integral curve is a parametric curve that represents a specific solution to an ordinary differential equation or system of equations...

 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

Affine connection

In the branch of mathematics called differential geometry, an affine connection is a geometrical object on a smooth manifold which connects nearby tangent spaces, and so permits tangent vector fields to be differentiated as if they were functions on the manifold with values in a fixed vector space...

) 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 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

Integral curve

In mathematics, an integral curve is a parametric curve that represents a specific solution to an ordinary differential equation or system of equations...

 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

Exponential map

In differential geometry, the exponential map is a generalization of the ordinary exponential function of mathematical analysis to all differentiable manifolds with an affine connection....

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.