mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...

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

vector field In vector calculus and physics, a vector field is an assignment of a vector to each point in a space, most commonly Euclidean space \mathbb^n. A vector field on a plane can be visualized as a collection of arrows with given magnitudes and dire ...

. These appear in a number of different contexts, including

differential topology In mathematics, differential topology is the field dealing with the topological properties and smooth properties of smooth manifolds. In this sense differential topology is distinct from the closely related field of differential geometry, which ...

Riemannian geometry Riemannian geometry is the branch of differential geometry that studies Riemannian manifolds, defined as manifold, smooth manifolds with a ''Riemannian metric'' (an inner product on the tangent space at each point that varies smooth function, smo ...

and

Lie group In mathematics, a Lie group (pronounced ) is a group (mathematics), group that is also a differentiable manifold, such that group multiplication and taking inverses are both differentiable. A manifold is a space that locally resembles Eucli ...

theory.

In differential topology

Let ''V'' be a smooth

on a smooth

manifold In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More precisely, an n-dimensional manifold, or ''n-manifold'' for short, is a topological space with the property that each point has a N ...

''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 In mathematics, an integral curve is a parametric curve that represents a specific solution to an ordinary differential equation or system of equations. Name Integral curves are known by various other names, depending on the nature and interpre ...

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.

In Riemannian geometry

, 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 A map is a symbolic depiction of interrelationships, commonly spatial, between things within a space. A map may be annotated with text and graphics. Like any graphic, a map may be fixed to paper or other durable media, or may be displayed on ...

to the

tangent space In mathematics, the tangent space of a manifold is a generalization of to curves in two-dimensional space and to surfaces in three-dimensional space in higher dimensions. In the context of physics the tangent space to a manifold at a point can be ...

, it represents the

tangent vectors In mathematics, a tangent vector is a vector that is tangent to a curve or surface at a given point. Tangent vectors are described in the differential geometry of curves in the context of curves in R''n''. More generally, tangent vectors are eleme ...

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 :exp : ''T''_''p''''M'' → ''M'' is defined as exp(''X'') = γ(1) where γ : ''I'' → ''M'' is the unique

geodesic In geometry, a geodesic () is a curve representing in some sense the locally shortest path ( arc) between two points in a surface, or more generally in a Riemannian manifold. The term also has meaning in any differentiable manifold with a conn ...

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 In mathematical physics, a pseudo-Riemannian manifold, also called a semi-Riemannian manifold, is a differentiable manifold with a metric tensor that is everywhere nondegenerate. This is a generalization of a Riemannian manifold in which the ...

(or any manifold with an

affine connection In differential geometry, an affine connection is a geometric object on a smooth manifold which ''connects'' nearby tangent spaces, so it permits tangent vector fields to be differentiated as if they were functions on the manifold with values i ...

) 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

\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 vector field on a

is complete. The

starting at the identity is a one-parameter subgroup of ''G''. There are one-to-one correspondences : ⇔ ⇔ g = ''T''_''e''''G''. Let ''G'' be a Lie group and g its

Lie algebra In mathematics, a Lie algebra (pronounced ) is a vector space \mathfrak g together with an operation called the Lie bracket, an alternating bilinear map \mathfrak g \times \mathfrak g \rightarrow \mathfrak g, that satisfies the Jacobi ident ...

. 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 In mathematics, a diffeomorphism is an isomorphism of differentiable manifolds. It is an invertible function that maps one differentiable manifold to another such that both the function and its inverse are continuously differentiable. Definit ...

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''.

References

{{Manifolds Geodesic (mathematics) Differential topology Lie groups

In differential topology

In Riemannian geometry

In Lie group theory

See also

References