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:

- Vector flow in differential topology
- Vector flow in Riemannian geometry
- Vector flow in Lie group theory
- See also

- exponential map (Riemannian geometry)
- infinitesimal generator (→ Lie group)
- integral curve (→ vector field)
- one-parameter subgroup
- flow (geometry)
- injectivity radius (→ glossary)

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.

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*.

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*.

In mathematics, a **Lie group** is a group that is also a differentiable manifold. A manifold is a space that locally resembles Euclidean space, whereas groups define the abstract, generic concept of multiplication and the taking of inverses (division). Combining these two ideas, one obtains a continuous group where points can be multiplied together, and their inverse can be taken. If, in addition, the multiplication and taking of inverses are defined to be smooth (differentiable), one obtains a Lie group.

In vector calculus and physics, a **vector field** is an assignment of a vector to each point in a subset of space. For instance, a vector field in the plane can be visualised as a collection of arrows with a given magnitude and direction, each attached to a point in the plane. Vector fields are often used to model, for example, the speed and direction of a moving fluid throughout space, or the strength and direction of some force, such as the magnetic or gravitational force, as it changes from one point to another point.

In geometry, a **geodesic** is commonly a curve representing in some sense the 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 connection. It is a generalization of the notion of a "straight line" to a more general setting.

In differential geometry, a **Riemannian manifold** or **Riemannian space**(*M*, *g*) is a real, smooth manifold *M* equipped with a positive-definite inner product *g*_{p} on the tangent space *T*_{p}*M* at each point *p*. A common convention is to take *g* to be smooth, which means that for any smooth coordinate chart (*U*, *x*) on *M*, the *n*^{2} functions

In Riemannian geometry, an **exponential map** is a map from a subset of a tangent space T_{p}*M* of a Riemannian manifold *M* to *M* itself. The (pseudo) Riemannian metric determines a canonical affine connection, and the exponential map of the (pseudo) Riemannian manifold is given by the exponential map of this connection.

In geometry, **parallel transport** is a way of transporting geometrical data along smooth curves in a manifold. If the manifold is equipped with an affine connection, then this connection allows one to transport vectors of the manifold along curves so that they stay *parallel* with respect to the connection.

In mathematics, a **symplectomorphism** or **symplectic map** is an isomorphism in the category of symplectic manifolds. In classical mechanics, a symplectomorphism represents a transformation of phase space that is volume-preserving and preserves the symplectic structure of phase space, and is called a canonical transformation.

In mathematics, particularly differential geometry, a **Finsler manifold** is a differentiable manifold *M* where a **Minkowski functional***F*(*x*, −) is provided on each tangent space T_{x}*M*, that enables one to define the length of any smooth curve *γ* : [*a*, *b*] → *M* as

This is a glossary of some terms used in Riemannian geometry and metric geometry — it doesn't cover the terminology of differential topology.

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 in a fixed vector space. Turns out that connections are the easiest way to define differentiation of the sections of vector bundles.

In mathematics, a **differentiable manifold** is a type of manifold that is locally similar enough to a linear space to allow one to do calculus. Any manifold can be described by a collection of charts, also known as an atlas. One may then apply ideas from calculus while working within the individual charts, since each chart lies within a linear space to which the usual rules of calculus apply. If the charts are suitably compatible, then computations done in one chart are valid in any other differentiable chart.

In mathematics, more particularly in the fields of dynamical systems and geometric topology, an **Anosov map** on a manifold *M* is a certain type of mapping, from *M* to itself, with rather clearly marked local directions of "expansion" and "contraction". Anosov systems are a special case of Axiom A systems.

In the theory of smooth manifolds, a **congruence** is the set of integral curves defined by a nonvanishing vector field defined on the manifold.

In the study of mathematics and especially differential geometry, **fundamental vector fields** are an instrument that describes the infinitesimal behaviour of a smooth Lie group action on a smooth manifold. Such vector fields find important applications in the study of Lie theory, symplectic geometry, and the study of Hamiltonian group actions.

In the mathematical field of differential geometry, a smooth map from one Riemannian manifold to another Riemannian manifold is called **harmonic** if its coordinate representatives satisfy a certain nonlinear partial differential equation. This partial differential equation for a mapping also arises as the Euler-Lagrange equation of a functional generalizing the Dirichlet energy. As such, the theory of harmonic maps encompasses both the theory of unit-speed geodesics in Riemannian geometry, and the theory of harmonic functions on open subsets of Euclidean space and on Riemannian manifolds.

In differential geometry, **normal coordinates** at a point *p* in a differentiable manifold equipped with a symmetric affine connection are a local coordinate system in a neighborhood of *p* obtained by applying the exponential map to the tangent space at *p*. In a normal coordinate system, the Christoffel symbols of the connection vanish at the point *p*, thus often simplifying local calculations. In normal coordinates associated to the Levi-Civita connection of a Riemannian manifold, one can additionally arrange that the metric tensor is the Kronecker delta at the point *p*, and that the first partial derivatives of the metric at *p* vanish.

In Riemannian geometry, the **cut locus** of a point in a manifold is roughly the set of all other points for which there are multiple minimizing geodesics connecting them from , but it may contain additional points where the minimizing geodesic is unique, under certain circumstances. The distance function from *p* is a smooth function except at the point *p* itself and the cut locus.

In mathematics, the **differential geometry of surfaces** deals with the differential geometry of smooth surfaces with various additional structures, most often, a Riemannian metric. Surfaces have been extensively studied from various perspectives: *extrinsically*, relating to their embedding in Euclidean space and *intrinsically*, reflecting their properties determined solely by the distance within the surface as measured along curves on the surface. One of the fundamental concepts investigated is the Gaussian curvature, first studied in depth by Carl Friedrich Gauss, who showed that curvature was an intrinsic property of a surface, independent of its isometric embedding in Euclidean space.

In mathematics, the **Riemannian connection on a surface** or Riemannian 2-manifold refers to several intrinsic geometric structures discovered by Tullio Levi-Civita, Élie Cartan and Hermann Weyl in the early part of the twentieth century: parallel transport, covariant derivative and connection form. These concepts were put in their current form with principal bundles only in the 1950s. The classical nineteenth century approach to the differential geometry of surfaces, due in large part to Carl Friedrich Gauss, has been reworked in this modern framework, which provides the natural setting for the classical theory of the moving frame as well as the Riemannian geometry of higher-dimensional Riemannian manifolds. This account is intended as an introduction to the theory of connections.

In the theory of Lie groups, the **exponential map** is a map from the Lie algebra of a Lie group to the group which allows one to recapture the local group structure from the Lie algebra. The existence of the exponential map is one of the primary reasons that Lie algebras are a useful tool for studying Lie groups.

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