Vector flow

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:

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

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_pM there exists a unique geodesic γ : I → M for which γ(0) = p and ${\displaystyle {\dot {\gamma }}(0)=V.}$ Let D_p be the subset of T_pM for which 1 lies in I.

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