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 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 : TpM → 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 TpM there exists a unique geodesic γ : I → M for which γ(0) = p and 
Let Dp be the subset of TpM for which 1 lies in I.
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