Vector Flow - Vector Flow in Riemannian Geometry

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) 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.