Vector Flow in Riemannian Geometry
Relevant concepts: (geodesic, exponential map, injectivity radius)
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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