Let be a differentiable manifold, let be an affine connection on , and let be a point in . Given a tangent vector at , there is are several geodesics on tangent to at . Note that is formally a differentiable map to from some interval , in particular a partial differentiable map to from the real line . (Here, is an upper real number so may be ; and is a lower real number so may be .)
If and are geodesics (on tangent to at ), then they agree on their common domain . Accordingly, If , then there is at most one extension of to that remains a geodesic. If is complete? (and perhaps in any case), then may be extended (uniquely) to all of . Regardless, there is a unique maximal geodesic (on tangent to at ). This is the maximal geodesic on tangent to at .
Last revised on April 30, 2013 at 18:05:38. See the history of this page for a list of all contributions to it.