nLab
maximal geodesic

Let MM be a differentiable manifold, let \Del be an affine connection on MM, and let pp be a point in MM. Given a tangent vector xx at pp, there is are several geodesics γ\gamma on MM tangent to xx at pp. Note that γ\gamma is formally a differentiable map to MM from some interval ]a,b[{]a,b[}, in particular a partial differentiable map to MM from the real line \mathbb{R}. (Here, aa is an upper real number so may be -\infty; and bb is a lower real number so may be \infty.)

If γ:]a,b[M\gamma\colon {]a,b[} \to M and γ:]a,b[M\gamma'\colon {]a',b'[} \to M are geodesics (on MM tangent to xx at pp), then they agree on their common domain ]max(a,a),min(b,b)[{]max(a,a'),min(b,b')[}. Accordingly, If ]a,b[]a,b[{]a,b[} \subseteq {]a',b'[}, then there is at most one extension of γ\gamma to ]a,b[{]a',b'[} that remains a geodesic. If MM is complete? (and perhaps in any case), then γ\gamma may be extended (uniquely) to all of =],[\mathbb{R} = {]-\infty,\infty[}. Regardless, there is a unique maximal geodesic (on MM tangent to xx at pp). This is the maximal geodesic on MM tangent to xx at pp.

Last revised on April 30, 2013 at 18:05:38. See the history of this page for a list of all contributions to it.