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Revision as of 21:41, 17 May 2007

this article defines a geodesically complete riemannian manifold (M,g) to be one that has the property (call it P) that any two points in the manifold M can be connected by a length-minimizing geodesic. i'm not sure this is the most precise definition of geodesic completeness: consider a proper open subset (say the unit open ball) in ${\displaystyle \mathbb {R} ^{n}}$ with standard euclidean metric g. then surely any two points in the ball can be connected by a length-minimizing geodesic (namely a straight line between the two points), but we don't consider the unit open ball to be geodesically complete because the spray of geodesics emanating from any point p in the ball runs out of the ball in finite time.

if we define geodesic completeness to mean that all geodesics have domain ${\displaystyle \mathbb {R} }$, rather than just an open subinterval of ${\displaystyle \mathbb {R} }$, then of course we get property P. but the example i show above seems to suggest the converse is not true. we want the more primitive notion of geodesic completeness; since geodesics can be understood as orbits of a hamiltonian vector field on the tangent bundle ${\displaystyle TM}$, this deeper definition gives a global hamiltonian flow on ${\displaystyle TM}$ and thus a smooth action of ${\displaystyle \mathbb {R} }$ on ${\displaystyle TM}$. merely assuming property P will not give this smooth action.

Mlord 21:41, 17 May 2007 (UTC)

Complete manifold (block log • contribs • deleted contribs • abuse filter log • creation log • change block settings • unblock • user rights management • checkuser (log))

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