Hi guys,


I am able to prove that any symmetric manifold is complete (Consider a local geodesic and use the symmetry to flip it, effectively doubling the length of the geodesic, ad infinitum). I want to use a similar procedure to prove that a manifold whose isometries act transitively is complete, i.e there is always an isometry which maps the start point of a local geodesic to its end point, preserving the geodesic. I am, however, unable to ensure that it is not `rotated' in the process, i.e I want the pushforward of the initial tangent, by the isometry, to be the final tangent, ensuring the resultant doubled geodesic is smooth.


My Lie group theory is a bit scratchy but I assume there is a method which allows me to construct the correct pushforward using only transitivity.


Any ideas would be great,


regards,


MK



Action of the group of isometries on a manifold
By the Hopf-Rinow theorem, you only have to prove that the manifold is a complete metric space. By homogeneity, the injectivity radius is bounded from below by a uniform positive constant. Using this and the compacity of balls whose radius is smaller than the injectivity radius of their center it is easy to check the convergence of Cauchy sequences.


Another way to do this is to interpret the bound on injectivity radius, $r$ say, in term of geodesic extension: a geodesic $\gamma$ defined on $[a,b]$ can be extended to a geodesic defined on $(a-r,b+r)$. From this the conclusion follows.


