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Is Geodesic a signature of Geometry of a Space and if it is then it should also have a co-ordinate independent definition perhaps should be incorporated in Tensor Mathematics with a associated name Geodesic tensor.

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What do you mean by signature? Also what do you mean by coordinate independent definition? A geodesic between to point is the curve that has the minimal (or the maximal sometimes) length (at least locally). Coordinates come into play when you try to compute it, not when you define it. –  tst Nov 25 '12 at 2:49
    
i wanted to put a meaning in the word signature which i found iwas much comfortable with, is something that is definition of spatial geometry.As any geometry can be described in many co-ordinate systems so tensor is the only tool to understand what they really are.And co-ordinate independent word that i used was i realized a loose word,it should be replaced with invariant in co-ordinate transformations.i hope now you understand my question.sorry for loose wording. –  Abhinav Anand Nov 25 '12 at 3:02
    
The geodesics are determined by the Riemmanian metric, so obviously are are coordinate independent in the sense that there is a smooth way to move them from one coordinate system to an other. As for the question of signature let me repeat myself, geodesics are determined by the Riemmanian metric. How familiar are you with these? Have you followed any course on differential geometry? –  tst Nov 25 '12 at 4:29
    
Now i am asking my question in a different way that is it possible for two different kind of geometric spaces defined by two different Riemann tensors with same geodesic? And if it is not so then Riemann tensor and geodesic are equivalent to each other in every possible sense and so geodesic tensor should also exist.I did not find this in my book of differential geometry by Willmore. –  Abhinav Anand Nov 25 '12 at 17:30
    
In which sense can 2 geodesics on different manifolds be the same or equal whatsoever? –  tst Nov 26 '12 at 0:00
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