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If a space has curvature, then the curvature can be seen intrinsically by finding sums of angles in triangles made of geodesics. Under general relativity, space-time is curved on local scales. On global scales, experiments have determined that space-time is almost totally flat. If one were to, for each of three points in a gravitational gradient (like near a black hole), find the angle between the sight lines to the other two points and then add up all of the angles, would it add up to 180 (degrees) or perhaps it would show intrinsic curvature?

Is the way of thinking about this good, because it is quite general and conceptual. I don't actually know the equations for general relativity, but this is what I thought of when I was thinking about intrinsic geometric properties.

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    I think that there is a problem in this view: in GR we are not dealing with Riemannian metrics, but instead with Lorentzian ones. That is, the signture of the metric is "(-,+,+,+)". The notion of "angle", something that varies between 0 and \Pi and denotes the inclination of one direction with respect to the other, is good for Riemannian metrics. There is no sense (as far as I know, please someone correct me if this is not true) in talking about this kind of angle in Lorentzian manifolds.2010-10-19
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    What is well-defined is the "hyperbolic angle" between two time-like vectors, but this is another thing. Maybe you can look at a Riemannian submanifold (with a positive-definite metric) to analyze angles, but I don't know if this helps.2010-10-19
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    @Ronaldo: Now looking at the comments I think I have just done what you had in mind :) Greets2011-01-18

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