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I am studying geodesic flows on manifolds and I have problem visualizing it. How should one think about geodesic vector field on tangent bundle? How can you visualize it? Any intuition is welcome.

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I like to think a geodesic flow as motion that follows the "path of least resistance", the natural motion that an object would follow in "real life" if, say, you roll a ball along some surface. But perhaps specialists have a better way of imagining it. –  Christopher A. Wong Oct 4 '12 at 7:26

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