I study differential geometry. Although there is a lot of study on the local theory, a global description lacks some explicit explanations. I mean, the study of surfaces describes curves, tangent plane, covariant derivative, the geodesic equation. However, I was not able to find a systematic manner to calculate the distance function rather than solve the geodesic equation with begin and end points and integrate its length with extreme as control points. Can you see any manner to find the distance function on an algebraic connected and complete (geodesic may pass everywhere) surface of type $z = f(x, y)$?
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I would not expect the distance function to be expressible in "closed form" in general, so the question becomes, what do you mean by "find" it? You should be able to find arbitrarily good upper and lower bounds using numerical methods. – Robert Israel Apr 25 '21 at 22:58
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Could you explain a little better upper limit? Lower limit is 0. :sweat_smile: – Bruno Peixoto Apr 26 '21 at 22:10
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Hi, Matt. Your answer suffices for me. – Bruno Lobo Apr 29 '21 at 15:05