TITLE: Curves have no intrinsic geometry
QUESTION [5 upvotes]: Please someone could explain me why the curves have no intrinsic geometry? With surfaces I can see that there are two kind of geometries, i.e. the euclidean one (related to euclidean isometries) and the intrinsic (related to isometries on surfaces), but I am not able to make a comparison/analogy with curves. Thanks.

REPLY [6 votes]: Curves have the same intrinsic geometry that a straight line has.  If $B$ is between $A$ and $C$ then the distance from $A$ to $B$ (measured along the curve, i.e. the arc length) plus the distance from $B$ to $C$, is the distance from $A$ to $C$.  If a string winds around through space and you pick it up and lay it down unrolled so that it's a straight line, you don't change any of the intrinsic distances; the length of the curve from $A$ to $B$ is still the same.  Hence the curvature doesn't matter.