I just finished learning the fundamental theorem of curves in 3 dimensions. As a reminder, this is the theorem that states that a continuous, C infinity, unit speed curve in 3d is uniquely determined by its curvature and torsion (up to actions by SE(3), that is rotations and translations). My question is, how does this generalize to higher dimensions? I suspect that in n dimensions, you would need n-1 Real functions ( in 3d we have the curvature and torsion) to uniquely determine a unit curve up to actions under SE(n). My reasoning is that curvature in a sense tells the curve how to move in a plane, and torsion tells it how to move off the plane, so for each extra dimension you would need one additional number, to tell it how to move in the extra "direction". Does anyone know if this is right, and how you could prove it? My prof had never heard of a generalization of this sort. Could anyone direct me to some books, journals etc... which deal with this? Thanks.