>>14180898Curvature is a euclidean invariant. In the plane you have only one function for curvature. If is a curve (parameterized by arclength), the we can define the frame at each point of the curve by and via and . Then implies so that . Moreover, the definition of implies that . Without loss of generality, we can choose since the equations are invariant under . Of course, represent the "change in direction" along the curve. is an infinitesimal measure of how much the curve deviates from a straight line as is tangent to the curve, hence the amount the curve deviates from a line at point can be measure by the magnitude which is called the of .
If you decide to work in , then you find a second invariant, Torsion. If Torsion is zero, then the curve is equivalent to a plane curve. If you want higher dimensions, the invariants start to become much more complicated to compute and hence it is more expedient to use the first and second fundamental forms.
For more on this (and other geometries), try "Cartan for Beginners: ..." by Ivey + Landsberg
For a completely formal and technical review, try "Foundations of Differential Geometry vol 1+2" by Kobayashi and Nomizu.
For foundations try "Introduction to Smooth manifolds" by Lee, "Foundations of Differentiable Manifolds and Lie Groups" by Warner (quite technical), and "An Introduction to Differential Manifolds and Riemannian Geometry" by Boothby.
