Curvature notions coincide for curve in Euclidean space
Suppose is a smooth curve in Euclidean space, parametrized by arc-length. Consider the following way of defining curvature for :
- View the image of as a 1-dimensional submanifold of Euclidean space
- Thus, give it the structure of a Riemannian manifold
- Use this to obtain a Levi-Civita connection on the image of
- Take the curvature of this connection
The curvature we get in this manner is a -tensor: it takes in three tangent vectors, and outputs one tangent vector. However, since the tangent space is one-dimensional, we can view the curvature as a real number: the real number obtained when we put all three input tangent vectors as unit vectors pointing along the -positive direction.
The other notion of curvature is the notion we usually define for a curve in Euclidean space.