Curvature notions coincide for curve in Euclidean space

Statement

Suppose $\gamma :(a,b)\to \mathbb {R} ^{n}$ is a smooth curve in Euclidean space, parametrized by arc-length. Consider the following way of defining curvature for $\gamma$:

• View the image of $\gamma$ 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 $\gamma$
• Take the curvature of this connection

The curvature we get in this manner is a $(1,3)$-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 $\gamma$-positive direction.

The other notion of curvature is the notion we usually define for a curve in Euclidean space.