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Diffgeom β

Curvature notions coincide for curve in Euclidean space

Statement

Suppose \gamma:(a,b) \to \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.