Curve (differential manifold)

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Definition

In the sense of differential manifolds, a curve is a connected one-dimensional differential manifold. Note that this differs somewhat from the notion of a curve in Euclidean space or a curve (Riemannian manifold).

There are two diffeomorphism types of a curve:

  • The real line: This is the non-compact case
  • The circle: This is the compact case

However, there is a wide range of possibilities for a curve as a Riemannian manifold, and an even wider range of possibilities for embeddings in the Euclidean plane. For instance, an embedding of the circle in the Euclidean plane, is equivalent to the notion of a simple closed curve in the plane.