Tangent spaces

Tangent vectors at a point form a linear space describing infinitesimal directions.

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Gaussian curvature
The product of principal curvatures measures intrinsic surface curvature.
Regular curves
A regular curve has a nonzero velocity, allowing local tangent geometry.
Riemannian metrics
A smoothly varying positive-definite inner product on tangent spaces measures lengths and angles.
Smooth manifolds
Compatible smooth coordinate charts support differentiation on a manifold.
Arc length
Integrating speed measures a sufficiently regular curve's length.