Curvature and torsion

Curvature measures tangent turning; torsion describes departure from a plane in space.

Different ways to understand it

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Arc length
Integrating speed measures a sufficiently regular curve's length.
Regular surfaces
Regular surface charts have linearly independent coordinate tangent vectors.
Covariant derivatives
A connection differentiates vector fields while accounting for changing tangent spaces.
Differential forms
Alternating multilinear objects encode integration over oriented geometric pieces.
Gaussian curvature
The product of principal curvatures measures intrinsic surface curvature.