Covariant derivatives

A connection differentiates vector fields while accounting for changing tangent spaces.

Different ways to understand it

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Differential forms
Alternating multilinear objects encode integration over oriented geometric pieces.
Gauss–Bonnet theorem
For a closed oriented surface, total Gaussian curvature relates to its Euler characteristic.
Arc length
Integrating speed measures a sufficiently regular curve's length.
Curvature and torsion
Curvature measures tangent turning; torsion describes departure from a plane in space.
Gaussian curvature
The product of principal curvatures measures intrinsic surface curvature.