Differential Geometry (10): Riemannian Geometry — Metrics, Connections, and Parallel Transport

Fri, 19 Nov 2021 09:00:00 +0000

Up to this point in the series we have studied smooth manifolds with their differentiable structure: charts, tangent vectors, differential forms, exterior calculus, integration. None of this required a notion of distance. We could differentiate functions, integrate top-degree forms, decide whether a distribution is integrable — all without ever measuring how long a curve is or what angle two tangent vectors make. The smooth structure is purely topological-with-derivatives. To do geometry in the classical sense — to measure, to compare, to speak of curvature, to recover the everyday meanings of “length” and “angle” — we need additional structure.

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Differential Geometry (10): Riemannian Geometry — Metrics, Connections, and Parallel Transport