= Riemannian geometry
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Riemannian geometry studies <smooth manifolds> equipped with a <Riemannian metric>: an <inner product> at each <tangent space>, varying smoothly with position. This local measuring rule determines <angles>, <arc length>, <Riemannian distance> and <geodesics>. An inhabitant of a <surface> can discover its geometry by measuring within it, without knowing how it sits in a larger <Euclidean space>.
A <circular cylinder> is a particularly useful example. Its surface bends in space, yet unrolling it preserves the <induced metric>: a small patch has the geometry of a flat sheet. By contrast, a round <sphere> has positive <Gaussian curvature>. The distinction between <intrinsic flatness of a circular cylinder> and <extrinsic curvature> separates the geometry an inhabitant measures from the way an outside observer sees the surface bend.
<Geodesics> supply the analogue of straight lines. On a round <sphere> they follow <great circles>; on a <circular cylinder> they become straight lines when lifted to an unrolled sheet. Their <arc length> lets us compare candidate routes, as in <great-circle distance> and <Riemannian distance on a circular cylinder>. The same measuring rule also defines differential operators such as the <Laplace-Beltrami operator>, connecting geometry with diffusion and <spectral geometry>.
Bernhard Riemann's 1854 inaugural lecture proposed studying spaces of arbitrary dimension through their infinitesimal measuring rules, including a positive quadratic expression for the squared line element. This moved the subject beyond a particular surrounding space: the measuring rule itself became part of the geometry. https://www.maths.tcd.ie/pub/HistMath/People/Riemann/Geom/[Riemann's original lecture and Clifford's English translation, collected by Trinity College Dublin]{ref}.
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