= Differential geometry
{wiki}
Differential geometry uses <calculus> to study <smooth curves>, <smooth surfaces>, and <smooth manifolds>. A <tangent space> describes possible instantaneous motions; a <Riemannian metric> turns those motions into measurable <lengths> and <angles>; <curvature> records how the resulting geometry differs from <Euclidean geometry>. Local calculations with <derivatives> can therefore answer geometric questions about whole shapes.
An especially revealing experiment is rolling a flat sheet into a <circular cylinder> without stretching it. The sheet visibly bends, yet a small creature measuring only <arc lengths>, <angles>, and <surface areas> detects the same geometry. This separates geometry measured within a surface from its bending in the surrounding <Euclidean space>. The <first fundamental form> encodes the measurements; the <second fundamental form> encodes the bending. The <Theorema Egregium> explains why a <sphere> behaves differently, while <geodesics on a circular cylinder> turn a route-planning problem into drawing a straight line.
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