a Riemannian metric
/ REE-mahn-ee-an /
A bare manifold knows its shape only in the loosest sense — it has no idea how long a path is or what angle two directions make. A Riemannian metric is the extra equipment that supplies exactly those measurements. It is, at each point, a tiny ruler-and-protractor for the tangent space: a rule for the length of every tangent vector and the angle between any two. Glue one such ruler to every point, varying smoothly, and the manifold becomes a place where you can actually measure distances, areas, and angles.
Precisely, a Riemannian metric g assigns to each point p an inner product on the tangent space T_p M: a way of multiplying two tangent vectors u, v to get a number g_p(u, v) that is symmetric, linear in each slot, and positive (g_p(v, v) > 0 for v not zero), all varying smoothly with p. In coordinates it is a symmetric positive-definite matrix g_ij(x) at each point, and the squared length of a tiny displacement (dx^1, ..., dx^n) is ds^2 = sum g_ij dx^i dx^j — the higher-dimensional cousin of the surface line element ds^2 = E du^2 + 2F du dv + G dv^2, the old first fundamental form made intrinsic. The length of a curve is found by integrating ds along it, and the distance between two points is the length of the shortest connecting curve.
The metric is the source of essentially all the geometry that follows: it determines angles, areas, the Levi-Civita connection, geodesics, and curvature. The same underlying manifold carries infinitely many different metrics — flat or curved, stretched or squashed — and they describe genuinely different geometries. A close relative, the Lorentzian metric of relativity, drops the positivity condition so that some directions have negative squared length (time versus space); that one sign change is the entire mathematical difference between Riemannian geometry and the geometry of spacetime.
On the upper half-plane (points with y > 0), the metric ds^2 = (dx^2 + dy^2) / y^2 stretches distance more and more as you approach the x-axis. With this single rule the plane becomes a model of hyperbolic geometry: straight Euclidean lines are no longer the shortest paths, and the shortest routes turn out to be semicircles meeting the x-axis at right angles.
Change only the metric, and the same flat plane becomes curved hyperbolic space — geometry lives in g, not in the points.
The metric is the intrinsic version of the surface theory's first fundamental form: it encodes all measurable lengths and angles but needs no surrounding space. Distance comes from minimizing curve length, so the shortest path may bend even when coordinates look straight.