From a metric to a length
On a Riemannian manifold you measured curves by integrating the speed against the metric g, and the distance between two points was the infimum of those lengths. Metric geometry asks a bolder question: what survives if all you keep is the distance function d, with no g, no charts, no smoothness? Surprisingly, a great deal. The first move is to recover a notion of length directly from d. Given a curve gamma: [0, 1] -> X, chop the interval into 0 = t_0 < t_1 < ... < t_n = 1, add up the hops d(gamma(t_{i-1}), gamma(t_i)), and take the supremum over all such partitions. That supremum is the length of gamma, written L(gamma).
Why a supremum and not an integral? Because finer partitions can only see more wiggling: refining a partition never decreases the sum, since the triangle inequality says a single hop is at most the sum of two hops through an inserted point. So the supremum is the honest, partition-independent total. If it is finite, gamma is called rectifiable; a curve can be continuous yet have infinite length, the way a fractal like the Koch curve does. This is exactly the construction you met for Riemannian distance, stripped of the smooth scaffolding.
L(gamma) = sup over partitions sum_{i=1}^{n} d( gamma(t_{i-1}), gamma(t_i) )
always: d(x, y) <= L(gamma) for any gamma from x to y
length metric: d(x, y) = inf { L(gamma) : gamma joins x to y }Length spaces: when distance is shortest length
Notice the inequality d(x, y) <= L(gamma): the straight-line distance can never exceed the length of any path, since the length already sums up hops that each obey the triangle inequality. A length space is a metric space where this slack is squeezed out to nothing: d(x, y) equals the infimum of lengths of curves joining x to y. In words, the distance is not some abstract number imposed from outside; it is the cost of the most efficient journey you can actually take inside the space.
The classic non-example sharpens the idea. Take the unit circle in the plane and give it the metric it inherits from R^2 (chord distance). The straight-line distance between two nearby points is the chord, but no curve inside the circle is shorter than the arc, and the arc is longer than the chord. So d(x, y) is strictly less than every L(gamma): the chord metric is not a length metric. The cure is to use arc length itself as the distance, and then the circle becomes a length space, with distance equal to the shorter arc.
Geodesic spaces and the metric Hopf-Rinow theorem
A length space tells you the distance is approached by ever-better curves, but the infimum need not be attained. A geodesic space demands more: between any two points there is an actual curve realizing the distance, a curve gamma with L(gamma) = d(x, y). Such a minimizing curve is a geodesic in the metric sense, and it is automatically parametrized so that distance along it is proportional to the parameter. Beware the clash of conventions: in Riemannian geometry a geodesic is any locally length-minimizing solution of the geodesic equation and may not be globally shortest, whereas in metric geometry it usually means a globally shortest segment. Always check which a book means.
When does a length space upgrade to a geodesic space? The bridge is the metric Hopf-Rinow theorem (Hopf-Rinow-Cohn-Vossen). It says: a length space that is complete and locally compact is a geodesic space, every closed bounded set in it is compact, and in fact the minimizing curve exists between every pair of points. This is the metric-space sibling of the smooth Hopf-Rinow theorem you used for Riemannian manifolds, where geodesic completeness, metric completeness, and the existence of minimizing geodesics all coincide.
Hausdorff distance: putting sets side by side
Before we can compare two whole spaces, warm up by comparing two subsets A and B that already sit inside one common metric space Z. The Hausdorff distance d_H(A, B) asks: how far must you fatten each set before it swallows the other? Thicken A by radius r to get its r-neighborhood; d_H(A, B) is the smallest r such that A's r-neighborhood contains B and B's r-neighborhood contains A. If a single point of B sticks out far from all of A, that gap alone forces d_H up. It is a genuine metric on the closed bounded subsets of Z.
N_r(A) = { z in Z : d(z, A) <= r } ( the r-neighborhood of A )
d_H(A, B) = inf { r >= 0 : A subset N_r(B) and B subset N_r(A) }Gromov-Hausdorff: a distance between spaces themselves
Hausdorff distance needs both sets inside one Z. Gromov's leap was to remove that crutch. Given two compact metric spaces X and Y that live nowhere near each other, the Gromov-Hausdorff distance d_GH(X, Y) is the infimum of d_H(f(X), g(Y)) over ALL ways of isometrically embedding both into ALL common spaces Z. You are free to choose the most flattering shared room in which to overlay X and Y; d_GH measures the best possible overlay. Two spaces are at distance 0 exactly when they are isometric, so d_GH is a metric on the set of compact spaces up to isometry.
The infimum-over-all-Z definition is conceptually clean but useless for computing. In practice you use the equivalent picture via correspondences: a correspondence pairs up points of X with points of Y (every x matched to at least one y and vice versa), and its distortion is the largest amount by which any matched pair distorts a distance, |d_X(x, x') - d_Y(y, y')|. Then d_GH(X, Y) equals one half the infimum of the distortion over all correspondences. This turns an abstract embedding problem into a concrete matching problem you can actually estimate.
- Bound d_GH from above by exhibiting one good correspondence and computing its distortion; you never need the optimal one to get an upper bound.
- Worked example: let X be a single point and Y a compact space of diameter D. Any correspondence pairs the point with all of Y; the distortion is the largest |0 - d_Y(y, y')| = D, so d_GH(point, Y) = D / 2.
- Sequences: X_n -> X in d_GH means the spaces look more and more alike at every scale, a notion strong enough to support Gromov's compactness theorem and the study of limits of manifolds with curvature bounds.
Gromov-Hausdorff convergence has one feature worth flagging honestly: a sequence of smooth manifolds can converge to a limit that is no longer a manifold. A thin flat torus pinched ever thinner converges to a circle; a manifold can collapse to a lower-dimensional space, or develop singular points where it is only an Alexandrov space rather than a smooth one. This is a feature, not a bug — it is the whole reason the coarse viewpoint matters, and it sets up the rougher comparison, quasi-isometry, that the later guides on the word metric and geometric group theory will build on.