An introduction to the geometry of metric spaces

These informal notes deal with some basic properties of metric spaces, especially concerning lengths of curves.

💡 Research Summary

The paper is a concise yet thorough introduction to the geometry of metric spaces, focusing on the notion of curve length. It begins by recalling the definition of a metric space (non‑negativity, identity of indiscernibles, symmetry, triangle inequality) and illustrates the concept with two elementary examples: the discrete metric and the standard metric on the real line.

Next, the author discusses norms on ℝⁿ, presenting the ℓ¹, ℓ² (Euclidean), ℓᵖ (for p≥1) and ℓ^∞ norms. By proving the usual equivalence inequalities (e.g., ‖x‖_∞ ≤ ‖x‖p ≤ n^{1/p}‖x‖∞) the paper shows that any norm induces a metric that is topologically equivalent to the Euclidean one. This groundwork is essential for later sections where intrinsic distances on the unit circle and sphere are introduced.

The unit circle S¹ is examined in detail. The intrinsic distance d(x,y) is defined as the length of the shorter arc joining x and y, which equals the central angle. The fundamental identity sin(d/2)=|x−y|/2 links the intrinsic distance to the ambient Euclidean distance. Consequently, the two metrics are bi‑Lipschitz equivalent: |x−y| ≤ d(x,y) ≤ (π/2)|x−y|, and for small separations d≈|x−y|. The maximum intrinsic distance π occurs precisely at antipodal points.

A parametrisation e(t)=(cos t, sin t) is introduced, showing that the arc length between e(a) and e(b) equals |b−a| whenever |b−a|≤π. This provides a concrete way to compute distances on S¹.

The paper then moves to a brief geometric lemma concerning orthogonal projection onto a plane, which later aids the analysis of the unit sphere.

For the unit sphere Sⁿ⁻¹ (n≥3) the intrinsic (spherical) distance is defined by the same sine formula sin(d/2)=|x−y|/2. To prove the triangle inequality, the author reduces the problem to the two‑dimensional case: any three points lie in a two‑dimensional subspace whose intersection with the sphere is a great circle, i.e., a copy of S¹. Since the triangle inequality holds on S¹, it holds on Sⁿ⁻¹. The same bi‑Lipschitz bounds as for the circle are obtained, and antipodal points again realize the maximal distance π.

The discussion of suprema and infima in ℝ serves to define the diameter of a bounded set E⊂M as diam E = sup{d(x,y):x,y∈E}. This notion is later used in the definition of curve length.

Lipschitz mappings are defined (ρ(f(x),f(y)) ≤ C d(x,y)). The paper notes that distance functions p↦d(p₀,p) are 1‑Lipschitz, that Lipschitz maps are uniformly continuous, and that the composition of Lipschitz maps multiplies their constants. These facts are crucial for estimating lengths of composed curves.

The central part of the paper defines the length of a continuous path p: