Manifolds 33 | Riemannian Metrics [dark version]

Riemannian geometry starts by turning an abstract smooth manifold into a space where distance, lengths, and angles actually make sense. The key move is to assign, at every point of a smooth manifold M, an inner product on the tangent space T_pM. That pointwise inner product—varying smoothly from point to point—upgrades the manifold from something with only neighborhoods and continuity into something with quantitative geometry.

On a general manifold, topology tells which points are “close” but it doesn’t provide numbers for how far apart two points are, nor does it supply a way to measure lengths or angles. The transcript connects this gap to familiar ideas from complex analysis: curve integrals depend on orientation (integrating from P to Q differs from integrating from Q to P). But even beyond orientation, the desire is more concrete—measuring minimal connections between points, i.e., distances. To do that, the manifold needs extra structure.

In Euclidean space R^n, inner products provide exactly what’s missing: given two vectors X and Y, an inner product produces a real number, and the length of a vector is computed as the square root of ⟨X, X⟩. Riemannian geometry generalizes this by replacing the single global inner product of R^n with a smoothly varying family of inner products on tangent spaces. Using the common notation G for the metric, the inner product at each point p is written as G_p, and the assignment p ↦ G_p must be continuous and differentiable in a smooth way.

The definition is formalized through charts. Around a point p, a chart map (often denoted by φ) and its inverse (a parameterization) provide local coordinates and induce a coordinate basis for tangent vectors in T_xM. In those coordinates, the metric becomes a collection of functions G_{ij}(x). Smoothness is expressed by requiring that, for every pair of indices i, j, the map x ↦ G_{ij}(x) is smooth in the manifold sense (equivalently, smooth as a function between coordinate domains).

A major practical payoff follows: in local coordinates, the abstract inner product G_x can be represented by an n×n symmetric matrix (G_{ij}). The symmetry comes from the symmetry of the inner product itself. Using the Einstein summation convention, the metric can be written in terms of one-forms dx^i and dx^j, so that feeding two tangent vectors into this expression reproduces the same numbers as the inner product. This matrix representation is what makes computations feasible: instead of manipulating the abstract object G_x directly, calculations often use the local symmetric matrix of metric components.

The transcript closes by postponing worked examples to the next installment, but the groundwork is clear: a Riemannian manifold is a smooth manifold M equipped with a smoothly varying inner product on each tangent space, encoded locally by a symmetric positive-definite metric matrix G_{ij}(x). That structure is what ultimately enables distance, angles, and length-based geometry on manifolds.