Manifolds 33 | Riemannian Metrics

Riemannian geometry starts by turning an abstract smooth manifold into a space where distances, lengths, and angles actually make sense. The key move is to attach an inner product to every tangent space of the manifold, and to require that these inner products vary smoothly from point to point. Without this extra structure, a manifold can tell what “nearby” means topologically, but it can’t measure how far apart two points are or how long a curve segment is.

The construction begins with a smooth manifold M. At each point P on M, there is a tangent space T_P M, a vector space where one can do linear-algebra-style calculations. A Riemannian metric assigns to each tangent space an inner product, denoted G_P. In practice, this means that for any two tangent vectors X and Y in T_P M, the metric produces a real number G_P(X, Y). From that inner product, one can define lengths (via the square root of G_P(X, X)) and angles, just as in Euclidean space R^n where an inner product determines geometry.

The “metric” is not just a collection of inner products; it must change continuously and differentiably across the manifold. Formally, the assignment P ↦ G_P is required to be smooth. This smoothness condition ensures that the geometric quantities derived from the metric behave well under coordinate changes—crucial for doing calculus on manifolds.

To make the definition concrete, the transcript uses local charts. Pick a point P and a neighborhood U with a chart map φ (with inverse giving coordinates). The chart induces a coordinate basis of tangent vectors {∂/∂x^i} (described as n vectors in each tangent space over points in U). Once those basis vectors are available, the metric can be expressed locally: the inner product values G_{ij}(x) = G_x(∂/∂x^i, ∂/∂x^j) become functions on the coordinate neighborhood. Smoothness then translates into the requirement that, for every pair of indices i, j, the function x ↦ G_{ij}(x) is smooth in the manifold sense.

This local viewpoint yields a matrix representation. In coordinates, the Riemannian metric becomes an n×n symmetric matrix G(x), whose entries are the numbers G_{ij}. The symmetry comes from the symmetry of the inner product itself. Equivalently, the metric can be written using differential one-forms dx^i and dx^j, so that the metric acts on tangent vectors by combining these one-forms with the inner product components. The upshot is practical: computations can use the symmetric matrix of components instead of the abstract inner product, with no loss of information.

A manifold equipped with such a smoothly varying inner product is called a Riemannian manifold, typically denoted as the pair (M, G). The transcript sets up the definition and its coordinate form, then points to the next installment for worked examples—especially for Riemannian metrics on submanifolds of R^n, where the ideas become more tangible.