Manifold Exploration

differential geometry blog

Manifold Exploration

What is a straight line on a curved surface? An interactive journey from smooth manifolds to latent space geometry.

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01

Smooth Manifolds

What does it mean for a space to be curved? We begin with the historical shift from Euclidean to Riemannian geometry, then formalize the notion of a smooth manifold using the sphere as our guiding example.

From Euclid to Riemann

For over two thousand years, geometry meant Euclidean geometry. Euclid's five postulates, formulated for the flat plane, provided the foundation. The fifth (the parallel postulate) states: given a line in the plane and a point not on it, there exists exactly one line through that point which never meets the first.

In the 19th century, mathematicians began questioning this assumption. Gauss, Bolyai, and Lobachevsky showed that consistent geometries exist where the parallel postulate fails. Bernhard Riemann took this further in his 1854 Habilitationsschrift, proposing a framework where geometry is not a fixed background but a property of the space itself.

On a sphere, for instance, "lines" (geodesics) are great circles. Any two great circles intersect, so there are no parallel lines at all. The geometry of the sphere is intrinsically different from that of the plane.

The Sphere as a Surface

Notation

We denote by S^2 the unit 2-sphere in \mathbb{R}^3:

S^2 = \left\{ (x, y, z) \in \mathbb{R}^3 \;\middle|\; x^2 + y^2 + z^2 = 1 \right\}

S^2 is a surface: two-dimensional, smooth, and curved. The most natural way to describe it is via spherical coordinates (\theta, \varphi):

(\theta, \varphi) \mapsto (\sin\theta\cos\varphi,\; \cos\theta,\; \sin\theta\sin\varphi), \quad \theta \in [0, \pi],\; \varphi \in [0, 2\pi)

Two parameters produce a point in \mathbb{R}^3. Near most points, this is a smooth bijection: the surface locally looks like a piece of \mathbb{R}^2. These coordinates will be our working coordinates for concrete computations throughout the blog (metric, Christoffel symbols, geodesics).

However, this parametrization fails at the poles (\theta = 0 and \theta = \pi): all values of \varphi map to the same point, and the Jacobian drops rank. No single coordinate system can cover all of S^2 without such degeneracies. We need a more general framework.

Surfaces as Parametrizations

Before introducing the abstract machinery, let us make precise what we just did. We described S^2 by giving a smooth map from a region of \mathbb{R}^2 into \mathbb{R}^3. This is the concrete notion of a parametric surface .

Definition

A parametric surface is a smooth map F : U \subset \mathbb{R}^2 \to \mathbb{R}^3 whose Jacobian has rank 2 everywhere. The image F(U) is the surface sitting in \mathbb{R}^3.

For the sphere, the parametrization is exactly the map we wrote above:

F(\theta, \varphi) = (\sin\theta\cos\varphi,\; \cos\theta,\; \sin\theta\sin\varphi)

The key observation is that a chart is the local inverse of a parametrization . Where F goes from coordinates to the surface, the chart \psi goes the other way:

\psi = F^{-1}_{\text{local}} : F(U) \subset \mathbb{R}^3 \longrightarrow U \subset \mathbb{R}^2, \quad (x,y,z) \mapsto (\theta, \varphi)

In practice, we often think in terms of the parametrization F (going "up" from coordinates to the surface) rather than the chart \psi (going "down" from the surface to coordinates). They carry the same information, just in opposite directions.

Topological Manifolds

Definition

A topological manifold M of dimension n is a topological space that is:

Hausdorff: distinct points have disjoint neighborhoods,

Second-countable: the topology has a countable basis,

Locally Euclidean: every point p \in M has a neighborhood homeomorphic to an open subset of \mathbb{R}^n.

S^2 satisfies all three conditions: it is a 2-dimensional topological manifold.

Charts and Atlases

Since no single coordinate system covers S^2 without singularities, we need several overlapping ones. The following construction provides a singularity-free atlas (we will not use it for calculations, only to prove that S^2 is a smooth manifold).

Definition

A chart on M is a pair (U_\alpha, \psi_\alpha) where U_\alpha \subset M is open and \psi_\alpha : U_\alpha \to \psi_\alpha(U_\alpha) \subset \mathbb{R}^n is a homeomorphism. An atlas is a collection of charts \{(U_\alpha, \psi_\alpha)\} that covers M.

Example. The stereographic projection from the north pole N = (0, 0, 1) maps S^2 \setminus \{N\} to \mathbb{R}^2:

\sigma_N(x, y, z) = \left( \frac{x}{1 - z},\; \frac{y}{1 - z} \right)

From the south pole S = (0, 0, -1):

\sigma_S(x, y, z) = \left( \frac{x}{1 + z},\; \frac{y}{1 + z} \right)

Each chart misses one point, but together (S^2 \setminus \{N\},\, \sigma_N) and (S^2 \setminus \{S\},\, \sigma_S) cover all of S^2.

Smooth Structure

On the overlap S^2 \setminus \{N, S\}, the transition map between our two charts is:

\sigma_S \circ \sigma_N^{-1} :...