Layer 3 / tier 1
What Is a Manifold?
Diagram: Local coordinate chart pasted onto a curved surface
Classify circle, sphere, graph of a function, and crossing lines as manifolds or non-manifolds.
Path-family build plan
GeometryPath
GeometryPath is the path-network sub-vertical for geometry as structure: Euclidean proof, topology, manifolds, Riemannian geometry, information geometry, optimal transport geometry, and geometric ML.
Launch target
- Topic pages
- 30
- Diagnostics
- 1
- Labs
- 6
Canonical reference stack
- Euclid, Elements.
- Hilbert, Foundations of Geometry.
- Coxeter, Introduction to Geometry.
- Munkres, Topology.
- Hatcher, Algebraic Topology.
- do Carmo, Differential Geometry of Curves and Surfaces.
- Lee, Introduction to Smooth Manifolds.
- Lee, Riemannian Manifolds: An Introduction to Curvature.
- Spivak, A Comprehensive Introduction to Differential Geometry.
- Amari and Nagaoka, Methods of Information Geometry.
- Villani, Optimal Transport: Old and New.
- Bronstein, Bruna, Cohen, and Velickovic, Geometric Deep Learning.
Modules
Layer 0
Euclidean and Axiomatic Geometry
Ground geometry in constructions, congruence, similarity, and proof.
- Euclid's Elements as method
- Axioms and incidence
- Congruence and similarity
- Circle geometry
- Transformations
Layer 1
Coordinates, Vectors, and Curves
Connect synthetic geometry to analytic and computational representations.
- Coordinate systems
- Vectors and affine spaces
- Curves and parametrization
- Arc length and curvature
- Implicit curves
Layer 2
Topology Basics
Teach continuity, connectedness, compactness, and holes before manifolds.
- Open and closed sets
- Connectedness
- Compactness
- Fundamental group
- Covering spaces
Layer 3
Manifolds and Differential Geometry
Introduce charts, tangent spaces, vector fields, differential forms, and maps.
- Manifolds and charts
- Tangent spaces
- Vector fields
- Differential forms
- Lie groups as manifolds
Layer 4
Riemannian Geometry
Teach metrics, geodesics, connections, and curvature with ML/physics links.
- Riemannian metrics
- Geodesics
- Connections
- Curvature
- Gradient flow on manifolds
Layer 5
Geometry for ML and Statistics
Connect geometry to information, transport, equivariance, and representation.
- Information geometry
- Optimal transport geometry
- Manifold learning
- Equivariant deep learning
- Hyperbolic embeddings
First topics
Layer 3 / tier 1
Tangent Spaces
Diagram: Tangent plane attached at a point on a surface
Compute the tangent space to a level set from a gradient constraint.
Layer 4 / tier 1
Riemannian Metrics
Diagram: Metric ellipses changing over a surface
Compare Euclidean and non-Euclidean path lengths under a position-dependent metric.
Layer 5 / tier 2
Information Geometry
Diagram: Statistical model as a curved parameter manifold
Compute the Fisher metric for a one-parameter Bernoulli family.
Linking and pedagogy
- Until standalone launch, GeometryPath is represented by TheoremPath geometry hubs plus the path-family graph.
- TheoremPath owns ML-facing information geometry, optimal transport, and manifold learning pages until the standalone corpus exists.
- ProofsPath owns olympiad proof technique; GeometryPath owns the geometric object and structure.
- AlgebraPath receives Lie-group and homological-algebra depth; ComputationPath receives homotopy-type-theory and computational topology handoffs.
- Every geometry page needs a diagram unless the page is purely historical or bibliographic.
- Introduce the object visually first, then state the invariant or theorem, then show how coordinates can mislead.
- Interactive labs should be lazy-loaded and limited to high-value cases: tangent plane, geodesic, curvature, hyperbolic model, and Fisher metric.