A space is locally Euclidean when every point has a small neighborhood that can be mapped one-to-one to a region of ℝᵈ without folding, twisting, or tearing. Locally, the space can therefore be treated as flat, even when its global shape is curved or difficult to visualize. section 12.1 section 12.4
Geometric examples make the property concrete:
- A small arc of a circle can be straightened into a line segment, making the circle a 1-manifold. section 12.1
- A small patch of a sphere can be flattened into a plane, making the sphere surface a 2-manifold. section 12.1 section 12.4
This local flatness supports calculus by allowing curves and surfaces to be approximated with linear pieces. In neural-network geometry, layers can be viewed as mappings between manifolds; successive transformations can produce a representation whose classes are easier to separate with a hyperplane. section 12.4
