Differentiability: Manifolds Require Smoothness
Differentiability Smoothness Calculus TangentLines
Mathematical rigor distinguishes “not pointy” through calculus—differentiability formalizes smoothness requirements for manifolds.
Manifold Dimensions: From Curves to Surfaces
ManifoldDimensions GeometricExamples OneManifolds TwoManifolds
Manifold dimension indicates the dimension of the locally flat neighborhood—how many coordinates specify position near any point.
Manifolds: Locally Flat, Globally Curved
Manifolds DifferentialGeometry LocallyFlat Topology
Manifolds are higher-dimensional analogs of curved surfaces appearing throughout general relativity, topology, number theory, and computer science, defined by the property of being locally flat despite global curvature.