What makes a manifold differentiable rather than just topological?

A topological manifold only requires charts to Euclidean space; a differentiable manifold additionally requires the transition maps between overlapping charts to be smooth, so that the notion of a smooth function on the manifold is well defined.

What is an exotic sphere?

It is a manifold homeomorphic but not diffeomorphic to the standard sphere; Milnor's discovery of such structures on the 7-sphere showed that smooth structures are not determined by the underlying topology.

Differentiable Manifolds

A differentiable manifold is a space that locally looks like Euclidean space and is glued together by smooth coordinate changes, making it the setting where calculus can be done on curved spaces.

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Definition

A differentiable (smooth) manifold of dimension n is a second-countable Hausdorff topological space equipped with an atlas of charts to open subsets of n-dimensional Euclidean space whose transition maps are infinitely differentiable.

Scope

This topic defines manifolds via atlases of charts with smooth transition maps, develops smooth structures, and treats the basic constructions: submanifolds, the rank and regular-value theorems giving level sets as manifolds, partitions of unity, and embeddings into Euclidean space (the Whitney embedding theorem). It introduces the distinction between topological and smooth structures, the surprising existence of exotic smooth structures, and Lie groups as manifolds with compatible group operations.

Core questions

How do charts and smooth transition maps let calculus be transported onto a curved space unambiguously?
When does a level set of a smooth map carry a natural manifold structure?
Why can every smooth manifold be embedded in some Euclidean space?
How can a single topological manifold admit inequivalent smooth structures?

Key concepts

Charts, atlases, and smooth transition maps
Smooth structures and submanifolds
Regular value theorem and level sets as manifolds
Partitions of unity and the Whitney embedding theorem
Topological versus smooth structure and exotic manifolds

Clinical relevance

Manifolds are the universal stage for modern geometry and physics: configuration and phase spaces in mechanics, spacetime in general relativity, and Lie groups in symmetry are all manifolds, and the smooth-structure subtleties uncovered by Milnor reshaped twentieth-century topology.

History

Riemann's 1854 notion of a manifold was made rigorous through the early-20th-century definition by atlases; Whitney's embedding theorems of the 1930s grounded the abstract theory, and Milnor's 1956 discovery of exotic 7-spheres revealed that smooth structure carries information beyond topology.

Key figures

Bernhard Riemann
Hassler Whitney
John Milnor

Seminal works

lee2012
milnor1956

Frequently asked questions

What makes a manifold differentiable rather than just topological?: A topological manifold only requires charts to Euclidean space; a differentiable manifold additionally requires the transition maps between overlapping charts to be smooth, so that the notion of a smooth function on the manifold is well defined.
What is an exotic sphere?: It is a manifold homeomorphic but not diffeomorphic to the standard sphere; Milnor's discovery of such structures on the 7-sphere showed that smooth structures are not determined by the underlying topology.