Smooth manifolds

Builds on

Required prerequisites, all the way down: 41

You will learn to

• Show that spheres, projective spaces and matrix groups are smooth manifolds, using the regular value theorem.
• Compute with tangent vectors, vector fields, differential forms and pullbacks.
• Prove Stokes's theorem for manifolds with boundary and compute the de Rham cohomology of simple manifolds.

Definition

• A smooth n-manifold is a second-countable Hausdorff space with a maximal atlas of charts to ℝⁿ whose transition maps are smooth.
• The tangent space TₚM can be defined as the space of derivations at p.
• A differential k-form is a smooth section of the k-th exterior power of the cotangent bundle.

Theorem

• Regular value theorem: the preimage of a regular value of a smooth map is an embedded submanifold.
• Stokes: ∫M dω = ∫{∂M} ω for a compactly supported (n − 1)-form ω on an oriented n-manifold M with boundary.
• Whitney: every smooth n-manifold embeds in ℝ²ⁿ.
• de Rham: de Rham cohomology is isomorphic to singular cohomology with real coefficients.

Common misconceptions

• Thinking every manifold comes with a preferred embedding in some ℝᴺ.
• Confusing vectors and covectors when changing coordinates.

Optional discoveries

• Functors The tangent construction is a functor from smooth manifolds to vector bundles, and the chain rule is precisely the statement that it preserves composition.
• ← Classical mechanics
• ← Differential geometry of curves and surfaces

Learning pathways

• Graduate preparation

Further reading

• John M. Lee, Introduction to Smooth Manifolds