Topological and smooth manifolds

2026-01-19T13:13:39Z

SmoothOperator: Topological and Smooth Manifolds start

A '''topological manifold''' is a topological space $M$ such that it is Hausdorff, locally Euclidean, second countable and paracompact.

A '''smooth manifold''' is a topological manifold $M$ with $A$-indexed family of charts $\mathscr{A} = \{(U_{\alpha}, \phi_{\alpha})\}_{\alpha \in A}$ such that,
# $M = \bigcup_{\alpha \in A} U_{\alpha}$,
# $\phi_{\alpha} \rightarrow \phi_{\alpha}(U_{\alpha}) \subseteq \mathbb{R}^{n}$ is a homeomorphism where $\phi_{\alpha} \subseteq \mathbb{R}^n$ is open (with respect to the standard topology of $\mathbb{R}^n$) for all $\alpha \in A$,
# $\phi_{\beta} \circ \phi_{\alpha} : \phi_{\alpha}(U_{\alpha\beta}) \rightarrow \phi_{\beta}(U_{\alpha\beta})$ is a diffeomorphism for all $\alpha,\beta \in A$, where $U_{\alpha\beta} := U_{\alpha} \cap U_{\beta}$.

Now we can go on to talk about maximal atlases, incompatible atlases, differential structures, etc.

Lecture Schedule

2026-01-19T12:54:03Z

SmoothOperator: Starting page focused on Topological and smooth manifolds

== Lectures ==
Tentative lecture plan is as follows:

=== Week 1 ===
Topics: Review of differential geometry

* [[Topological and smooth manifolds]],
* Vector and tensor fields,
* Algebra of differential forms,
* De Rham Cohomology

Reading:

* Taubes Ch 1
* Frenkel Sections 1-3
* Nakahara Sections 5.1, 5.2, 5.4. 6.
* Sontz Chapters 2, 3, 5

=== Week 2 ===
Topics: Review of differential geometry

* vector fields
* Lie brackets
* diffeomorphisms
* Lie derivatives
* Lie groups and Lie algebras.
* Exponential maps
* adjoint representation
* Basics of Symplectic Geometry
* Hamiltonian vector fields
* Hamiltonian group actions
* Moment maps and co-moment maps.

Reading:

* Rudolph-Schmidt (Differential Geometry and Mathematical Physics. Part I) Section 5

=== Week 3 ===
Topics:

* Further details on smooth actions,
* Symplectic structure on cotangent bundles
* Quotients of free, proper, smooth actions
* Principal bundles.
* Frame bundles.
* Associated fibre bundles.

Reading:

=== Week 4 ===
Topics:

* Associated fibre bundles: vector bundles, adjoint bundle.
* Reduction and extension of structure group.
* Group of gauge transformations.
* Linear connections.

Reading:

=== Week 5===
Topics:

* Connections: Koszul connections, principal connections, Ehresmann connections.
* Equivalences between them.
* Atiyah sequence and differential operators.

Reading:

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Topological and smooth manifolds

Lecture Schedule