General Manifolds: A First Look

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General Manifolds and Topology

Covers manifolds, topology, smooth maps, and tangent vectors in detail.

Connections: motivation and definition

Explores the definition of connections for smooth vector fields on manifolds.

Topology of Riemann Surfaces

Covers the topology of Riemann surfaces, focusing on orientation and orientability.

Manifolds: Charts and Compatibility

Covers manifolds, charts, compatibility, and submanifolds with smooth analytic equations.

Differential Forms on Manifolds

Introduces differential forms on manifolds, covering tangent bundles and intersection pairings.

Riemannian connections

Explores Riemannian connections on manifolds, emphasizing smoothness and compatibility with the metric.

Smooth sets and functions: Example - atlas for a circle

Covers smooth sets and functions on manifolds, with a focus on constructing an atlas for a circle.

Differential Forms Integration

Covers the integration of differential forms on smooth manifolds, including the concepts of closed and exact forms.

Local Homeomorphisms and Coverings

Covers the concepts of local homeomorphisms and coverings in manifolds, emphasizing the conditions under which a map is considered a local homeomorphism or a covering.

Retractions vector fields and tangent bundles: Tangent bundles

Covers retractions, tangent bundles, and embedded submanifolds on manifolds with proofs and examples.