Retractions vector fields and tangent bundles: Tangent bundles

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

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

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

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

Covers smooth maps, vector fields, and retractions on manifolds, emphasizing the importance of smoothly varying curves.

Introduces retractions and vector fields on manifolds, providing examples and discussing smoothness and extension properties.

Covers smooth maps on manifolds, defining functions, tangent spaces, and differentials.

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

Introduces differentiating vector fields along curves on manifolds with connections and the unique operator satisfying specific properties.

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