Lectures related to Smooth maps and differentials: Smooth maps

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

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

Explores smooth functions on manifolds, emphasizing continuity and atlas topologies.

Explores smooth manifolds through diffeomorphisms and embedded submanifolds in a linear space.

Covers embedded submanifolds, Stiefel manifold, tangent spaces, and differential ranks.

Delves into topological entropy in compact manifolds and Reeb flows, emphasizing the forcing of entropy via cylindrical contact homology.

Explores how submanifolds can be deformed into linear patches using diffeomorphisms and the inverse function theorem.

Discusses the challenges in differentiating vector fields on submanifolds and the importance of choosing the right method.

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

Explores the transition from embedded to general manifolds, upgrading foundational concepts and discussing mathematical reasons for both approaches.

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

Explores tangent spaces as free movement directions on submanifolds, offering a geometrically satisfying linearization notion.

Explores fixed-rank matrices as an embedded submanifold, focusing on constructing local defining functions and efficiently computing tangent vectors.

Introduces smooth manifolds, emphasizing the importance of submanifolds of linear spaces.

Explores smooth manifolds through defining functions and submanifolds, highlighting the importance of smoothness.

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

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

Covers tangent spaces and submersions in differential geometry, emphasizing vector spaces and differentiable structures.