3-dimensional Anosov flows: Stable Manifolds and Foliations

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Differential Forms on Manifolds

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

Connections: motivation and definition

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

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.

Topology of Riemann Surfaces

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

Optimization on Manifolds

Covers optimization on manifolds, focusing on smooth manifolds and functions, and the process of gradient descent.

General Manifolds and Topology

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

Geodesics in Wasserstein Space

Explores geodesics in the Wasserstein space, emphasizing constant speed geodesics and their properties.

Algebraic Topology and Differential Geometry

Explores algebraic topology and differential geometry in understanding robot dynamics and global behavior.

Retractions vector fields and tangent bundles: Tangent bundles

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

Optimality conditions: second order

Explores necessary and sufficient optimality conditions for local minima on manifolds, focusing on second-order critical points.