Tangent Bundles and Vector Fields

Related lectures (32)

Page 3 of 4

Delves into tangent vectors as equivalence classes on manifolds, highlighting their abstract nature and role in optimization.

Covers second-order retractions in optimization on manifolds, focusing on smooth curves and their relation to the gradient and Hessian of a function.

Explores upgrading foundations from embedded to general manifolds in optimization, discussing smooth sets and tangent vectors.

Introduces Manopt, a toolbox for optimization on manifolds, focusing on solving optimization problems on smooth manifolds using the Matlab version.

Discusses linear Lie groups, their definitions, properties, and the relationship between integral curves and vector fields.

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

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

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

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

Explains acceleration along curves and geodesics on manifolds, generalizing straight lines to spheres.