Smooth maps and differentials: Differentials

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Tangent Bundles and Vector Fields

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

Optimality conditions: second order

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

Manopt: Optimization Toolbox for Manifolds

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

From embedded to general manifolds: Why?

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

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: Retractions and vector fields

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

Smooth sets and functions: Smooth functions, topology, and manifolds

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

Smooth Manifolds: Diffeomorphisms

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

Riemannian metrics and gradients: Why and definition of Riemannian manifolds

Covers Riemannian metrics, gradients, vector fields, and inner products on manifolds.

Differentiating vector fields: Why do it?

Explores the importance of differentiating vector fields and the correct methodology to achieve it, emphasizing the significance of going beyond the first order.