Riemannian metrics and gradients: Why and definition of Riemannian manifolds

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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.

Differentiating Vector Fields: How Not to Do It

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

Optimality Conditions: First Order

Covers optimality conditions in optimization on manifolds, focusing on global and local minimum points.

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.

Tangent Bundles and Vector Fields

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

Gradients on Riemannian submanifolds, local frames

Discusses gradients on Riemannian submanifolds and the construction of local frames.

Taylor expansions: second order

Explores Taylor expansions and retractions on Riemannian manifolds, emphasizing second-order approximations and covariant derivatives.

Optimization on Manifolds: Context and Applications

Introduces optimization on manifolds, covering classical and modern techniques in the field.

Differential Forms Integration

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

Comparing Tangent Vectors: Parallel Transport

Explores the definition, existence, and uniqueness of parallel transport of tangent vectors on manifolds.