Tangent vectors without embedding space: Revisiting the embedded case

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Tangent vectors without embedding space: Equivalence classes

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

Connections: Axiomatic Definition

Explores connections on manifolds, emphasizing the axiomatic definition and properties of derivatives in differentiating vector fields.

Topology of Riemann Surfaces

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

Geodesic convexity: why, and what we need

Explores geodesic convexity and its extension to optimization on manifolds, emphasizing the preservation of the key fact that local minima imply global minima.

Riemannian metrics and gradients: Why and definition of Riemannian manifolds

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

Hessians, Symmetry and Examples: Sphere, Stiefel

Covers Hessians, symmetry, and examples related to vector fields, functions, and manifolds.

Manifolds and Tangent Space

Introduces manifolds, charts, atlases, tangent space, tensors, and the metric in curved spaces.

Riemannian connections: What they are and why we care

Covers Riemannian connections, emphasizing their properties and significance in geometry.

Differential Forms on Manifolds

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

Tangent spaces: Linearization of Embedded Submanifolds

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