Gradients on Riemannian submanifolds, local frames

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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.

Riemannian connections

Explores Riemannian connections on manifolds, emphasizing smoothness and compatibility with the metric.

Smooth Manifolds: Diffeomorphisms

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

Tangent vectors without embedding space: Revisiting the embedded case

Explores defining tangent vectors without an embedding space, focusing on creating tangent spaces at every point of a manifold through equivalence classes of curves.

Local Frames

Covers the concept of local frames, their construction, and limitations.

Comparing Tangent Vectors: Parallel Transport

Explores comparing tangent vectors and parallel transport on manifolds.

Differential Forms on Manifolds

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

Riemannian connections: What they are and why we care

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

Hessians, Symmetry and Examples: Sphere, Stiefel

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

Riemannian metrics and gradients: Why and definition of Riemannian manifolds

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