Optimality Conditions: First Order

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Covers the definition of retraction, open submanifolds, local defining functions, tangent spaces, and Riemannian metrics.

Explores geodesic convexity, focusing on properties of convex functions on manifolds.

Explores Riemannian metrics on manifolds and the concept of Riemannian submanifolds in Euclidean spaces.

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

Explains Riemannian submanifolds, metrics, and gradients computation on manifolds.

Introduces geodesic convexity on Riemannian manifolds and explores its properties.

Covers the structure of Riemannian manifolds, geodesic convexity, and the Riemannian distance function.

Covers manifolds, topology, smooth maps, and tangent vectors in detail.

Presents the fundamental theorem of Riemannian geometry and demonstrates the uniqueness of the Riemannian connection.

Explores matrix-based approaches for computing the Newton step on a Riemannian manifold.