Riemannian metrics and gradients: Riemannian gradients

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Page 3 of 4

Explores symmetries, Riemannian connection, vector fields, and Lie bracket in geometry.

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

Covers connections on manifolds, symmetric connections, Lie brackets, and compatibility with the metric in Riemannian geometry.

Explores computing gradients on Riemannian manifolds through extensions and retractions, emphasizing orthogonal projectors and smooth extensions.

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

Covers optimization on manifolds, smoothness, tools needed for optimization, and advanced algorithms using Hessians and Riemannian connections.

Covers the definition and computation of Riemannian Hessians on manifolds.

Delves into the rigidity of negatively curved manifolds and the interplay between curvature and symmetry.

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

Explores algebraic topology and differential geometry in understanding robot dynamics and global behavior.