Covariant derivatives along curves

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Page 3 of 4

Introduces retractions and vector fields on manifolds, providing examples and discussing smoothness and extension properties.

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

Covers the differentiation of vector fields along curves with examples and finite differences.

Explores fundamental groups, homotopy classes, and coverings in connected manifolds.

Delves into topological entropy in compact manifolds and Reeb flows, emphasizing the forcing of entropy via cylindrical contact homology.

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

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

Explores geodesics in the Wasserstein space, emphasizing constant speed geodesics and their properties.

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

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