Riemannian metrics and gradients: Examples and Riemannian submanifolds

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Page 2 of 4

Covers retractions, tangent bundles, and embedded submanifolds on manifolds with proofs and examples.

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

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

Covers the Grassmann manifold and retractions on submanifolds.

Explores the transition from embedded to general manifolds, upgrading foundational concepts and discussing mathematical reasons for both approaches.

Covers smooth maps on manifolds, defining functions, tangent spaces, and differentials.

Covers Newton's method on Riemannian manifolds, focusing on second-order optimality conditions and quadratic convergence.

Explores conformal symmetries in Euclidean and AdS spaces, isometries, induced metric, Poincaré coordinates, and boundary structure.

Introduces optimization on manifolds, covering classical and modern techniques in the field.

Explores the definition of connections for smooth vector fields on manifolds.