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Related lectures (32)

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Comparing Tangent Vectors: Parallel Transport

Explores the definition, existence, and uniqueness of parallel transport of tangent vectors on manifolds.

Smooth maps on manifolds and differentials

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

Differentiating vector fields: Why do it?

Explores the importance of differentiating vector fields and the correct methodology to achieve it, emphasizing the significance of going beyond the first order.

From embedded to general manifolds: upgrading our foundations

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

Comparing Tangent Vectors: Parallel Transport

Explores comparing tangent vectors and parallel transport on manifolds.

Retractions, vector fields and tangent bundles: Retractions and vector fields

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

From embedded to general manifolds: Why?

Explores upgrading foundations from embedded to general manifolds in optimization, discussing smooth sets and tangent vectors.

Tangent Bundles and Vector Fields

Covers smooth maps, vector fields, and retractions on manifolds, emphasizing the importance of smoothly varying curves.

Manifolds and Tangent Space

Introduces manifolds, charts, atlases, tangent space, tensors, and the metric in curved spaces.

All things Riemannian: metrics, (sub)manifolds and gradients

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