Lectures related to Differentiating Vector Fields: How Not to Do It

Retractions vector fields and tangent bundles: Tangent bundles

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

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

Riemannian metrics and gradients: Why and definition of Riemannian manifolds

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

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

Riemannian connections

Explores Riemannian connections on manifolds, emphasizing smoothness and compatibility with the metric.

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.

Riemannian connections: What they are and why we care

Covers Riemannian connections, emphasizing their properties and significance in geometry.

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

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

Connections: Axiomatic Definition

Explores connections on manifolds, emphasizing the axiomatic definition and properties of derivatives in differentiating vector fields.

Covariant derivatives along curves

Explores covariant derivatives along curves and second-order optimality conditions in vector fields and manifolds.

Differentiating Vector Fields: Definition

Introduces differentiating vector fields along curves on manifolds with connections and the unique operator satisfying specific properties.

Manopt: Optimization Toolbox for Manifolds

Introduces Manopt, a toolbox for optimization on manifolds, focusing on solving optimization problems on smooth manifolds using the Matlab version.

Dynamics of Steady Euler Flows: New Results

Explores the dynamics of steady Euler flows on Riemannian manifolds, covering ideal fluids, Euler equations, Eulerisable flows, and obstructions to exhibiting plugs.

Differential Forms on Manifolds

Introduces differential forms on manifolds, covering tangent bundles and intersection pairings.

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.

Symmetry Property: Riemannian Connection in Geometry

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

Hessians, Symmetry and Examples: Sphere, Stiefel

Covers Hessians, symmetry, and examples related to vector fields, functions, and manifolds.

Local Frames

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

Differential Forms Integration

Covers the integration of differential forms on smooth manifolds, including the concepts of closed and exact forms.

Hands on with Manopt: Optimization on Manifolds

Introduces Manopt, a toolbox for optimization on smooth manifolds with a Riemannian structure, covering cost functions, different types of manifolds, and optimization principles.