Riemannian metrics and gradients: Why and definition of Riemannian manifolds

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

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

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

Newton's method on Riemannian manifolds

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

Riemannian metrics and gradients: Examples and Riemannian submanifolds

Explores Riemannian metrics on manifolds and the concept of Riemannian submanifolds in Euclidean spaces.

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.

Topology of Riemann Surfaces

Covers the topology of Riemann surfaces, focusing on orientation and orientability.

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.

From embedded to general manifolds: Why?

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

Smooth maps on manifolds and differentials

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

Differentiating Vector Fields: Definition

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