Symmetry Property: Riemannian Connection in Geometry

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Riemannian metrics and gradients: Why and definition of Riemannian manifolds

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

Differential Forms on Manifolds

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

Riemannian Hessians: Definition and Example

Covers the definition and computation of Riemannian Hessians on manifolds.

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.

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

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

Rigidity in Negative Curvature

Delves into the rigidity of negatively curved manifolds and the interplay between curvature and symmetry.

Differentiating Vector Fields: How Not to Do It

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

Retractions vector fields and tangent bundles: Tangent bundles

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

Bird's-eye view and aims

Covers optimization on manifolds, smoothness, tools needed for optimization, and advanced algorithms using Hessians and Riemannian connections.

Optimality Conditions: First Order

Covers optimality conditions in optimization on manifolds, focusing on global and local minimum points.