Lectures related to Riemannian Hessians: Connections and Symmetry

Riemannian connections

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

Riemannian connections: What they are and why we care

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

Symmetry Property: Riemannian Connection in Geometry

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

Connections: motivation and definition

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

Hessians, Symmetry and Examples: Sphere, Stiefel

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

Riemannian connections: Proof sketch

Presents the fundamental theorem of Riemannian geometry and demonstrates the uniqueness of the Riemannian connection.

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.

Rigidity in Negative Curvature

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

Comparing Tangent Vectors: Parallel Transport

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

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.

Optimality Conditions: First Order

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

Riemannian Hessians: Definition and Example

Covers the definition and computation of Riemannian Hessians on manifolds.

Bird's-eye view and aims

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

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

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

Algebraic Topology and Differential Geometry

Explores algebraic topology and differential geometry in understanding robot dynamics and global behavior.

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.