Concept

Tangent bundle

Related lectures (30)

Differential Forms on Manifolds

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

Retractions vector fields and tangent bundles: Tangent bundles

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

Trigonometric Functions

Covers the concept of trigonometric functions and their applications in geometry.

Differential Geometry: Parametric Curves & Surfaces

Introduces the basics of differential geometry for parametric curves and surfaces, covering curvature, tangent vectors, and surface optimization.

General Manifolds and Topology

Covers manifolds, topology, smooth maps, and tangent vectors in detail.

Curves and Surfaces

Covers the representation of plane curves and the nature of obtained curves.

Manopt: Optimization on Manifolds

Introduces Manopt, a toolbox for optimization on manifolds, covering gradient and Hessian checks, solver calls, and manual caching.

Tangent vectors without embedding space: Making tangent spaces linear

Explores making tangent spaces linear, defining tangent vectors without an embedding space and their operations, as well as the equivalence of different tangent space notions.

Curves: Parameterized Curves and Tangent Vectors

Explores the definition of curves, parameterized curves, and tangent vectors in relation to open intervals and continuous functions.

Tangent spaces to submanifolds

Introduces tangent spaces to submanifolds and their properties in optimization on manifolds.

Differentiable Functions: Definitions and Interpretation

Covers the definitions of differentiable and derivable functions and their geometric interpretation.

Differentiability of Functions of Several Variables

Covers the differentiability of functions of multiple variables and the significance of directional derivatives and gradients.

Elliptical Arch Construction

Explores the construction of elliptical arches using a single ellipse arc.

Interpolation and Calculus

Covers revisions on calculus, derivation rules, interpolation, and tangent vectors.

Parametric Curves - Introduction

Introduces parametric curves and discusses tangent vectors and symmetry properties.

Differential Geometry: Surfaces

Explores the differential geometry of parametric surfaces, covering tangent space, normal curvature, principal curvatures, and asymptotic curves.

Unit Tangent Bundle of Hyperbolic Spaces

Covers the unit tangent bundle of hyperbolic spaces and geodesic flows.

Tangent Vectors and Bézier Curves

Explains tangent vectors of curves and the construction and applications of Bézier curves in computer graphics.

Parametric Curves Study Elements

Covers the study of parametric curves and tangent vectors.

Tangent vectors without embedding space: Equivalence classes

Delves into tangent vectors as equivalence classes on manifolds, highlighting their abstract nature and role in optimization.