# Bundles, curvature and holonomy

Thirty-four chapters, self-checked by the writing AI, develop vector bundles, characteristic classes, connections, curvature, holonomy and Riemannian and Kähler geometry.

- [Local tools for bundles and transport](src/local-tools-for-bundles-and-transport.md)
- [Integral Thom classes and Euler indices](supporting/DG-CHAR-8e0b5f71efec/src/DG-CHAR-06.md)
- [Manifold duality and the Euler characteristic](supporting/DG-CHAR-8e0b5f71efec/src/DG-CHAR-07.md)
- [Projective bundles, Gysin sequences and line classes](supporting/DG-CHAR-8e0b5f71efec/src/DG-CHAR-08.md)
- [Chern classes and the Whitney formula](supporting/DG-CHAR-8e0b5f71efec/src/DG-CHAR-09.md)
- [Curvature, characteristic classes and Gauss–Bonnet](supporting/DG-CHAR-8e0b5f71efec/src/DG-CHAR-17.md)
- [Principal bundles and associated bundles](src/principal-bundles-and-associated-bundles.md)
- [Connections and parallel transport](src/connections-and-parallel-transport.md)
- [Curvature and holonomy groups](src/curvature-and-holonomy-groups.md)
- [Reduction and the holonomy theorem](src/reduction-and-the-holonomy-theorem.md)
- [Flat connections and infinitesimal holonomy](src/flat-connections-and-infinitesimal-holonomy.md)
- [Invariant connections on homogeneous bundles](src/invariant-connections-on-homogeneous-bundles.md)
- [Linear and affine connections](src/linear-and-affine-connections.md)
- [Geodesics, normal coordinates and curvature](src/geodesics-normal-coordinates-and-curvature.md)
- [Riemannian connections and convex neighbourhoods](src/riemannian-connections-and-convex-neighbourhoods.md)
- [Completeness and the Hopf–Rinow theorem](src/completeness-and-the-hopf-rinow-theorem.md)
- [Holonomy and the de Rham decomposition theorem](src/holonomy-and-the-de-rham-decomposition-theorem.md)
- [Sectional curvature and space forms](src/sectional-curvature-and-space-forms.md)
- [Affine transformations and the isometry group](src/affine-transformations-and-the-isometry-group.md)
- [Holonomy, Killing fields and analytic extension](src/holonomy-killing-fields-and-analytic-extension.md)
- [Submanifolds and hypersurfaces](src/submanifolds-and-hypersurfaces.md)
- [Jacobi fields, conjugate points and the Morse index theorem](src/jacobi-fields-conjugate-points-and-the-morse-index-theorem.md)
- [Comparison theorems, cut loci and curvature and topology](src/comparison-theorems-cut-loci-and-curvature-and-topology.md)
- [Complex manifolds and Kähler metrics](src/complex-manifolds-and-kahler-metrics.md)
- [Kähler curvature and Hermitian vector bundles](src/kahler-curvature-and-hermitian-vector-bundles.md)
- [Homogeneous spaces and invariant connections](src/homogeneous-spaces-and-invariant-connections.md)
- [Symmetric spaces](src/symmetric-spaces.md)
- [Symmetric Lie algebras and Hermitian symmetric spaces](src/symmetric-lie-algebras-and-hermitian-symmetric-spaces.md)
- [Characteristic classes and the Weil homomorphism](src/characteristic-classes-and-the-weil-homomorphism.md)
- [Vector bundles and their constructions](supporting/DG-CHAR-8e0b5f71efec/src/DG-CHAR-01.md)
- [Fundamental classes from finite chart gluing](supporting/DG-CHAR-8e0b5f71efec/src/DG-CHAR-02.md)
- [Grassmannians and classifying maps](supporting/DG-CHAR-8e0b5f71efec/src/DG-CHAR-03.md)
- [Schubert cells and universal Grassmannian cohomology](supporting/DG-CHAR-8e0b5f71efec/src/DG-CHAR-04.md)
- [Pontryagin classes and the Euler square](supporting/DG-CHAR-8e0b5f71efec/src/DG-CHAR-10.md)

[Course contents](index.html).

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