Will J. Merry

Differential Geometry I Lecture Notes

26. Bundle-valued forms

We explain how horizontal equivariant vector-valued forms on a principal bundle correspond to bundle-valued forms on the associated bundle.

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25. Associated bundles

We show how to create associated fibre bundles from a principal bundle. This reproves the vector bundle construction results from earlier lectures.

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24. Principal bundles

We define principal bundles. Examples of principal bundles include homogeneous spaces (as the base space) and the frame bundle of a vector bundle.

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23. The Poincaré Lemma

We prove the global Stokes' Theorem. We then prove that de Rham cohomology is a homotopy invariant, and use this to prove the Poincaré Lemma.

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21. Manifolds with boundary

We define topological and smooth manifolds with boundary, and show how an orientation of a smooth manifold with boundary induces one on its boundary.

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19. Differential forms

We define differential forms. We construct the exterior differential as a graded derivation of degree one, and prove it commutes with the Lie derivative.

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18. Tensor fields

We define tensor fields on manifolds, and prove that the Lie derivative extends to a tensor derivation that commutes with all contractions.

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17. Sheaves and manifolds

We define (pres)heaves, and give the sheaf-theoretic definition of a manifold. We explain the relation between vector bundles and locally free sheaves.

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10. The exponential map

We define the exponential map of a Lie group, then investigate Lie group actions on manifolds. We conclude by looking at the adjoint representation.

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9. Lie groups

We define Lie groups, and explain why the tangent space to the identity of a Lie group can naturally be seen as a Lie algebra.

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