Differentiable Manifolds


Differentiable manifolds: local structure, submanifolds, Sard's Theorem, transversality, vector fields and flows. Fibre bundles, tangent and cotangent bundles of a manifold. Lie derivative of vector fields. Lie algebras. Lie groups (classical). Homogeneous spaces. Differential forms, exterior derivative. Symplectic forms. Integration on manifolds. Stokes' theorem. One or more of the following additional topics may also be covered:
- Riemannian manifolds. Curvature. Symmetric spaces. Classical examples.
- de Rham cohomology, singular cohomology and de Rham's theorem.
- Degree of a map. Index of a vector field. Applications.
- Distributions. Frobenius and Stefan-Sussmann theorems.

Research and Events


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Defended Theses

  • Structure, algorithmics and dynamics of endomorphisms for certain classes of groups
      André da Cruz Carvalho (April 2023)
      Pedro V. Silva
  • A point-free study of z-embeddings, more general classes of localic maps, and uniform continuity
      Ana Belén Avilez García (April 2023)
      Jorge Picado
  • Analysis of equations of motion of inextensible strings and networks
      Ayk Telciyan (April 2023)
      Dmitry Vorotnikov
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