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Calculus on Manifolds


Tuesday, 1 October 2019 to Sunday, 15 March 2020
Total hours: 40
Hours of lectures: 40

Examination procedure

  • oral exam


Multivariable calculus, basic notions in topology


Differentiable Manifolds

Vector fields and and Lie derivatives

Tensors and differential forms

Exterior algebras and derivatives

Frobenius Theorem

Metric and curvature

Moving frames and structure equations

Rigidity results

Time permitting, Hodge's Theory


Educational goals: the aim of the course is to use calculus tools on "curved objects", and to relate local/differential concepts to global ones, with applications to the study of differentiable and Riemannian manifolds.


Bibliographical references

M. Spivak, A comprehensive introduction to Differential Geometry, Publish or Perish.

R. Abraham, J. Marsden, T. Ratiu, Manifolds, tensor analysis, and applications, Springer.

F. Warner, Foundations of differentiable manifolds and Lie groups, Springer.