This subject is taught exclusively in Spanish.



Both the differential and the integral calculus in Euclidean spaces will be generalised to certain topological spaces known as differentiable manifolds. These spaces can be locally identified with Euclidean spaces by means of suitable local coordinate systems. Thus, the local geometry of manifolds is reduced to classical analysis, while the notions and relations which do not depend on the chosen local coordinates system are those proper to Differential Geometry.



The concept of the smooth manifold and smooth map will be introduced, and students will learn to work with coordinates. The tangent space, vector fields and differential forms on manifolds will be considered. The exterior differential of differential forms and the integral calculus with differential forms will be defined eventually proving a general version of Stokes' Theorem, and showing some classical applications and particular cases such as Green's and Stokes' Theorem as studied in classical Calculus.

COMPETENCIES:



M12CM01- Learn the notions, tools and methods of the geometry of smooth manifolds.

M12CM02- Know the differential, integral and tensor calculus on smooth manifolds.

M12CM03- Know certain important basic results of the geometry of smooth manifolds.

M12CM04- Use tensor and exterior calculus, both in intrinsic form and in coordinates. Apply the calculus methods of Differential Geometry.



LEARNING OUTCOMES:



1. Use tensor and exterior calculus, both in intrinsic form and in coordinates.

2. Apply the calculus methods of Differential Geometry.

1. SMOOTH MANIFOLDS: Smooth manifolds. Basic notions and examples. Topology of a manifold. Smooth maps between manifolds. Diffeomorphisms. Tangent and cotangent space. Differential of a smooth map. Chain rule. Classification of smooth maps by the rank of its differential.



2. VECTOR FIELDS OVER A MANIFOLD: Tangent bundle. Vector Fields as derivations. Lie algebra of vector fields. Calculus in coordinates. Vector fields related by a smooth map. Integral curves of a vector field. Flow.



3. DIFFERENTIAL FORMS: Differential forms on manifolds. Exterior product. Exterior algebra of a manifold. Exterior differential of differential forms. Closed and exact forms. Notions about the de Rham cohomology groups. Betti numbers and invariance by diffeomorphisms. Lie derivative and interior product.



4. INTEGRATION IN MANIFOLDS. Volume forms and orientation. Integration in manifolds. Regular domains. Stokes' Theorem. Applications.

The more relevant facts will be exposed in the lectures following the basic references listed in the Bibliography.



Lectures will be supplied with classroom practices (problem sessions) and seminars.

The problem sessions will require students to solve problems by applying the concepts and results learned in the lectures.



In the seminar sessions, students will work on previously posed problems and relevant examples relative to the contents of the theorical lectures, to motivate reflection and academic discussion during the session.

• Continuous Assessment System
• Final Assessment System
• Tools and qualification percentages:
  • See GUIDELINES (Orientaciones) (%): 100

Written exam (mandatory to pass in order to apply the other parts with their percenteages): 60%

Seminars: 25%

Assignments (written problems): 15%







Article 8.3 of the Student Assessment Regulations for official degrees, "students shall be entitled to be assessed by the final assessment system, regardless of whether or not they took part in the continuous assessment system. To that end, students shall submit a written waiver of continuous assessment to the lecturer responsible for the subject within 9 weeks of the beginning of the semester [...] That final assessment will be a written final exam".

Written exam: 100%.

Basic bibliography

W. M. BOOTHBY, An introduction to differentiable manifolds and Riemannian Geometry, Academic Press, 1975.



F. BRICKELL y R. S. CLARK, Differentiable manifolds, an introduction, Van Nostrand, 1970.



P.M. GADEA y J. MUÑOZ, Analysis and algebra on differentiable manifolds: a workbook for students and teachers, Kluwer Academic Publishers, 2001.



J.M. GAMBOA y J.M. RUIZ, Iniciación al estudio de las variedades diferenciables, 2ª Edición, Sanz y Torres, 2006.



J. M. LEE, Introduction to smooth manifolds, Springer Verlag, 2002.



F. WARNER, Foundations of differentiable manifolds and Lie groups, Springer Verlag, 1983.

Web addresses

https://www.ime.usp.br/~gorodski/teaching/mat5799-2015/hitchin-manifolds2012.pdf