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.

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".

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.