Differentiable manifolds

Recommended Prerequisites

Advanced calculus. Topology. Geometry of curves and surfaces (not essential).

Teaching Methods

Classes are expository, but also demanding the participation of students, who are expected to present proofs of theorems and solve proposed exercises.

Learning Outcomes

In this curricular unit the basic theory of differential manifolds is introduced, namely the concepts of tangent vector space, vector fields, differential forms and Riemannian manifolds.

The following generic competences are developed: calculus skills; knowing mathematical results, ability to generalize and abstract; logic argumentation; written and oral rigorous and clear expression; ability to do research; ability to do autonomous learning;  imagination, creativity and critical thinking.

Work Placement(s)

No

Syllabus

Differentiable manifolds and differentiable maps. Topological aspects. Tangent vector space and the linear map induced by a differentiable map. Immersions. Submersions. The Sard and Whitney theorems. Vector fields. Integral curves and flows. Lie groups. Differentiable forms. Orientable manifolds. Exterior differentiation. Lie derivative. Integration on manifolds. Riemannian manifolds. The Levi-Civita connection. Geodesics.

Head Lecturer(s)

Margarida Maria Lopes da Silva Camarinha

Assessment Methods

Final assessment
Exam: 100.0%

Continuous assessment
Two mid-term exams : 100.0%

Bibliography

A. Salgueiro, Variedades Diferenciáveis, Departamento de Matemática da FCTUC, 2009.

 

F. Brickell, R.S. Clark, Differentiable Manifolds, Van Nostrand Reinhold, 1970.

 

E.L. Lima, Variedades Diferenciáveis, IMPA, 1973.