Differentiable manifolds
1
2017-2018
02021954
Mathematics
Portuguese
Face-to-face
SEMESTRIAL
6.0
Compulsory
2nd Cycle Studies - Mestrado
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)
NoSyllabus
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.