Differentiable Manifolds

Recommended Prerequisites

Advanced calculus. Topology. Curve and surface geometry (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

Distinguishable variations and applications. Topological features. Tangent vector space and linear application induced by a distinguishable application. Immersions. Submersions. Sard and Whitney theorems. Vector fields. Integral curves and fluxes. Lie groups. Differential shapes. Steerable varieties. Exterior differentiation. Lie derivative. Integration in varieties. Riemannian varieties. The Levi-Civita connection. Geodesics.

Assessment Methods

Final Assessment
Exam: 100.0%

Continuous Assessment
Two mid-term exams (weight of 50% each): 100.0%

Bibliography

SALGUEIRO, A. (2009). Variedades diferenciáveis. Universidade de Coimbra.

 

BRICKELL, F. & CLARK, R. S. Differentiable manifolds.

 

LIMA, Elon Lages. Variedades diferenciáveis.