Lectures are held in Italian or English, at the students' choice. The course is addressed to students in mathematics. but it can also be also attended by students in physics.
Basic introduction to the concepts and methods of modern differential geometry.
The fundamental notions discussed in the course are those of smooth structure, tangent and cotangent bundle, vector field and tensor field, algebra of differential forms, and de Rham cohomology. Connections between the topics covered and mathematical analysis, topology, algebra, mathematical physics, algebraic topology and algebraic geometry are highlighted.
The course follows a traditional approach.
1. Smooth atlases; smooth structures; topological issues
2. Smooth manifolds
3. The inverse mapping theorem
4. Partitions of unity
5. Quotient manifolds
6. Sheaf of smooth functions
7. Cotangent space and tangent space; differential of a smooth map
8. Tangent bundle and vector fields
9. Flow of a vector field; Lie derivative
10. Multilinear algebra (tensor product of R-Modules; exterior algebra)
11. Tensor fields
12. Differential forms and Cartan differential
13. de Rham cohomology
14. Orientability; integration of differential forms and Stokes' theorem.
Detailed notes will be made available to students on the Aulaweb site.
Ricevimento: By appointment (email address: bartocci@dima.unige.it)
CLAUDIO BARTOCCI (President)
MATTEO PENEGINI
According to the academic calendar.
Oral exam (2 questions).
