The course is an introduction to the basic concepts and methods of differential geometry.

Lectures

Oral Exam

The contents of the courses Algebra 1, Geometry 1 and 2, Linear Algebra, and of the courses in Analysis of the Laurea in Mathematics curriculum.

Differentiable manifolds: tangent and cotangent spaces, vector fields and differential forms, vector fields and coordinates: the Frobenius theorem, Lie groups and Lie algebras.
Topics in differential topology: Sard’s lemma, the deRham theorem.
Riemannian geometry: Riemannian manifolds and Levi-Civita connections, curvature, geodesics, completeness, the theorems of Hopf-Rinow and Whitehead; Jacobi fields.
Complex manifolds (if time allows): holomorphic functions of several complex variables and their basic properties, meromorphic functions, complex manifolds, Kähler manifolds.

Notes by Gian Pietro Pirola.
Frank Warner: "Foundations of differentiable manifolds and Lie groups". Graduate Texts in Mathematics, 94. Springer-Verlag, New York-Berlin.
Manfredo Perdigao Do Carmo: "Riemannian Geometry", Birkhaeuser.
Boothby, William M.: "An introduction to differentiable manifolds and Riemannian geometry". Pure and Applied Mathematics, No. 63. Academic Press, New York-London, 1975.
Th. Broecker and K. Jaenich: "Introduction to differential topology".
Milnor, J.: "Morse theory". Annals of Mathematics Studies, No. 51 Princeton University Press, Princeton, N.J. 1963.
D. Huybrechts: "Complex geometry. An introduction". Universitext. Springer-Verlag, Berlin, 2005.