The course intends to give an introduction to the basic concepts and methods of differential Geometry

Lectures

Oral examination

The contents of the courses Algebra 1, Geometry 1 and 2, Linear Algebra, and of the courses in Analysis of the first two years in maths courses

Differentiable varieties:
Tangent and cotangent spaces, vector fields and forms. Froebenius theorem, Lie groups and Lie algebras.

Elements of differentiable topology:
Sard lemma , De Rahm theorem

Riemannian Geometry:
Riemannian varieties, Levi Civita connection,
curvature, geodesics, Hopf-Rinow and Whithehead thorems, Jacobi fields.

Complex and Algebraic varieties.
Holomorphic and meromorphic functions, Kaeheler and projective varieties. Zariski topology

Gian Pietro Pirola: dispense.
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.
P.A. Griffiths, J. Harris: "Principles of algebraic geometry". John Wiley &
Sons, Inc., New York, 1994. Wiley & sons.
I.R. Shafarevich: "Basic Algebraic Geometry 1" (Second Edition), Springer, 1994.
J. Harris, "Algebraic Geometry - A First Course", Graduate Texts in Mathematics 133,
Springer, 1992.