The course intend to give an introduction of the
basic concept and of the methods of differential Geometry

Lectures

Oral examination

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

Differentiable varieties,Element of differentiable topology,Riemannian Geometry,Complex and Algebraic varieties.



Differentiable varieties:

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



Element of differentiable topology:

Sard lemma , De Rhm theorem



Riemannian Geometry

Riemannian variety Levi Civita connection,

curvatiure, geodetics, 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.