1.Differential Forms in Rn
4.Integration on Manifolds; Stokes Theorem and Poincare's Lemma
1.Integration of Differential Forms
5.Differential Geometry of Surfaces
1.The Structure Equations of R
2.Surfaces in R3
3.Intrinsic Geometry of Surfaces
6.The Theorem of Gauss-Bonnet and the Theorem of Morse
1.The Theorem of Gauss-Bonnet
2.The Theorem of Morse
This is a free translation of a set of notes published originally in Portuguese in1971. They were translated for a course in the College of Differential Geometry, ICTP, Trieste, 1989. In the English translation we omitted a chapter onthe Frobenius theorem and an appendix on the nonexistence of a completehyperbolic plane in euclidean 3-space （Hilbert's theorem）. For the presentedition, we introduced a chapter on line integrals.
In Chapter 1 we introduce the differential forms in Rn. We only assumean elementary knowledge of calculus, and the chapter can be used as a basisfor a course on differential forms for "users" of Mathematics.
In Chapter 2 we start integrating differential forms of degree one alongcurves in Rn. This already allows some applications of the ideas of Chapter 1.This material is not used in the rest of the book.
In Chapter 3 we present the basic notions of differentiable manifolds. Itis useful （but not essential） that the reader be familiar with the notion of aregular surface in R3.
In Chapter 4 we introduce the notion of manifold with boundary andprove Stokes theorem and Poincare's lemma.
Starting from this basic material, we could follow any of the possi-ble routes for applications: Topology, Differential Geometry, Mechanics, LieGroups, etc. We have chosen Differential Geometry. For simplicity, we restricted ourselves to surfaces.
Thus in Chapter 5 we develop the method of moving frames of Elie Cartanfor surfaces. We first treat immersed surfaces and next the intrinsic geometryof surfaces
Finally, in Chapter 6, we prove the Gauss-Bonnet theorem for compactorientable surfaces. The proof we present here is essentially due to S.S.Chern.We also prove a relation, due to M. Morse, between the Euler characteristicof such a surface and the critical points of a certain class of differentiablefunctions on the surface.