Università Roma Tre

1. Topological classification of curves and compact surfaces. Triangulations, Euler characteristic.

2. Smooth and regular curves in Euclidean space. Immersions and imbeddings. Arc length. Curvature and the Fundamental Theorem of local geometry of plane curves.

3. Regular surfaces in R^3, local coordinates. Inverse image of a regular value. Maps and diffeomorphisms. Tangent plane and derivative of a map. Normal unit vector, orientation

and the Gauss map of a surface in R^3. Orientable surfaces, examples. The Moebius band is non-orientable.

4. Riemannian metric, examples. The shape operator is self-adjoint. Principal and asymptotic directions. The Mean and Gauss curvatures.

5. The geometry of the Gauss map. The sign of the Gauss curvature and position of the tangent plane. Theorems of Meusnieur and Olinde Rodrigues. Geometric properties of compact surfaces, Minimal surfaces and Ruled surfaces.

6. Isometries and local isometries. The Shape operator in isothermal coordinates. Proof of Gauss' Theorema Egregium. Examples, counter-examples and applications.

7. Homeworks.

8. 12 hours of lab for the visualization and computation on curves and surfaces.

[1] J.M. Lee, Introduction to topological manifolds. Springer, (2000). http://dx.doi.org/10.1007/b98853

[2] E. Sernesi, Geometria 2. Boringhieri, (1994).

[3] M. Do Carmo , Differential Geometry of Curves and Surfaces. Prentice Hall, (1976).

[4] M. Abate, F. Tovena, Curves and Surfaces. Springer, (2006).