Topology: topological classification of curves and surfaces. Differential geometry: study of the geometry of curves and surfaces in R^3 to provide concrete and easily calculable examples on the concept of curvature in geometry. The methods used place the geometry in relation to calculus of several variables, linear algebra and topology, providing the student with a broad view of some aspects of mathematics.
teacher profile teaching materials
2. Smooth and regular curves in Euclidean space. Immersions and imbeddings. Arc lenght. 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] M. Do Carmo , Differential Geometry of Curves and Surfaces. Prentice Hall, (1976).
[3] E. Sernesi, Geometria 2. Boringhieri, (1994).
[4] M.Abate, F.Tovena, Curve e Superfici. Springer, (2006).
Programme
1. Topological classification of curves and compact surfaces. Triangulations, Euler carachetristic.2. Smooth and regular curves in Euclidean space. Immersions and imbeddings. Arc lenght. 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.
Core Documentation
[1] J.M. Lee, Introduction to topological manifolds. Springer, (2000). - – http://dx.doi.org/10.1007/b98853
[2] M. Do Carmo , Differential Geometry of Curves and Surfaces. Prentice Hall, (1976).
[3] E. Sernesi, Geometria 2. Boringhieri, (1994).
[4] M.Abate, F.Tovena, Curve e Superfici. Springer, (2006).
Type of evaluation
The final grade is based on a written exam, homeworks and lab project. teacher profile teaching materials
Programme
Weekly meetings in the computer science laboratory for the use of Wolfram Mathematica software, the goal being the graphic representation of curves and surfacesCore Documentation
Alfred Gray, Elsa Abbena, Simon Salamon, "Modern Differential Geometry of Curves and Surfaces with Mathematica"Reference Bibliography
Alfred Gray, Elsa Abbena, Simon Salamon, "Modern Differential Geometry of Curves and Surfaces with Mathematica"Type of delivery of the course
Computer science LaboratoryType of evaluation
Project in Mathematica