SPECIFIC COMPETENCES



M02CM03 - Knowing the main theorems of the local theory of curves and surfaces and being able to apply them in order to solve geometric problems.

M02CM04 - Differentiate between local and global, intrinsic and extrinsic concepts.

M02CM16 - Set up relationships between local theory and global properties of curves and surfaces in R3.

M02CM17 - Assimilate the most outstanding properties and theorems of the global differential geometry of curves and surfaces.

M02CM18 - Use differential and integral calculus and topology to study the global properties of curves and surfaces.

M02CM19 - Apply differential equations and line and surface integrals to shape global properties of curves and surfaces.



LEARNING OUTCOMES

Use differential and integral calculus and topology to study the global properties of curves and surfaces.

Apply differential equations and line and surface integrals to shape global properties of curves and surfaces.

Calculate rotation indices on flat curves.

Characterize convex curves in terms of curvature.

Work with curve and surface curvature integrals.

Know the orientability problem on surfaces.

Know the rigidity problem of the sphere.

Know how to apply Gauss-Green formulas on surfaces.

Know how to classify compact surfaces.

Calculate the Euler-Poincaré characteristic.

Work with the minimizing property of geodesic curves.

1. GLOBAL GEOMETRY OF PLANAR AND SPACE CURVES: Jordan Curve Theorem. Isoperimetric inequality. Four Vertex Theorem. Cauchy-Crofton Formula. The Turning Tangent Theorem. Fenchel's Theorem. Fary-Milnor Theorem.



2. A CHARACTERIZATION OF COMPACT ORIENTABLE SURFACES: Tubular neighborhoods. Characterization of compact orientable surfaces.



3. THE GAUSS-BONNET THEOREM: The local Gauss-Bonnet theorem. The Euler-Poincaré characteristic. The global Gauss-Bonnet theorem and applications.



4. RIGIDITY OF THE SPHERE: Theorem of Liebmann. Formulas of Minkowski and Herglotz. Theorem of Cohn-Vossen.



5. COMPLETE SURFACES: Geodesic completeness and metric completeness. The Hopf-Rinow theorem.



6. VARIATIONAL TECHNIQUES AND GEOMETRIC APPLICATIONS: First variation of the arc-length, geodesics. Second variation of the arc-length, Bonnet's theorem. Jacobi vector fields and conjugate points. Surfaces with non-positive Gaussian curvature.

The theoretical content will be presented in lectures following basic references in the Bibliography. These lectures will be complemented with problems classes (classroom practices) in which students will apply the knowledge acquired in lectures to resolve issues. In the seminars, issues and examples representative of course content content will be developed, which generally have been provided in advance to the students, to work on them and encourage subsequent reflection and discussion in the session dedicated to it.

Written exam with questions and problems: 100%

Basic bibliography

M. P. DO CARMO, Geometría diferencial de curvas y superficies, Alianza Universidad Textos 135, Alianza Editorial, 1990.

L.A. CORDERO, M. FERNÁNDEZ y A. GRAY, Geometría diferencial de curvas y superficies con Matemática©, Addison-Wesley Iberoamericana, 1995.

S. MONTIEL y A. ROS, Curvas y superficies, Proyecto Sur, 1998.

M. ABATE, F. TOVENA, Curves and Surfaces, Springer Verlag, 2012.

A.F. COSTA, M. GAMBOA y A.M. PORTO, Notas de Geometría diferencial de curvas y superficies, Sanz y Torres, 1996.

A.S. FEDENKO, Problemas de geometría diferencial, Editorial MIR, 1991.

R. S. MILLMAN y G. D. PARKER, Elements of Differential Geometry, Prentice Hall Inc., 1977.

A. PRESSLEY, Elementary Differential Geometry, Springer Verlag, 2001.

In-depth bibliography

S. S. CHERN, Curves and Surfaces in Euclidean Spaces, Studies in Global Geometry and Analysis, MAA Studies in Math., The Mathematical Association of America, 1967.
W. KLINGENBERG, Curso de Geometría diferencial, Alhambra, 1978.