Geometry 4

A.Y. 2019/2020
9
Max ECTS
89
Overall hours
SSD
MAT/03
Language
Italian
Learning objectives
The aim of the course is to present an introduction to differentiable manifolds, with a particular reference to curves and surfaces in R^3.
Expected learning outcomes
Knowledge of some elementary properties of differentiable varieties and ability to use them in some concrete instances.
Course syllabus and organization

Unique edition

Responsible
Lesson period
Second semester
Prerequisites for admission
The arguments of the courses of the first three semesters.
Assessment methods and Criteria
The final examination consists of two parts: a written exam and an oral exam.

- During the written exam, the student must solve some exercises in the format of open-ended questions, with the aim of assessing the student's ability to solve problems in basic differential geometry. The duration of the written exam will be proportional to the number of exercises assigned, also taking into account the nature and complexity of the exercises themselves (however, the duration will not exceed three hours). In place of a single written exam given during the first examination session, the student may choose instead to take two midterm exams. The outcomes of these tests will be available in the SIFA service through the UNIMIA portal.
- The oral exam can be taken only if the written component has been successfully passed. In the oral exam, the student will be required to illustrate results presented during the course in order to evaluate her/his knowledge and comprehension of the arguments covered as well as the capacity to apply them.

The complete final examination is passed if all two parts (written, oral) are successfully passed. Final marks are given using the numerical range 0-30, and will be communicated immediately after the oral examination.
Geometria 4 (prima parte)
Course syllabus
Topological manifolds and topological manifolds with boundary. Topological surfaces.
Differentiable manifolds; examples (n-spheres, tori, projective spaces). Differentiable mappings. Rank theorem. Submanifolds.
Local theory of curves in R3. Curvature and torsion. Frenet Serret formulas.
Local theory of surfaces in R3. First fondamental form. Examples. Arc lengh of a curve on a surface. Gauss map. Second fondamental form. Normal curvature of a curve on a surface. Principal curvatures. Gaussian curvature, average curvature. Elliptic, parabolic and hyperbolic points.
Tangent spaces. Differential map.
Teaching methods
Lessons and exercises
Teaching Resources
M.Abate, F.Tovena, Curve e superfici, New York Springer-Verlag 2006
M.Abate, F.Tovena, Geometria Differenziale, New York Springer-Verlag 2011
W.M. Boothby, An introduction to differentiable manifolds and Riemannian geometry. Orlando Academic Press, Inc. 1986
R. Bott, L. Tu, Differential Forms in Algebraic Topology. New York Springer Verlag 1982
E. Sernesi, Geometria 2, Bollati Boringhieri 1994
Geometria 4 mod/02
Course syllabus
Partition of Unity. Multilinear algebra: tensor and wedge product.
Vector bundles and sections. Tangent and cotangent bundles.
Differential forms. Exterior differentiation. Volume form and orientation. Integration on manifolds. Stokes' theorem.
Teaching methods
Lessons and exercises
Teaching Resources
M.Abate, F.Tovena, Geometria Differenziale, New York Springer-Verlag 2011
W.M. Boothby, An introduction to differentiable manifolds and Riemannian geometry. Orlando Academic Press, Inc. 1986
R. Bott, L. Tu, Differential Forms in Algebraic Topology. New York Springer Verlag 1982
E. Sernesi, Geometria 2, Bollati Boringhieri 1994
Geometria 4 (prima parte)
MAT/03 - GEOMETRY - University credits: 6
Practicals: 33 hours
Lessons: 27 hours
Geometria 4 mod/02
MAT/03 - GEOMETRY - University credits: 3
Practicals: 11 hours
Lessons: 18 hours
Professor: Bertolini Marina
Educational website(s)