GEOMETRY 2

Degree course: 
Corso di First cycle degree in MATHEMATICS
Academic year when starting the degree: 
2015/2016
Year: 
2
Academic year in which the course will be held: 
2016/2017
Course type: 
Compulsory subjects, characteristic of the class
Credits: 
8
Period: 
Second semester
Standard lectures hours: 
64
Detail of lecture’s hours: 
Lesson (64 hours)
Requirements: 

Basic notions of pointset topology and of calculus of many real variables.

Final Examination: 
Orale
Assessment: 
Voto Finale

Aims and outcomes

The course aims at introducing the student to:

  • the fundamental notions related to the extrinsic theory, both local and global, of the regular surfaces in Euclidean 3-space;
  • the integration theory on surfaces with some geometric applications.

Moreover, by means of exercises on concrete examples:

  • the student is led to develop abilities in the computation of the main quantities describing the geometry of the surface, such as the mean and the Gaussian curvatures. Exercises of more theoretical nature will also lead the student to develop abilities in the abstract investigation of geometric properties of regular surfaces.

It is expected that the student acquires:

  • the basic notions on the extrinsic theory, both local and global, of regular surfaces in the Euclidean 3-space;
  • the basic notions from the integration theory on surfaces;
  • computational skills concerning the main quantities that describe the extrinsic geometry of the surface and investigation abilities on more theoretical properties of surfaces in Euclidean 3-space.

Program

Differentiable curves

  • Smooth curves and their length. Minimizing properties of segments.
  • Regular curves, tangent line and arc-length parameter.
  • Plane curves, Frenet frame, signed curvature and Frenet formulas.
  • Fundamental theorem of the local geometry of plane curves.
  • Plane curves of constant curvature.

Differentiable surfaces

  • Regular surfaces in the Euclidean 3-space.
  • Implicit function theorem and level surfaces.
  • Smooth maps between regular surfaces.
  • Tangent plane and differential of a smooth map.
  • First fundamental theorem and isometries.
  • Orientable surfaces and Gauss map.
  • Second fundamental form and curvatures.
  • Totally umbilical surfaces.
  • The Hilbert-Liebmann rigidity theorems.
  • The tubular neighborhood theorem.
  • Integration on surfaces.

Teaching methods
Frontal lectures. Homeworks.

Textbooks

  • M. Abate, F. Tovena. Curve e Superfici. Springer 2006.
  • S. Montiel, A. Ros. Curves and Surfaces. GSM 69, A.M.S. 2009.

Final examination
Written examination based on homeworks and oral examination.

Professors

PIGOLA STEFANO

Borrowers