GEOMETRY 2
- Overview
- Assessment methods
- Learning objectives
- Contents
- Full programme
- Bibliography
- Teaching methods
Elements of general topology.
Calculus on functions of several variables.
Written and oral examination.
the students will learn the fundamentals of the theory of regular surfaces in euclidean space, in their local and global aspects. They should also acquire ability to calculate local and global invariants of concrete surfaces. Moreover they should acquire ability in the abstract study of their geometrical properties.
The course is divided into two parts: the first is dedicated to differentiable curves in the plane and provides an overview of concepts and results that we will try to extend, in the second part, to differentiable surfaces in the space.
A) Differentiable curves
1) Smooth curves in space and their length. Minimizing property of line segments.
2) Regular curves, tangent line and arc parameter.
3) Plane curves, Frenet's frame, Curvature with sign and Frenet's formulas.
4) Fundamental theorem of the local geometry of plane curves.
5) Plane curves with constant curvature.
B) Differentiable surfaces
1) Regular surfaces in Euclidean space.
2) Implicit function Theorem and level surfaces.
3) Smooth functions between regular surfaces.
4) Tangent and differential plane of a smooth map.
5) First fundamental form and isometries.
6) Orientability and Gauss map.
7) Second fundamental form and curvatures.
8) Totally umbilical surfaces.
9) The Hilbert-Liebmann rigidity theorems.
10) Integration on compact surfaces.
The course is divided into two parts: the first is dedicated to differentiable curves in the plane and provides an overview of concepts and results that we will try to extend, in the second part, to differentiable surfaces in the space.
A) Differentiable curves
1) Smooth curves in space and their length. Minimizing property of line segments.
2) Regular curves, tangent line and arc parameter.
3) Plane curves, Frenet's frame, Curvature with sign and Frenet's formulas.
4) Fundamental theorem of the local geometry of plane curves.
5) Plane curves with constant curvature.
B) Differentiable surfaces
1) Regular surfaces in Euclidean space.
2) Implicit function Theorem and level surfaces.
3) Smooth functions between regular surfaces.
4) Tangent and differential plane of a smooth map.
5) First fundamental form and isometries.
6) Orientability and Gauss map.
7) Second fundamental form and curvatures.
8) Totally umbilical surfaces.
9) The Hilbert-Liebmann rigidity theorems.
10) Integration on compact surfaces.
M. Abate, F. Tovena. Curve e Superfici. Springer 2006.
S. Montiel, A. Ros. Curves and Surfaces. GSM 69, A.M.S. 2009.
M. Postnikov,
Lectures in Geometry, semester II
MIR 1982
Class lectures and homework.