GEOMETRY 2
- Overview
- Assessment methods
- Learning objectives
- Contents
- Full programme
- Bibliography
- Teaching methods
- Contacts/Info
Linear algebra. Basic notions of general topology. Calculus for functions in many variables
In accordance with the objectives of the course, the verification of the learning will be carried out through:
- a written test, in which the student must show that he has acquired the ability to verify the main properties of curves and surfaces on concrete examples, chosen from the list assigned during the course. The writing will last two hours and will consist of two problems. The written test mark will have a maximum score of 16. If you get at least 8 you can access the oral exam.
- an oral exam, during which the student must show that he has acquired the main notions and the proofs of the most relevant theorems of the course. The maximum oral mark is 16. The final mark is given by the sum of the written and oral marks. A grade greater than 30 automatically brings the “lode”.
This course aims to provide an elementary introduction to some basic notions of differential geometry by developing the theory of curves and surfaces in Euclidean spaces. By its very nature, the course allows to put in action almost all the mathematical skills that students have acquired up to this stage of their curriculum. The concepts introduced in this course are fundamental for further studies in geometry and algebraic topology, and also in mathematical and theoretical physics.
The course is divided into two parts: the first is dedicated to differentiable curves in the plane and in space and provides an overview of concepts and results that we will try to extend, in the second part, to the differentiable surfaces of space.
A) Differentiable curves
1) Smooth curves in space and their length. 2) Regular curves, tangent line and arc parameter.
3) Curves in space, Frenet trihedron, Curvature with sign and Frenet formulas.
4) Fundamental theorems of the global geometry of plane curves.
B) Abstract differentiable manifolds immersed in Euclidean spaces.
1) Regular surfaces of the Euclidean space.
2) Level surfaces.
3) Smooth functions between regular surfaces.
4) Tangent and differential plane of a smooth map.
5) Vector fields on surfaces, their integral curves
5) First fundamental form and isometries.
6) Orientability and Gauss map.
7) Second fundamental form and curvatures.
8) Orientability and integration on compact surfaces.
9) Parallelism and geodesics.
9) Local Gauss Bonnet theorem.
10) Triangulations and global Gauss Bonnet theorem for orientable compact surfaces.
The course is divided into two parts: the first is dedicated to differentiable curves in the plane and in space and provides an overview of concepts and results that we will try to extend, in the second part, to the differentiable surfaces of space.
A) Differentiable curves
1) Smooth curves in space and their length. 2) Regular curves, tangent line and arc parameter.
3) Curves in space, Frenet trihedron, Curvature with sign and Frenet formulas.
4) Fundamental theorems of the global geometry of plane curves.
B) Abstract differentiable manifolds immersed in Euclidean spaces.
1) Regular surfaces of the Euclidean space.
2) Level surfaces.
3) Smooth functions between regular surfaces.
4) Tangent and differential plane of a smooth map.
5) Vector fields on surfaces, their integral curves
5) First fundamental form and isometries.
6) Orientability and Gauss map.
7) Second fundamental form and curvatures.
8) Orientability and integration on compact surfaces.
9) Parallelism and geodesics.
9) Local Gauss Bonnet theorem.
10) Triangulations and global Gauss Bonnet theorem for orientable compact surfaces.
M. Abate, F. Tovena. Curve e Superfici. Springer 2006.
J. Milnor, Topology from a differentiable viewpoint. 1966
List of problems to do at home, with tips and traces provided during class exercises. 56 hours theory lessons and 16 hours of exercises in classroom. The lectures and exercises will be carried out by the teacher using a tablet, and the notes of each lesson or frontal exercise will be available on e-learning at the end of each week.
The teacher is available to meet students for clarification or further information by e-mail appointment.