GEOMETRIA II
In is required the knowledge of the basic notions of pointset topology and of calculus of many real variables.
According to the objectives of the teaching, the verification of learning will be carried out through:
- a written test in which the student must demonstrate that she/he has acquired the ability to verify the principal properties of smooth surfaces on concrete examples;
- an oral test during which the student will have to show that she/he has acquired the most important notions and the proofs of the most relevant theorems of the course.
The course aims at introducing the students to:
1) the fundamental notions related to the extrinsic theory, both local and global, of the regular surfaces in Euclidean 3-space;
2) the integration theory on surfaces with some geometric applications.
Moreover, by means of exercises on concrete examples:
3) 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:
1) the basic notions on the extrinsic theory, both local and global, of regular surfaces in the Euclidean 3-space;
2) the basic notions from the integration theory on surfaces;
3) 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.
The course is divided into two parts: the first is devoted to differentiable planar curves and provides an overview of concepts and results that will be extended, in the second part, to differentiable surfaces in the 3-space.
A) Differentiable curves
1) Smooth curves and their length. Minimizing properties of segments.
2) Regular curves, tangent line and arc-length parameter.
3) Plane curves, Frenet frame, signed curvature and Frenet formulas.
4) Fundamental theorem of the local geometry of plane curves.
5) Plane curves of constant curvature.
B) Differentiable surfaces
1) Regular surfaces in the Euclidean 3-space.
2) Implicit function theorem and level surfaces.
3) Smooth maps between regular surfaces.
4) Tangent plane and differential of a smooth map.
5) First fundamental theorem and isometries.
6) Orientable surfaces and Gauss map.
7) Second fundamental form and curvatures.
8) Totally umbilical surfaces.
9) The Hilbert-Liebmann rigidity theorems.
10) Integration on surfaces.
M. Abate, F. Tovena. Curve e Superfici. Springer 2006.
S. Montiel, A. Ros. Curves and Surfaces. GSM 69, A.M.S. 2009.
Frontal lectures and homework.
Borrowed from
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