Knowledge and understanding: understanding the differential geometry of curves and surfaces in the Euclidean 3-space.

Applying knowledge and understanding: carrying out exercises about regular curves and surfaces in the Euclidean 3-space; computing curvature, torsion and Frenet trihedron of a curve; computing curvatures and coefficients of fundamental forms of a surface.

Learning skills: reading and understanding introductory texts concerning the differential geometry of curves and surfaces.

Communication skills: presenting in a correct and appropriate manner definitions and theorems of the differential geometry of curves and surfaces.

Calculus in one and more variables. Linear algebra, affine and Euclidean geometry. Elementary notions of algebra.

The exams of Analysis 1 and 2, Algebra 1, Geometry 1 and 2 are propedeutical.

Regular curves in R^3. Frenet formulas. Regular surfaces in R^3. First and second fundamental forms. Principal curvatures. Main curvature. Gaussian curvature. Theorema Egregium. Geodesics.

M.P. Do Carmo: Differential geometry of curves and surfaces, Prentice-
Hall, 1976

M. Abate – F. Tovena: Curve e superfici, Springer Italia, 2006

Regular curves in R^3. Curvature and torsion. Frenet formulas.

Regular surfaces in R^3. Tangent plane. Differential of maps. Tangent and normal vector fields. First and second fundamental forms. Normal
curvatures. Gaussian curvature. Mean curvature. Euler formula. Elliptic, hyperbolic, planar and parabolic points. Isometries. Christoffel symbols. Gauss' Theorema Egregium.
Geodesics.
Ruled surfaces. Minimal surfaces.

Frontal lecture and group work. Exercises and relevant examples will be presented in the classroom or submitted by the students. Exercise sheets to be solved at home will be distributed. Attending students will have the opportunity to carry out projects that will form part of the final assessment.

Homework assignments can be found on TEAMS platform.

Any changes to the methods described here, which become necessary to
ensure the application of the safety protocols related to the COVID19
emergency, will be communicated on the websites of the Department of
Mathematics and Geoscience - DMG and of the Study Program in
Mathematics.

Six possible dates for the final exam, each consisting of a written three hours exam, and of an oral exam, focused on theory and exercises.

This course explores topics closely related to one or more goals of the United Nations 2030 Agenda for Sustainable Development (SDGs)