Geometry 4

A.Y. 2022/2023
9
Max ECTS
93
Overall hours
SSD
MAT/03
Language
Italian
Learning objectives
The aim of the course is to present an introduction to differentiable manifolds, with a particular reference to curves and surfaces in R^3.
Expected learning outcomes
Knowledge of some elementary properties of differentiable varieties and ability to use them in some concrete instances.
Single course

This course cannot be attended as a single course. Please check our list of single courses to find the ones available for enrolment.

Course syllabus and organization

Single session

Responsible
Lesson period
Second semester
Prerequisites for admission
Geometry 1,2 and 3
Assessment methods and Criteria
Oral and written examination
Geometria 4 (prima parte)
Course syllabus
COMPLEMENTS OF TOPOLOGY

- Metric spaces

- Compactness, countability axioms, paracompactness, separability


DIFFERENTIABLE MANIFOLDS

- Differentiable manifolds

- Maps between manifolds

- The tangent space

- Rank theorem, inverse function theorem and applications; immersions, submersions, embeddings and submanifolds

- The tangent and cotangent bundles; vector fields

- Vector bundles (some notes)

- Multilinear algebra and tensor fields

- Differential forms and exterior differential

- Orientable manifolds and integration on manifolds

- Riemannian manifolds (some notes)
Teaching methods
Oral Lessons and exercises
Teaching Resources
"Introduction to Smooth Manifolds", 2nd ed., J. M. Lee (Springer)
- "Differential Geometry of Curves and Surfaces", M. P. do Carmo (Prentice-Hall)
- "An introduction to differentiable manifolds and Riemannian geometry", W.M. Boothby
(Orlando Academic Press)
Geometria 4 (seconda parte)
Course syllabus
CURVES
Curves in the plane
Vector product and curves in the space. Arc Length
Frenet Triedron. Rigidity theorem of curves in the plane and in the space.
SURFACES
Definitions of regular surface, tangent space. First Fundamental form, length, angles and area on a surface. Differential maps between surfaces, the differential map, the Gauss map, Weingarten operator, principal curvatures, Principal directions, Second Fundamental form, Gauss curvature. Nature of points on a surface. Normal curvature, Meusnier Theorem. Asymptotic curves, lines of curvature, Rigidity theorem of a surface. Isometries between surfaces. Egregium Theorem of Gauss. Intrinsic concepts. Covariant derivatives, geodetic curvatures. Ideas of the Gauss Bonnet Theorem and applications.
Teaching methods
Oral lessons and exercises
Teaching Resources
- "Differential Geometry of Curves and Surfaces", M. P. do Carmo (Prentice-Hall)
Geometria 4 (prima parte)
MAT/03 - GEOMETRY - University credits: 6
Practicals: 24 hours
Lessons: 36 hours
Geometria 4 (seconda parte)
MAT/03 - GEOMETRY - University credits: 3
Practicals: 24 hours
Lessons: 9 hours
Professor: Gori Anna
Educational website(s)