Aim of the course is to give to the students some basic tools and topics in Riemmannian Geometry.

1. Preliminaries
1.1. General topology
1.2. Algebraic topology
1.3. Multivariable analysis
1.4. Projective geometry
2. Tensors
2.1. Multilinear algebra
2.2. Tensors
2.3. Scalar products
2.4. The symmetric and exterior algebras
2.5. Grassmannians
2.6. Orientation
3. Smooth manifolds
3.1. Smooth manifolds
3.2. Smooth maps
3.3. Partitions of unity
3.4. Tangent space
3.5. Smooth coverings
3.6. Orientation
3.7. Submanifolds
3.8. Immersions, embeddings, and submersions
3.9. Examples
3.10. Homotopy and isotopy
3.11. The Whitney embedding

There are no mandatory prerequisites.

Knowledge and understanding: at the end of the course the student will learn the basic notions of Riemmannian Geometry.

Applying knowledge and understanding: at the end of the course the student will learn the methodologies of Riemmannian Geometry and will be able to apply them to the study of various problems.

Making judgements: at the end of the course the student will be able to apply the techniques of Riemmannian Geometry in order to solve new problems, also coming from real-world applications.

Communications skills: at the end of the course the student will have the ability to express the fundamental notions of Riemmannian Geometry using a rigorous terminology.

Learning skills: during the course the student will learn the ability to study the notions of Riemmannian Geometry, also in order to use it in solving different kind of problems.

The teaching material prepared by the lecturer in addition to recommended textbooks (such as for instance slides, lecture notes, exercises, bibliography) and communications from the lecturer specific to the course can be found inside the Moodle platform › blended.uniurb.it

Teaching

Theorical and practical lessons.

Attendance

Although strongly recommended, course attendance is not mandatory.

Course books

M. Abate and F. Tovena, Geometria differenziale, Springer–Verlag, 2011.

M. Lee, Riemannian manifolds, Graduate Texts in Mathematics, vol. 176, Springer, 1997.

M. Spivak, A comprehensive introduction to differential geometry. Vol. III and Vol. V, Publish or Perish, 1979.

Assessment

The exam of Differential Geometry consists of a written exam on the topics of the course.  

Disability and Specific Learning Disorders (SLD)

Students who have registered their disability certification or SLD certification with the Inclusion and Right to Study Office can request to use conceptual maps (for keywords) during exams.

To this end, it is necessary to send the maps, two weeks before the exam date, to the course instructor, who will verify their compliance with the university guidelines and may request modifications.

Teaching

As for attending students.

Attendance

As for attending students.

Course books

As for attending students.

Assessment

As for attending students.

Disability and Specific Learning Disorders (SLD)

Students who have registered their disability certification or SLD certification with the Inclusion and Right to Study Office can request to use conceptual maps (for keywords) during exams.

To this end, it is necessary to send the maps, two weeks before the exam date, to the course instructor, who will verify their compliance with the university guidelines and may request modifications.