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DIFFERENTIAL GEOMETRY 1 (532SM)

A.Y. 2020 / 2021

Period 
Second semester
Credits 
6
Duration/Length 
48
Type of Learning Activity 
Programme specific subjects
Study Path 
[PDS0-2018 - Ord. 2018] common
Syllabus 
Teaching language 

English

Learning objectives 

KNOWLEDGE AND UNDERSTANDING: becoming familiar with several
foundational results of Riemannian and Lorentzian geometry.
APPLYING KNOWLEDGE AND UNDERSTANDING: developing the ability to
solve independently exercises in Riemannian and Lorentzian geometry as
well as that of drawing simple corollaries from theoretical results proved
in lectures.
MAKING JUDGEMENTS: recognising and applying the basic techniques of
Riemannian and Lorentzian geometry to relevant problems and situations.
COMMUNICATION SKILLS: achieving proficiency in the language of
Riemannian geometry and curvature.
LEARNING SKILLS: developing the ability to use with profit the main
textbooks in Riemannian geometry

Prerequisites 

Basic notions in differential and algebraic topology and tensor calculus.

Contents 

Riemannian metrics. Geodesics and curvature: sectional, Ricci and scalar curvatures. Topology of Riemannian manifolds with a curvature bound. Lorentzian metrics.

Teaching format 

Traditional lectures will be complemented by example in class that focus on the application of theoretical results.
Student partecipation is encouraged.

Extended Programme 

.

End-of-course test 

Oral exam. Initially candidates will be required to show their understanding of the context, structure and proof of one or more of the main theoretical results proved in lectures. Subsequently they will be
tested with respect to their ability to solve simple problems and to produce relevant examples or counterexamples. The final mark will reflect the outcome of these two steps.

Other information 

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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.
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Texts/Books 

Partial list of references:
J. Jost, Riemannian Geometry and Geometric Analysis, Springer-Verlag,
2011;
I. Chavel, Riemannian Geometry, Cambridge studies in advanced mathematics, 2010


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