Differential geometry (2015/2016)

Course code
4S003196
Name of lecturer
Giuseppe Mazzuoccolo
Coordinator
Giuseppe Mazzuoccolo
Number of ECTS credits allocated
6
Academic sector
MAT/03 - GEOMETRY
Language of instruction
English
Period
I semestre dal Oct 1, 2015 al Jan 29, 2016.

Lesson timetable

I semestre
Day Time Type Place Note
Monday 9:30 AM - 11:30 AM lesson Lecture Hall M from Oct 5, 2015  to Jan 29, 2016
Thursday 2:30 PM - 4:30 PM lesson Lecture Hall M from Oct 22, 2015  to Jan 29, 2016

Learning outcomes

REVIEW IN GENERAL TOPOLOGY;
SURFACES EMBEDDED IN THE EUCLIDEAN 3-SPACE;
TENSOR CALCULUS;
DIFFERENTIAL MANIFOLDS;
RIEMANNIAN MANIFOLDS;
CONNECTIONS AND CURVATURE TENSOR.

Syllabus

-REVIEW GENERAL TOPOLOGY
-SURFACES EMBEDDED IN THE EUCLIDEAN 3-SPACE:
• Differentiable Atlas
• Orientable Atlas
• Tangent plane
• Normal versor
• First Fundamental Form: lengths and area
• Geodesic curvature and normal curvature
• Normal sections and Meusnier Theorem
• Principal Curvatures, Gaussian curvature, Mean curvature: minimal surfaces
• Theorema Egregium
• Geodetics
- TENSOR CALCULUS
• Free vector space
• Tensor product of two vector spaces
• Tensor product of n vector spaces
• Tensor Algebra
• Transformation of the componenents of a tensoriale
• Mixed tensors
• Symmetric tensors
• Antysimmetric (alternating) tensors
• Exterior Algebra
• Determinant
• Area and Volume
-DIFFERENTIAL MANIFOLDS
• Definition and examples
• Classification of 1-manifolds
• Classification of simply-connected 2-manifolds
• Product and quotient spaces
• Differentiable maps
• Tangent space and tangent bundle
• Vector field on a manifold
• Tensor field
• Exterior Algebra on manifolds
• Riemannian Manifolds
• Metric Tensor
• Orientations
• Volume
• Exterior derivative
• De Rham Cohomology
• Homotopy
-AFFINE CONNECTION AND CURVATURE TENSOR
• Affine connection
• Parallel transport
• Levi-Civita connection
• Geodetics
• Riemann curvature tensor
• Bianchi identities

Assessment methods and criteria

Written Test (2 hours).
Test format: the student select one essay (answer with 2/3 pages) in a list of two AND two “short”
questions ( answer within 10 lines) in a list of three.

STUDENT MODULE EVALUATION - 2015/2016