The course aims to provide students with the basic concepts on Differential Geometry of manifolds. At the end of the course the student will know the main terminology and definitions about manifolds and Riemannian manifolds, and some of the main results. He/she will be able to produce rigorous arguments and proofs on these topics and he/she will be able to read articles and texts of Differential Geometry.
The course consists of lectures. Notes for each lecture will be provided.
-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
Reference books | |||||
Author | Title | Publisher | Year | ISBN | Note |
Do Carmo | Differential Geometry of Curves and Surfaces (Edizione 2) | 2016 | |||
Do Carmo | Riemannian Geometry | 1992 | |||
Jürgen Jost | Riemannian Geometry and Geometric Analysis (Edizione 5) | Springer | 2008 |
During the exam, students must show that:
- they know and understand the fundamental concepts of differential geometry
- they have analytical and abstraction abilities
- they support their argumentation with mathematical rigor.
The exam consists of a written test in which the student will have to choose one of two essays in which they provide a broad discussion of one of the topics presented during the lectures (answer approximately 2/3 pages ) and two of three short questions (answer approximately 10 rows).
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