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Department of Mathematics and Applications
"Renato Caccioppoli"


Differential Geometry

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Differential Geometry

Credits

6

Prerequisites

None.

Scientific-disciplinary sector (SSD)

MAT/03 Geometry.

Examination method

Oral exam.

Learning
objectives

The aim of the course is to provide an introduction to the study of certain structures on differentiable manifolds: mainly connections on the tangent bundle, and Riemannian and pseudo-Riemannian metrics. The fundamental tools for the study of such manifolds will be provided, the most important results will be discussed, and the main techniques of proof and problem solving will be illustrated.

Syllabus

(i) Charts and atlases, differentiable structures, topology induced by an atlas, differentiable manifolds. Differentiable maps. Tangent and cotangent vectors. Tangent map. Differential of a function. Vector fields. Local frames. Tensors and tensor fields. Differential forms. Exterior product. (ii) (Pseudo-)Riemannian manifolds. Examples: Euclidean spaces, hyperbolic plane, Poincaré disc, spheres. Submanifolds and restriction of a metric. Existence theorem for Riemannian metrics. Isometries and conformal transformations. Local orthonormal frames. Gradient, divergence, curl, Laplacian. Outline of integration. Divergence theorem and Green’s identities. (iii) Linear connections: existence, Christoffel symbols, torsion, curvature, 1st Bianchi identity. Geodesics and parallel transport. The geodesic field. The exponential map. (iv) Levi-Civita connection and Riemannian geodesics. Koszul formula. Normal coordinates. The Riemannian distance. Minimising and locally minimising curves. First variation formula for arc length. Geodesically complete manifolds. Hopf-Rinow theorem. (v) Riemann tensor of the Levi-Civita connection; 2nd Bianchi identity; invariance under local isometries. Ricci tensor and scalar curvature. Flat manifolds. Weyl tensor and conformally flat manifolds; characterisation. Sectional curvature. Manifolds with constant sectional curvature. Outline of the Killing-Hopf and Cartan-Hadamard theorems.

Expected learning
outcomes

By the end of the course, students must demonstrate that they

  • know and understand the fundamental elements of differential geometry and of its proof techniques;
  • are able to apply the knowledge acquired to the study and solution of concrete examples;
  • are able to communicate ideas and solutions clearly, rigorously and effectively to both specialist and non-specialist audiences;
  • are able to identify the most appropriate methods to analyse and solve a problem relating to the course topics and to interpret the results correctly.

Learning outcomes
to be assessed

Knowledge and skills acquired on the course topics, the student’s presentation skills and command of language, the ability to apply the knowledge acquired to the solution of simple problems, the ability to support a discussion with examples and counterexamples, and command of the mathematical tools used in the course.