Geometry and Topology (21/22)

Faculteit Science and Engineering
Jaar 2021/22
Vakcode WMMA018-05
Vaknaam Geometry and Topology (21/22)
Niveau(s) master
Voertaal Engels
Periode semester II a
ECTS 5
Rooster rooster.rug.nl

Uitgebreide vaknaam Geometry and Topology (tweejaarlijks 2021/2022)
Leerdoelen At the end of the course, the student is able to:
1) derive or reproduce the formula for Christoffel symbols of a Levi-Civita connection associated with a Riemannian metric on a smooth manifold
2) calculate lengths of trajectories or their intersection angles, areas of surfaces and volumes of three-dimensional domains
3) parallel-transport (co)vectors along a given curve, to find the (classes of) geodesics and outline their properties, and to distinguish between manifolds by using their invariants such as the Riemann, Ricci, or scalar curvatures.
4) recognize the smoothness class of manifolds and maps, introduce coordinate charts and reparametrizations, calculate the dimension and prove compactness of homogeneous spaces studied in the course.
5) apply the notion of homotopy of a map to approximate it by a smooth map; calculate the index of a smooth map, and classify smooth maps to an n-sphere up to homotopy.
6) calculate the Whitney number, the genus of a surface, and apply the Gauss-Bonnet and Poincare theorems; prove the hedgehog lemma, and relate the Euler characteristic of triangulated surface to the sum of indexes of singular points of a generic vector field on it.
Omschrijving This master course is equally oriented towards mathematicians and physicists. Its content interlaces between Riemannian geometry (tensor calculus on manifolds) when it is read before / in parallel with General Relativity, and Poincare topology of manifolds (analysis situs) otherwise, e.g., in 2019/20.

The goals of topological face are (1) familiarity with the concepts of manifold, Lie group, homogeneous and symmetric spaces; (2) familiarity with the construction of vector bundle, normal vector bundle, tangent bundle and tangent mapping; (3) acquaintance with various properties of smooth maps of manifolds (Sard's theorem, Morse function); (4) approximation of continuous homotopies by smooth ones, construction of the degree of mapping, and classification of homotopy classes of maps to a sphere; (5) construction of the degree of a vector field, the Whitney number, and the Gauss-Bonnet and Poincare theorems; (6) definition of the index of a singular point of a vector field, the intersection index, and linking coefficient; (7) time-permitting, applications in complex analysis and/or knots, links, and braids.
All these mathematical constructions are extremely important in modern theoretical physics, astronomy and mechanics.

The material of advanced course in Riemannian geometry covers standard notions, objects, and structures such as manifolds, tensor fields, metric tensor, Levi-Civita connections and Christoffel symbols, parallel transport, geodesics (as the locally shortest), and the Riemann, Ricci, and scalar curvature tensors. The notions of vector bundles beyond the tangent bundle and smooth fibre bundles conclude the course. The lectures aim is to communicate a firm theoretical background, including familiarity
with the proofs of the structures' properties, and ability to apply the formalism in General Relativity or Field Theory. (A familiarity with basics of General Relativity would facilitate the study yet it is not a compulsory pre-requisite.)
Uren per week
Onderwijsvorm Hoorcollege (LC), Werkcollege (T)
Toetsvorm Opdracht (AST), Schriftelijk tentamen (WE)
(max(100% exam, 45% from homeworks + 55% exam).)
Vaksoort master
Coördinator A.V. Kiselev
Docent(en) A.V. Kiselev
Verplichte literatuur
Titel Auteur ISBN Prijs
Tensor calculus on manifolds (Lecture notes, 2009); will be available during the lectures A.V.Kiselev
Lecture notes in General Relativity (Lectures 1-3, 1997);
on-line arXiv:gr-qc/9712019
S.M.Carroll 978-5-89482-560-1
Modern geometry -- methods and applications. Part II. The geometry and topology of manifolds. 1985 B.A.Dubrovin, A.T.Fomenko, S.P.Novikov
The geometry of Physics: an introduction (CUP, Cambridge, 1997, revised 2001) Th.Frankel
Morse theory. 1969 J.Milnor
Entreevoorwaarden A familiarity with basics of General Relativity and/or Analysis on Manifolds would facilitate the study yet it is not a compulsory prerequisite.
Opmerkingen This course unit prepares for Geometry and Differential Equations, General Relativity, Differential Geometry, Lie Groups and Algebras.

This course was registered last year with course code WMMA13000
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