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M4042 Differential geometry
Aims:
Manifolds are "topological spaces with local coordinates" (e.g. surfaces) i.e. spaces locally homeomorphic to the euclidean m-space. A differentiable structure on such a manifold makes it possible to generalize to these "curved spaces" many basic concepts of calculus, connected with differentiation or integration of functions and maps defined on the "flat" euclidean m-space.
This is an introductory course on differentiable manifolds and related basic concepts, which are the common ground for differential geometry, differential topology, global analysis, (i.e calculus on manifolds including geometric theory of integration) and modern mathematical physics.
Syllabus:
Smooth manifolds; coordinate systems. Tangent vectors; tangent and cotangent bundles of a manifold, tensor fields; connections and covariant differentiation. Metrics. Geodesics. Curvature tensor. Relation between geometry and topology of manifolds. Applications to general relativity.
Prerequisites:
MTH2111 / MTH3111 Mathematical Structures or MTH3132 Analysis and Geometry or MTH2021 Linear Algebra with Applications. M4052 Topology is recommended as a co-requisite.
Recommended texts:
Informal lecture notes may be provided.
Do Carmo, M., Introduction to Riemannian geometry, Birkhauser, 1992.
References:
Spivak, M.A., A Comprehensive introduction to differential geometry, Vol. I, Publish or Perish Inc., 1979 (1970).
Spivak, M, Calculus on manifolds, W.I. Benjamin, Inc., 1965.
Lecturer: TBA
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