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Differentiable manifolds 2 (BM)


This is a continuation of the course Differentiable manifolds 1, which considered manifolds as subsets in Euclidian space with certain properties. We will generalise this concept to manifolds as topological spaces which locally look like Euclidean space, and on which differentiability of functions and maps is well defined. A standarde example is the n-dimensional projective space P^n: the topological quotient of the n-dimensional unit sphere S^n by the antipodal map; a function on P^n is differentiable if its composition with the quotient map is differentiable on S^n. We will see that many concepts from Differentiable manifolds 1 generalise to these abstract manifolds in a natural way: tangent spaces, vector fields, differential forms, integration, the Theorem of Stokes, (semi-)Riemannian metrics etc. Naturally associated to manifolds are vector bundles and sheaves, and we will give an introduction to these.

The theory of differentiable manifolds is the basis of very different fields, e.g. differential geometry, differential topology, global analysis, and general relativity, and is also useful for understanding algebraic and arithmetic geometry.

Parts of Differentiable manifolds 1 (it is not strictly necessary that you have followed that course, but it would be very helpful) and basic topology

Homework exercises (1/3) and oral exam (2/3)

Course notes available via Blackboard