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# Introduction to manifolds

Vak
2011-2012

This is a course on DIFFERENTIABLE MANIFOLDS, the abstract generalization of smooth curves and surfaces in euclidean space. Such a manifold has a topology and a certain dimension n, and locally it is homeomorphic with a piece of n-dimensional euclidean space, such that these pieces (in general in a non-trivial way) “are differentiably glued together”. Examples are the n-dimensional sphere and projective space, and n-dimensional smooth quadratic hypersurfaces in (n+1)-dimensional euclidean space; the latter are the n-dimensional analogues of the smooth quadratic curves and surfaces treated in linear algebra 2.
In particular, it is possible to define differentiable functions on and differentiable maps between differentiable manifolds.
A crucial ingredient in this theory is the TANGENT SPACE of a manifold at a point, the generalization of the tangent line resp. tangent plane of a curve resp. surface in euclidean space; this concept enables one to define a kind of Jacobian of a smooth map between manifolds, and to generalize the Inverse and Implicit Function Theorem.
Using the dual of the tangent space we further introduce DIFFERENTIAL FORMS which are used for integration on manifolds. Here the aim is to prove the THEOREM OF STOKES, the general version of the classical integral theorems of Gauss and Greene.

Prerequisites
Differentiation and integration of functions of several real variables; Topology
Literature