1. Differentiable Manifolds 3 From the following drawing in the 2-plane through 0, x, and a it is easily seen that u+ is the usual stereographic projection. We also get v--+\y) = ( j ^ a + ^ y for y € {a}1" \ {0} and (u- o u+ )(y) = T\. The latter equation can directly be seen from the drawing using the intercept theorem. 1.3. Smooth mappings. A mapping f : M — N between manifolds is said to be Ck if for each x G M and one (equivalently: any) chart (V, v) on N with f(x) G V there is a chart (U,u) on M with x G U, f(U) C V, and v o f o u~x is Ck. We will denote by Ck(M, N) the space of all C^-mappings from M to N. A Cfc-mapping / : M — N is called a Ck-diffeomorphism if / _ 1 : N ^ M exists and is also C'0. Two manifolds are called diffeomorphic if there exists a diffeomorphism between them. From differential topology (see [84]) we know that if there is a C1-diffeomorphism between M and N1 then there is also a C°°-diffeomorphism. There are manifolds which are homeomorphic but not diffeomorphic: On M4 there are uncountably many pairwise nondiffeomorphic differentiable struc- tures on every other Rn the differentiable structure is unique. There are finitely many different differentiable structures on the spheres Sn for n 7. A mapping f : M — N between manifolds of the same dimension is called a local diffeomorphism if each x G M has an open neighborhood U such that f\U:U—* f(U) C N is a diffeomorphism. Note that a local diffeomorphism need not be surjective.

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