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Manifolds and Vector Bundles
We are now ready to study manifolds and the differential calculus of maps between manifolds. Manifolds
are an abstraction of the idea of a smooth surface in Euclidean space. This abstraction has proved useful
because many sets that are smooth in some sense are not presented to us as subsets of Euclidean space. The
abstraction strips away the containing space and makes constructions intrinsic to the manifold itself. This
point of view is well worth the geometric insight it provides.
Manifolds
Charts and Atlases. The basic idea of a manifold is to introduce a local object that will support
differentiation processes and then to patch these local objects together smoothly. Before giving the formal
definitions it is good to have an example in mind. In Rn+1 consider the n-sphere Sn; that is, the set of
x ∈ Rn+1 such that x =1(· denotes the usual Euclidean norm). We can construct bijections from
subsets of Sn to Rn in several ways. One way is to project stereographically from the south pole onto a
hyperplane tangent to the north pole. This is a bijection from Sn, with the south pole removed, onto Rn.
Similarly, we can interchange the roles of the poles to obtain another bijection. (See Figure .)
With the usual relative topology on Sn as a subset of Rn+1, these maps are homeomorphisms from their
domain to Rn. Each map takes the sphere minus the two poles to an open subset of Rn. If we go from Rn
to the sphere by one map, then back to Rn by the other, we get a smooth map from an open subset of Rn
to Rn. Each map assigns a coordinate system to Sn minus a pole. The union of the two domains is Sn, but
no single homeomorphism can be used between Sn and Rn; however, we can cover Sn using two of them.
In this case they patible; that is, in the region covered by both coordinate systems, the change of
coordinates is smooth. For some studies of the sphere, and for other manifolds, two coordinate systems will
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