Unformatted text preview: a topological manifold M is a collection F of local coordinate systems on M such that: 1. The union of the domains of the local coordinate systems is all of M . 2. If 1 and 2 are in F , then 1 and 2 are C compatible.
15 3. F is maximal with respect to 2. That is, if is C compatible with all F , then F . A C differentiable manifold of dimension n is a topological manifold M of dimension n, together with a C differentiable structure F on M . Notes: 1. It is possible for a topological manifold to have more than one distinct differentiable structures. 2. In this discussion, we have limited ourselves to C differentiable structures. With somewhat more work, we could define C k structures for k < . 3. We have limited the domains of our local coordinate systems to be open
16 subsets of M. This means that the usual spherical and cylindrical coordinate systems on R3 do not count as local coordinate systems by our definition. 4. With somewhat more work, we could define differentiable manifolds with boundaries. 5. We have limited...
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