An -dimensional topological manifold is a second countable, Hausdorff topological space11For connected manifolds, the assumption that is second-countable is logically equivalent to being paracompact, or equivalently to being metrizable. The topological hypotheses in the definition of a manifold are needed to exclude certain counter-intuitive pathologies. Standard illustrations of these pathologies are given by the long line (lack of paracompactness) and the forked line (points cannot be separated). These pathologies are fully described in Spivak. See this page (http://planetmath.org/BibliographyForDifferentialGeometry). that is locally homeomorphic to open subsets of .
A differential manifold is a topological manifold with some additional structure information. A chart, also known as a system of local coordinates, is a mapping , such that the domain is an open set, and such that is homeomorphic to the image . Let and be two charts with overlapping domains (http://planetmath.org/Function). The continuous injection
Note that each transition function is really just real-valued functions of real variables, and so we can ask whether these are continuously differentiable. The atlas defines a differential structure on , if every transition function is continuously differentiable.
More generally, for , the atlas is said to define a differential structure, and is said to be of class , if all the transition functions are -times continuously differentiable, or real analytic in the case of . Two differential structures of class on are said to be isomorphic if the union of the corresponding atlases is also a atlas, i.e. if all the new transition functions arising from the merger of the two atlases remain of class . More generally, two manifolds and are said to be diffeomorphic, i.e. have equivalent differential structure, if there exists a homeomorphism such that the atlas of is equivalent to the atlas obtained as -pullbacks of charts on .
The atlas allows us to define differentiable mappings to and from a manifold. Let
be a continuous function. For each we define
called the representation of relative to chart , as the suitably restricted composition
We judge to be differentiable if all the representations are differentiable. A path
is judged to be differentiable, if for all differentiable functions , the suitably restricted composition is a differentiable function from to . Finally, given manifolds , we judge a continuous mapping between them to be differentiable if for all differentiable functions on , the suitably restricted composition is a differentiable function on .
|Date of creation||2013-03-22 12:20:22|
|Last modified on||2013-03-22 12:20:22|
|Last modified by||matte (1858)|
|Defines||change of coordinates|