流形定向的拓扑定义.pdf

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Wiki: Orientability
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For orientation of vector spaces, see orientation (mathematics). Quality
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For other uses, see orientation.
In mathematics, orientability is a property of surfaces in Euclidean
space measuring whether or not it is possible to make a consistent choice
of surface normal vector at every point. A choice of surface normal
allows one to use the right-hand rule to define a "clockwise" direction of
loops in the surface, as needed by Stokes' theorem for instance. More
generally, orientability of an abstract surface, or manifold, measures
whether one can consistently choose a "clockwise" orientation for all
loops in the manifold. Equivalently, a surface is orientable if a two-
dimensional figure such as in the space cannot be moved
(continuously) around the space and back to where it started so that it
looks like its own mirror image .


The torus is an orientable surface
The notion of orientability can be generalized to higher dimensional
manifolds as well. A manifold is orientable if it has a consistent choice
of orientation, and a connected orientable manifold has exactly two
different possible orientations. In this setting, various equivalent
formulations of orientability can be given, depending on the desired
application and level of gen