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2012 | OriginalPaper | Chapter
21. Quotient Manifolds
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Abstract
In Chapter 4, we studied surjective smooth submersions, and emphasized their relationships with quotient maps. But one question we did not address there was which quotients of smooth manifolds are themselves smooth manifolds. The quotients that are easiest to analyze are those resulting from smooth Lie group actions. The main theorem of the chapter is the quotient manifold theorem, which asserts that a smooth Lie group action yields a quotient space with a natural smooth manifold structure provided that it is free (meaning that the group acts without fixed points) and proper (which means roughly that each compact subset is moved away from itself by most elements of the group). After proving the theorem, we apply it to the study of actions by discrete groups, which under suitable conditions yield covering maps, and homogeneous spaces, which are smooth manifolds endowed with smooth transitive Lie group actions. At the end of the chapter we describe a number of applications to the theory of Lie groups.