Appendix: Submanifolds
2000
This appendix collects some basic definitions and results on submanifolds of finitedimensional spaces in a form that is readily applicable for computations. For proofs we refer, e.g., to Abraham, Marsden, and Ratiu [2]. Throughout this appendix, the dimension n of the ambient space is given, d is an integer with , and ρ denotes a positive integer or ∞. Recall that, when is of class on an open subset , then G is called an immersion or submersion at a point if its first derivative is a one to one mapping or a mapping onto , respectively. More generally, G is an immersion or submersion on a subset S ⊂ E if it has that property at each point of S. Note that these definitions require for G to be an immersion and for it to be a submersion. Clearly, if and has maximal rank m, then G is a submersion at . Definition A.1. A subset is a d-dimensional -submanifold of if ℳ. is nonempty and for every point there exists an open neighborhood of in and a submersion of class such that . The following result is frequently used. Theorem A.1 (Submersion Theorem). Suppose that for the mapping , , on the open set E, the set is not empty and G is a submersion on ℳ. Then ℳ. is an (n −; m)-dimensional -submanifold of . An essential property of manifolds is the concept of a local parametrization. Definition A.2. Let ℳ be a nonempty subset of . A local d-dimensional parametrization of ℳ. is a pair where is a nonempty open subset and is a mapping of class such that (i) is an open subset of ℳ (under the topology induced by the standard topology of ) and φ is a homeomorphism of onto . (ii) φ is an immersion on . If and is a local d-dimensional parametrization of ℳ such that , then is called a local d-dimensional parametrization of ℳ near .