Abstract
Even though it is possible to prove that each Lie group, up to covering, is isomorphic to a linear Lie group of the type discussed in Part I, the natural setting for Lie groups is the category of smooth manifolds in which Lie groups can be viewed as the group objects. Thus we will use linear Lie groups rather as a source of examples and will start building the theory of Lie groups from scratch in Chapter 9 by defining them as groups which are smooth manifolds for which the group operations are smooth.
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- 1.
For each manifold M the identity \(\mathop {\rm id}\nolimits _{M} \colon M \to M\) is a smooth map, so that this lemma leads to the “category of smooth manifolds”. The objects of this category are smooth manifolds and the morphisms are the smooth maps. In the following, we shall use consistently category theoretical language, but we shall not go into the formal details of category theory.
- 2.
Note that the assumption that M be immersed is not redundant (see [KM97], §27.11).
References
Kriegl, A., and P. W. Michor, “The Convenient Setting of Global Analysis”. AMS Surveys and Monographs 53, AMS, Providence, 1997
Lang, S., “Fundamentals of Differential Geometry”, Grad. Texts Math. 191, Springer-Verlag, Berlin, 1999
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Hilgert, J., Neeb, KH. (2012). Smooth Manifolds. In: Structure and Geometry of Lie Groups. Springer Monographs in Mathematics. Springer, New York, NY. https://doi.org/10.1007/978-0-387-84794-8_8
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DOI: https://doi.org/10.1007/978-0-387-84794-8_8
Publisher Name: Springer, New York, NY
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