Abstract
In many areas of mathematics, Lie groups appear naturally as symmetry groups. Examples are groups of isometries of Riemannian manifolds, groups of holomorphic automorphisms of complex domains, or groups of canonical transformations in hamiltonian mechanics. In all these cases, one considers group actions on manifolds by smooth maps. Even though the focus of this book is the geometry and structure theory of Lie groups rather than their applications, we have to study the concept of a smooth action of a Lie group on a manifold in some detail since it is an essential tool in the smooth versions of group theoretic considerations like the study of quotient groups and conjugacy classes.
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- 1.
Christian Felix Klein (1849–1925) held the chair of geometry in Erlangen for a few years and the “Erlanger Programm” was his “Programmschrift”, where he formulated his research plans when he came to Erlangen. Later he was a professor for mathematics in Munich, Leipzig, and eventually in Göttingen.
- 2.
A typical example is the group \(\mathop {\rm Mot}\nolimits (E_{2})\) of motions (orientation preserving isometries) of the euclidian plane. The length of an interval or the area of a triangle are properties preserved by this group, hence geometric quantities. It was an important conceptual step to observe that changing the group means changing the geometry, resp., the notion of a geometric quantity. For example, the automorphism group \(\mathop {\rm Aff}\nolimits (A_{2})\) of the two-dimensional affine plane A 2 does not preserve the area of a triangle (it is larger than the euclidian group), hence the area of a triangle cannot be considered as an affine geometric quantity.
References
Duistermaat, J. J., and J. A. C. Kolk, “Lie groups”. Springer, Berlin, 2000
Greub, W., Halperin, S., and R. Vanstone, “Connections, Curvature, and Cohomology,” Academic Press, New York and London, 1973
Kobayashi, S., “Transformation Groups in Differential Geometry”. Springer, Berlin, 1995
Rudin, W., “Real and Complex Analysis,” McGraw Hill, New York, 1986
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Hilgert, J., Neeb, KH. (2012). Smooth Actions of Lie Groups. 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_10
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DOI: https://doi.org/10.1007/978-0-387-84794-8_10
Publisher Name: Springer, New York, NY
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