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A Survey on Regularity Theorems in Differentiable Compact Transformation Groups

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Proceedings of the Conference on Transformation Groups

Abstract

In differentiable transformation groups, one mainly studies the geometric behavior of differentiable actions of compact Lie groups on certain given types of manifolds. Since the existence of some interesting differentiable action on a smooth manifold M is itself a non-trivial fact which is not universally enjoyed by general manifolds (i.e., not every manifold has interesting symmetries), it seems rather logical to consider firstly smooth actions on those manifolds with sufficiently rich natural symmetries, such as euclidean spaces, spheres, homogeneous spaces, etc. For manifolds with “natural actions”, a simple-minded but rather fruitful approach is to compare the behavoir of general differentiable actions with the behavior of “natural actions”. From the above viewpoint, it is quite fair to say that euclidean spaces, spheres and discs are among the best testing spaces for the study of differentiable transformation groups. For these testing spaces, linear actions are clearly the “natural actions” and the comparisons between general differentiable actions and linear actions consist of the following two complementary efforts. Namely, one tries to prove more and more resemblances between differentiable actions and linear actions on the one hand, and on the other hand one tries to construct more and more varieties of differentiable examples to see how differentiable actions may differ from linear actions.

Dedicated to Professor Deane Montgomery.

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  1. Wu-Yi Hsiang
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  1. Department of Mathematics, Tulane University, New Orleans, Louisiana, USA

    Paul S. Mostert

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Hsiang, WY. (1968). A Survey on Regularity Theorems in Differentiable Compact Transformation Groups. In: Mostert, P.S. (eds) Proceedings of the Conference on Transformation Groups. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-46141-5_3

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  • DOI: https://doi.org/10.1007/978-3-642-46141-5_3

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