A general theory of canonical forms

Richard S. Palais, Chuu-Lian Terng
1987 Transactions of the American Mathematical Society  
If G is a compact Lie group and M a Riemannian G-manifold with principal orbits of codimension k then a section or canonical form for M is a closed, smooth k-dimensional submanifold of M which meets all orbits of M orthogonally. We discuss some of the remarkable properties of G-manifolds that admit sections, develop methods for constructing sections, and consider several applications. O. Introduction. Let G be a compact Lie group acting isometrically on a Riemannian manifold M. Then the image S
more » ... of a small ball in the normal plane v{Gx)x under the exponential map is a smooth, local Gx-slice, which in general cannot be extended to a global slice for M. A section E for M is defined to be a closed, smooth submanifold of M which meets every orbit of M orthogonally. A good example to keep in mind is perhaps the most important of all canonical form theorems; namely for M we take the Euclidean space of symmetric k x k matrices with inner product (A, B) = tr(AB), and for G the orthogonal group O{k) acting on M by conjugation. Then the space E of diagonal matrices is a section. Moreover the symmetric group Sk acts on E by permuting the diagonal entries and the orbit spaces M/G and E/Sk are isomorphic as stratified sets. Quite generally it is good intuition to think of a section E as representing a "canonical form" for elements of M; hence our title. Riemannian G-manifolds which admit sections are definitely the exception rather than the rule and they have many remarkable properties. The existence of sections for M has important consequences for the invariant function theory, submanifold geometry, and G-invariant variational problems associated to M. While we do not know of earlier papers treating sections in generality, we have found several which treat important special cases. In particular when we showed G. Schwarz an early version of our results he pointed out to us a preprint of an important paper [Da2] by J. Dadok in which a detailed study is made (including a complete classification theorem) of orthogonal representations of compact connected Lie groups which admits sections (Dadok calls these polar representations). Later still we discovered two very interesting and much earlier papers by L. Conlon [Col, Co2] in which he considers Riemannian G-manifolds which admit fiat, totally geodesic sections. This includes the case of polar representations, and Conlon came close to conjecturing Dadok's classification result. We will discuss in more detail later the results in these papers and how they relate to our own. We would like to thank Dadok for a number of helpful comments. It is clear not only that he
doi:10.1090/s0002-9947-1987-0876478-4 fatcat:3yyq36gnbfg7lonmgag2d55zry