By Bennett Chow; Peng Lu; and Lei Ni

ISBN-10: 0821842315

ISBN-13: 9780821842317

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Iµ ] ΩO = i∗µ Ω. Since [iµ ] ◦ πµ = πO ◦ iµ , this says that i∗µ πO By (ii), we have − − ∗ ΩO = i∗µ (i∗O Ω + J∗O ωO ) = (iO ◦ iµ )∗ Ω + (JO ◦ iµ )∗ ωO = i∗µ Ω i∗µ πO since iO ◦ iµ = iµ and JO ◦ iµ = µ on J−1 (µ). Remarks. 1. A similar result holds for right actions. 2. In this proof the freeness and properness of the Gµ -action on J−1 (µ) were only used indirectly. In fact these conditions are suﬃcient but not necessary for Pµ to be a manifold. All that is needed is for Pµ to be a manifold and πµ to be a submersion and the above proof remains unchanged.

Poincar´e realized that the equations of ﬂuids, free rigid bodies, and heavy tops could all be described in Lie algebraic terms in a beautiful way. The importance of these equations was realized by Hamel [1904, 1949] and Chetayev [1941], but to a large extent, the work of Poincar´e lay dormant until it was revived in the Russian literature in the 1980’s. 3 Reduction Theory: Historical Overview 31 The more recent developments of Lagrangian reduction were motivated by attempts to understand the relation between reduction, variational principles and Clebsch variables in Cendra and Marsden [1987] and Cendra, Ibort, and Marsden [1987].

Both stratiﬁcations are cone spaces. The stratiﬁcation of M/G is minimal among all Whitney stratiﬁcations of M/G. This statement is the so called Stratiﬁcation Theorem. Later we shall need the following result for a proper Lie group action. If M/G is connected, there is a unique maximal conjugacy class (H) ∈ I(G, M ), that is, (Gx ) (H) for all x ∈ M . Its associated orbit type manifold M(H) is open and dense in M and the orbit space M(H) is connected. The Symplectic Stratiﬁcation Theorem. With these preparations in mind we can state now the precise statement of the Symplectic Stratiﬁcation Theorem.

### Hamilton’s Ricci Flow by Bennett Chow; Peng Lu; and Lei Ni

by Ronald

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