By Bennett Chow; Peng Lu; and Lei Ni
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Additional info for Hamilton’s Ricci Flow
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 , 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  and Cendra, Ibort, and Marsden .
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
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