1549055384-Symplectic_Geometry_and_Topology__Eliashberg_

LECTURE l. REDUCTION THEORY 345

developed symplectic reduction; related results, but with a different motivation and

construction were found by Meyer [1973]. The construction is now well known: let

(P, D) be a symplectic manifold and l : P ----+ g* be an equivariant momentum map;

then avoiding singularities, 1-^1 (μ) / G μ = Pμ is a symplectic manifold in a natural

way. For example, for P = T*G, one gets coadjoint orbits.

Kazhdan, Kostant and Sternberg [1978] showed how Pμ can be realized in

terms of orbit reduction Pμ S=! 1-^1 (0)/G and from this it follows that Pμ are the

symplectic leaves in P/G. This paper was also one of the first to notice deep links
between reduction and integrable systems, a subject continued by, for example,
Bobenko, Reyman and Semenov-Tian-Shansky [1989].
The way in which the Poisson structure on Pμ is related to that on P/G was
clarified in a generalization of Poisson reduction due to Marsden and Ratiu [1986],
a technique that has also proven useful in integrable systems (see, for example,
Pedroni [1995] and Vanhaecke [1996]).

The mechanical connection. A basic construction implicit in Smale [1970],

Abraham and Marsden [1978] and explicit in Kummer [1981] is the notion of the
mechanical connection. The geometry of this situation was used to great effect in
Guichardet [1984] and Iwai [1987, 1990].
Assume Q is Riemannian (the metric often being the kinetic energy metric)
and that G acts on Q freely by isometries, so 7f : Q ----+ Q/G is a principal bundle.

If we declare the horizontal spaces to be metric orthogonal to the group orbits,

this uniquely defines a connection called the mechanical connection. There
are explicit formulas for it in terms of the locked inertia tensor; see for instance,
Marsden [1992] for details. The space Q/G is called shape space and plays a
critical role in the theory.^4

Tangent and cotangent bundle reduction. The simplest case of cotangent
bundle reduction is reduction at zero in which case one has (TQ)μ=O = T(Q/G),
the latter with the canonical symplectic form. Another basic case is when G is
abelian. Here, (TQ)μ S=! T(Q/G) but the latter has a symplectic structure modi-
fied by magnetic terms; that is, by the curvature of the mechanical connection.
The Abelian version of cotangent bundle reduction was developed by Smale
[1970] and Satzer [1975] and was generalized to the nonabelian case in Abraham

and Marsden [1978]. It was Kummer [1981] who introduced the interpretations of

these results in terms of the mechanical connection.
The Lagrangian analogue of cotangent bundle reduction is called Routh re-
duction and was developed by Marsden and Scheurle [1993a,b]. Routh, around
1860 investigated what we would call today the Abelian version.
The "bundle picture" begun by the developments of the cotangent bundle re-
duction theory was significantly developed by Montgomery, Marsden and Ratiu
[1984] and Montgomery [1986] motivated by work of Weinstein and Sternberg on
Wong's equations (the equations for a particle moving in a Yang-Mills field).
This bundle picture can be viewed as follows. Choosing a connection, such as
the mechanical connection, on Q ----+ Q / G , one gets a natural isomorphism

TQ/G S=! T (Q/G) EB g*

(^4) Shape space and its geometry plays a key role in computer vision. See for example, Le and
Kendall [1993].