Dirac manifolds. (English) Zbl 0850.70212

Courant, Theodore James

Trans. Am. Math. Soc. 319, No. 2, 631-661 (1990).

Summary: A Dirac structure on a vector space \(V\) is a subspace of \(V\) with a skew form on it. It is shown that these structures correspond to subspaces of \(V\oplus V^*\) satisfying a maximality condition, and having the property that a certain symmetric form on \(V\oplus V^*\) vanishes when restricted to them. Dirac structures on a vector space are analyzed in terms of bases, and a generalized Cayley transformation is defined which takes a Dirac structure to an element of \(O(V)\). Finally a method is given for passing a Dirac structure on a vector space to a Dirac structure on any subspace.
Dirac structures on vector spaces are generalized to smooth Dirac structures on a manifold \(P\), which are defined to be smooth subbundles of the bundle \(TP\oplus T^*P\) satisfying pointwise the properties of the linear case. If a bundle \(L\subset TP \oplus T^*P\) defines a Dirac structure on \(P\), then we call \(L\) a Dirac bundle over \(P\). A \(3\)-tensor is defined on Dirac bundles whose vanishing is the integrability condition of the Dirac structure. The basic examples of integrable Dirac structures are Poisson and presymplectic manifolds; in these cases the Dirac bundle is the graph of a bundle map, and the integrability tensors are \([B,B]\) and \(d\Omega\), respectively. A function \(f\) on a Dirac manifold is called admissible if there is a vector field \(X\) such that the pair \((X,df)\) is a section of the Dirac bundle \(L\); the pair \((X,df)\) is called an admissible section. The set of admissible functions is shown to be a Poisson algebra.
A process is given for passing Dirac structures to a submanifold \(Q\) of a Dirac manifold \(P\). The induced bracket on admissible functions on \(Q\) is in fact the Dirac bracket as defined by Dirac for constrained submanifolds.

MSC:

53D17	Poisson manifolds; Poisson groupoids and algebroids
53D05	Symplectic manifolds (general theory)
70G45	Differential geometric methods (tensors, connections, symplectic, Poisson, contact, Riemannian, nonholonomic, etc.) for problems in mechanics
70H45	Constrained dynamics, Dirac’s theory of constraints

Keywords:

Poisson manifold; symplectic manifold; Dirac brackets; constrained dvnamics; Lie algebroid

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