Tangent Dirac structures of higher order. (English) Zbl 1240.53058

Kouotchop Wamba, P. M.; Ntyam, A.; Wouafo Kamga, J.

Arch. Math., Brno 47, No. 1, 17-22 (2011).

An almost Dirac structure on the \(n\) dimensional smooth manifold \(M\) is the rank \(n\) vector subbundle \(L \subseteq TM \oplus T^*M\) which is isotrophic with respect to the natural (neutral signature) scalar product on \(TM \oplus T^{*}M\). The Courant bracket \([\dot ,\dot ]_C\) defined by \([(X_1,\alpha _1),(X_2,\alpha _2)] = ([X_1,X_2], L_{X_1} \alpha _1 - \iota _{X_2} d \alpha _1)\) for \((X_i,\alpha _i)\in \Gamma (TM \oplus T^{*}M)\) generalizes the Lie bracket on \([,]\) on \(M\). The Dirac structure is an almost Dirac structure \(L\) which is closed under \([\dot ,\dot ]_C\) (the “integrability condition”).
In [J. Phys. A, Math. Gen. 23, No. 22, 5153–5168 (1990; Zbl 0715.58013)] Ted Courant defined the tangent lift \(L^1\) of \(L\) on the manifold \(TM\) and proves this lift is integrable if \(L\) is integrable. In the present work, authors define the lift \(L^r\) of \(L\) on \(T^rM\) and, as the main result, show that \(L\) is integrable if and only if \(L^r\) is integrable.

Reviewer: Josef Šilhan (Brno)

MSC:

53C15	General geometric structures on manifolds (almost complex, almost product structures, etc.)
53C75	Geometric orders, order geometry
53D05	Symplectic manifolds (general theory)

Keywords:

Dirac structure; almost Dirac structure; tangent functor of higher order; natural transformation

Citations:

Zbl 0715.58013