L9a. The definition of a Courant algebroid.

We defined the Dorfman bracket in L7a. We call it bracket although it is not skew-symmetric. One may want to have a proper skew-symmetric bracket. To do so, we just have to skew-symmetrize: $$[[ v, w ]] = \frac{1}{2}( [v,w] - [w,v] ),$$   in order to get the Courant bracket. They are indeed closely related: $$[v,w] = [[v,w]] + D\langle v,w\rangle$$

In Problem Sheet 3 we have checked many properties that the Dorfman and the Courant bracket satisfy. They all fit in an abstract object called Courant algebroid. Let us give two definitions of this object, one using a non-skew-symmetric but well behaved Dorfman bracket, and the other one using a skew-symmetric Courant bracket.

A Courant algebroid $(E,\langle\cdot,\cdot\rangle,[\cdot,\cdot],\pi)$ over a manifold $M$ consists of a vector bundle $E\to M$ together with a non-degenerate symmetric bilinear form $\langle\cdot,\cdot\rangle$ on $E$, a DORFMAN bracket $[\cdot,\cdot]$ on the sections $\mathcal{C}^\infty(E)$ and a bundle map $\pi:E\to TM$ such that the following properties are satisfied:
 (D1): $[v,[w,w']]=[[v,w],w'] + [w,[v,w']],$
(D2): $\pi([v,w])=[\pi(v),\pi(w)]$,
(D3): $[v,fw]=f[v,w]+(\pi(v)f)w$,
(D4): $\pi(v)\langle w,w'\rangle = \langle [v,w], w' \rangle + \langle w, [v,w'] ,w'\rangle$,
(D5): $[v,v]=\pi^* d\langle v, v\rangle$,
  for any $v,w,w' \in \Gamma(E)$, $f,g\in \mathcal{C}^\infty(M)$, where $D: \mathcal{C}^\infty(M)\to \mathcal{C}^\infty(E)$ is defined by $$\langle D f, v \rangle=\frac{1}{2}\pi(v)(f).$$

A Courant algebroid $(E,\langle\cdot,\cdot\rangle,[[\cdot,\cdot]],\pi)$ over a manifold $M$ consists of a vector bundle $E\to M$ together with a non-degenerate symmetric bilinear form $\langle\cdot,\cdot\rangle$ on $E$, a skew-symmetric COURANT bracket $[[\cdot,\cdot]]$ on the sections $\mathcal{C}^\infty(E)$ and a bundle map $\pi:E\to TM$ such that the following properties are satisfied:
 (C1): (sorry about how this looks with the double bracket) $[[v,[[w,w']]]]=[[[[v,w]],w']] + [[w,[[v,w']]]] - \frac{1}{3} D( \langle [[v,w]],w'\rangle + \langle [[w,w']], v \rangle + \langle [[w',v]],w \rangle),$
(C2): $\pi([[v,w]])=[[\pi(v),\pi(w)]]$,
(C3): $[[v,fw]]=f[[v,w]]+(\pi(v)f)w - \langle v,w \rangle D f$,
(C4): $\pi(v)\langle w,w'\rangle = \langle [[v,w]]+D\langle v,w\rangle, w' \rangle + \langle w, [[v,w']] + D\langle v,w'\rangle \rangle$,
(C5): $\pi\circ D =0$, and consequently, $\langle D f, D g \rangle = 0$,
  for any $v,w,w' \in \Gamma(E)$, $f,g\in \mathcal{C}^\infty(M)$, where $D: \mathcal{C}^\infty(M)\to \mathcal{C}^\infty(E)$ is defined by $$\langle D f, v \rangle=\frac{1}{2}\pi(v)(f).$$