TITLE: Restricting the Courant Bracket
QUESTION [2 upvotes]: Let $M$ be a manifold and $E$ be a subbundle of $\mathbb{T}M:= TM\oplus T^{*}M$ such that $\Gamma(E)$ is closed under the Courant bracket. The Courant bracket $[[\cdot,\cdot]]$ is an operation on $\Gamma(\mathbb{T}M)$ given by
$$
[[X\oplus\alpha,Y\oplus\beta]] = [X,Y]\oplus(L_{X}\alpha - i_{Y}d\beta),
$$
where $L_{X}, i_{Y}, d$ are the usual Lie derivative, insertion and exterior differential, respectively.
Q: Suppose that $\sigma,\tau\in\Gamma(\mathbb{T}M)$ have the following property: there exists a (regular,embedded) submanifold $N$ of $M$ such that $\sigma_{p},\tau_{p}\in E_{p}$ $\forall p\in N$.
Is it true that $[[\sigma,\tau]]_{p}\in E_{p}$ $\forall p\in N$?
This property holds, for example, in the case of vector fields and their Lie bracket: if two vector fields are tangent to a submanifold, then their Lie bracket is also tangent to it.
My guess is that this is true since the Courant bracket satisfy the folllwing property:
$$
[[\sigma,f\tau]] = f[[\sigma,\tau]] + L_{\mathrm{pr}_{TM}(\sigma)}f\cdot\tau,
$$
where $\mathrm{pr}_{TM}:\mathbb{T}M\rightarrow TM$ is the canonical projection $X\oplus\alpha\mapsto X$.

REPLY [1 votes]: It seems that it is not necessarily true...
Take a basis of (local) sections $\{e_{1},\ldots,e_{k}\}$ of $E$ and complete it to a basis $\{e_{1},\ldots,e_{k+l}\}$ of $\mathbb{T}M$. Then, every $\sigma\in\Gamma(\mathbb{T}M)$ such that $\sigma_{p}\in E_{p}$ for all $p\in N$ is of the form
$$
\sigma = \sum_{i=1}^{k+l} f^{i}e_{i},
$$
where $f^{i}(p)=0$ $\forall p\in N$ and $i>k$.
Assume that $E$ has the property as stated in the question. Then, $[[e_{j},\sigma]]_{p}\in E_{p}$ for all $p\in N$ and $j\leq k$. By $\mathbb{R}$-linearity and applying the property of the Courant bracket presented in the question, we get
\begin{align}
[[e_{j},\sigma]]&=\sum_{i=1}^{k+l}f^{i}[[e_{j},e_{i}]]+L_{a(e_{j})}f^{i}\cdot e_{i}\\
&= \sum_{i=1}^{k}f^{i}[[e_{j},e_{i}]]+L_{a(e_{j})}f^{i}\cdot e_{i} + \sum_{i=k+1}^{k+l}f^{i}[[e_{j},e_{i}]]+L_{a(e_{j})}f^{i}\cdot e_{i}.
\end{align}
Since $j\leq k$ and $e_{i}\in\Gamma(E)$ $\forall i\leq k$, we have
$$
\sum_{i=1}^{k}f^{i}[[e_{j},e_{i}]]+L_{a(e_{j})}f^{i}\cdot e_{i}\in\Gamma(E).
$$
On the other hand, for $i>k$ and $p\in N$, we have $f^{i}(p)[[e_{j},e_{i}]]_{p}=0$. Furthermore, since $\{e_{i}\}_{i=1}^{k+l}$ is a basis such that it extends a basis of $\Gamma(E)$, it follows that $L_{a(e_{j})}f^{i}(p)$ must be equal to zero for all $p\in N$. Since $\sigma$ is arbitrary, then $f^{i}$ is arbitrary. Hence, we conclude that $e_{j}$ has the following property: For all $f\in C^{\infty}(M)$ such that $f|_{N}=0$, we have $L_{a(e_j)}f|_{N}=0$.
In other words, $a(e_{j})$ must be a vector field in $M$ which is tangent to the submanifold $N$ for all $j\leq k$.
Conclusion: A necessary condition to have the property of the question is: For every $p\in N$ and $\sigma\in E_{p}$, we must have $a(\sigma)\in T_{p}N$.
For sufficiency, we must ask further compatibility conditions on $E$ and $N$, because of the non-skew-symmetry of the Courant bracket.
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