${\EuScript{J}}_x:E_x\to{{\mathbb R}}$ which are non-degenerate and have index $0<q<\dim(E)=n$, where $U\subset M$ is an open set such that $X_t(U) \subset \overline{U}, \forall t \geq 0,$ for a vector field $X$. We also assume that $({\EuScript{J}}_x)_{x\in U}$ is continuously differentiable along the flow.

The continuity assumption on ${\EuScript{J}}$ just means that for every continuous section $Z$ of $E_U$ the map $U\to{{\mathbb R}}$ given by $x\mapsto {\EuScript{J}}(Z(x))$ is continuous. The $C^1$ assumption on ${\EuScript{J}}$ along the flow means that the map $x\mapsto
{\EuScript{J}}_{X_t(x)} (Z(X_t(x)))$ is continuously differentiable for all $x\in U$ and each $C^1$ section $Z$ of $E_U$.

The assumption that $M$ is a compact manifold enables us to globally define an inner product in $E$ with respect to which we can find the an orthonormal basis associated to ${\EuScript{J}}_x$ for each $x$, as follows. Fixing an orthonormal basis on $E_x$ we can define the linear operator $$\begin{aligned}
  J_x:E_x\to E_x \quad\text{such that}\quad {\EuScript{J}}_x(v)=<J_x
  v,v> \quad \text{for all}\quad v\in T_xM,\end{aligned}$$ where $<,>=<,>_x$ is the inner product at $E_x$. Since we can always replace $J_x$ by $(J_x+J_x^*)/2$ without changing the last identity, where $J_x^*$ is the adjoint of $J_x$ with respect to $<,>$, we can assume that $J_x$ is self-adjoint without loss of generality. 