also let ${\EuScript{C}}_0=\{C_0(x)\}_{x\in U}$ be the corresponding family of zero vectors $C_0(x)={\EuScript{J}}_x^{-1}(\{0\})$ for all $x\in U$. In the adapted coordinates obtained above we have $$\begin{aligned}
  C_0(x)=\{v=\sum_{i}\alpha_iv_i\in E_x :
  \sum_{j=q+1}^n \alpha_j^2 = \sum_{i=1}^q
  \alpha_i^2\}\end{aligned}$$ is the set of *extreme points* of $C_\pm(x)$.

The following definitions are fundamental to state our main result.

\[def:J-separated\] Given a continuous field of non-degenerate quadratic forms ${\EuScript{J}}$ with constant index on the trapping region $U$ for the flow $X_t$, we say that the flow is

-   ${\EuScript{J}}$-*separated* if $DX_t(x)(C_+(x))\subset
      C_+(X_t(x))$, for all $t>0$ and $x\in U$;

-   *strictly ${\EuScript{J}}$-separated* if $DX_t(x)(C_+(x)\cup
      C_0(x))\subset C_+(X_t(x))$, for all $t>0$ and $x\in U$;

-   ${\EuScript{J}}$-*monotone* if ${\EuScript{J}}_{X_t(x)}(DX_t(x)v)\ge {\EuScript{J}}_x(v)$, for each $v\in
      T_xM\setminus\{0\}$ and $t>0$;

-   *strictly ${\EuScript{J}}$-monotone* if $\partial_t\big({\EuScript{J}}_{X_t(x)}(DX_t(x)v)\big)\mid_{t=0}>0$, for all $v\in T_xM\setminus\{0\}$, $t>0$ and $x\in U$;

-   ${\EuScript{J}}$-*isometry* if ${\EuScript{J}}_{X_t(x)}(DX_t(x)v) = {\EuScript{J}}_x(v)$, for each $v\in T_xM$ and $x\in U$.

Thus, ${\EuScript{J}}$-separation corresponds to simple cone invariance and strict ${\EuScript{J}}$-separation corresponds to strict cone invariance under the action of $DX_t(x)$.

\[rmk:J-separated-C-\] If a flow is strictly ${\EuScript{J}}$-separated, then for $v\in T_xM$ such that ${\EuScript{J}}_x(v)\le0$ we have ${\EuScript{J}}_{X_{-t}(x)}(DX_{-t}(v))<0$ for all $t>0$ and $x$ such that $X_{-s}(x)\in U$ for every $s\in[-t,0]$. Indeed, otherwise ${\EuScript{J}}_{X_{-t}(x)}(DX_{-t}(v))\ge0$ would imply ${\EuScript{J}}_x(v)={\EuScript{J}}_x\big(DX_t(DX_{-t}(v))\big)>0$, contradicting the assumption that $v$ 