subspace $L_x \subset F_x$ one has $$\begin{aligned}
\label{eq:def-sec-exp}
      \vert \det (DX_t \vert_{L_x})\vert > C e^{{\lambda}t},
      \forall t>0.
    \end{aligned}$$

In [@Salg2016], this author give another definition of singular hyperbolicity encompassing two previous as follow.

\[new-def-singhyp\] A compact invariant set $\Lambda \subset M$ is *$p$-sectional hyperbolic or singular hyperbolic of order $p$* for $X$ if all singularities in $\Lambda$ are hyperbolic, there exists a partially hyperbolic splitting of the tangent bundle on $T_{\Lambda}M = E \oplus F$ and constants $C,\lambda > 0$ such that for every $x
  \in \Lambda$ and every $t>0$ we have

1.  $\Vert DX_t\vert_{E_x} \Vert \leq C e^{- \lambda t}$;

2.  $\vert \wedge^p DX_t \vert_{L_x}\vert > C^{-1} e^{{\lambda}t}$, for every $p$-dimensional linear subspace $L_x \subset
      F_x$.

In our applications here, we only deal with two dimensional singular hyperbolic case, but we conjecture that analogous results hold for singular hyperbolic sets of any order $p$, with $2 \leq p \leq \dim{F}$.

From now on, we consider $M$ a connected compact finite dimensional riemannian manifold and all singularities of $X$ (if they exist) are hyperbolic.

Fields of quadratic forms {#sec:fields-quadrat-forms}
-------------------------

From now, we introduce the quadratic forms and its properties.

Let ${\EuScript{J}}:E_U\to{{\mathbb R}}$ be a continuous family of quadratic forms 