Fix a compact connected smooth manifold M. Denote by $PH(M)$ the set of Riemannian metrics $g$ of $M$ such that the geodesic flow $g_t$, given by
$$ g^t:T^1M\rightarrow T^1M$$
$$v\mapsto \gamma_v'(t),$$ where $\gamma_v(t)$ is the only geodesic with $\gamma_v(0)=\pi(v)$ and $\gamma_v'(0)=v$, is partially hyperbolic. By partially hyperbolic I mean that there exist constants $c>0$, $0<\lambda<1$ and a continuous non-trivial $Dg^t-$invariant splitting of $TT^1M=E^s\oplus <X> \oplus E^c\oplus E^u$ such that
- $\|D_x g^t\cdot v\|\leq c\lambda^t\|v\|$, for $v\in E^s$ and all $t>0$;
2)$\|D_x g^t\cdot v\|\leq c\lambda^{t}\|v\|$, for $v\in E^u$ and all $t>0$;
- $c\lambda^t\|v\|\leq\|D_xg^t\cdot v\|\leq c\lambda^{-t}\|v\|$, for $v\in E^c$ and all $t>0$.
For instance, all Anosov flows are partially hyperbolic flows, so $PH(M)$ contains all negatively curved metrics.
Questions: 1) Given two metrics $g_0, g_1\in PH(M)$ is there a continuous path $\alpha:[0,1]\rightarrow PH(M)$ such that $\alpha(0)=g_0$ and $\alpha(1)=g_1$?
- More specifically we could ask $PH(M)$ is convex, i.e., the path $\alpha(t)=(1-t)g_0+tg_1$ is fully contained in $PH(M)$.
I believe the answer may be positive because it seems to me that we can not break the partially hyperbolic structure in a continuous way.