Let $(M,g)$ be a closed Riemannian manifold and $f:M\to M$ be a $C^r$ ($r\ge2$) Anosov diffeomorphism, that is, there is a continuous hyperbolic splitting $TM=E^s\oplus E^u$ with respect to the Riemannian metric $g$ over $M$.

We know there exists stable and unstable foliations $\mathcal{W}^s$ and $\mathcal{W}^u$ tangent to $E^s$ and $E^u$.

Moreover, all leaves $W^s(x)$ and $W^u(x)$ are $C^r$ submanifolds for every $x\in M$.

So we can talk about the sectional/Ricci/scalar curvatures of these submanifolds (endow with the restricted metrics $g|_{W^s(x)}\text{ and } g|{W^u(x)}$).

I want to know if there are some results and references about this topic.

This should be an interesting question, and some results in decay of correlations do assume the sectional curvatures are bounded.

Maybe there are some results for

more general classes of maps (say partially hyperbolic diffeo's),

more general submanifolds (say foliations with continuous tangent $E=T\mathcal{W}$).

Thank you!

Edit: the relation between the metric $g$ and the hyperbolic splitting $TM=E^s\oplus E^u$.

The Riemannian metric $g$ induces a norm $\|\cdot\|_x$ on each tangent space $T_xM$ by $\|v\|^2_x=g_x(v,v)$ for every $v\in T_xM$.

The map $f:M\to M$ is said to be hyperbolic if there exists a continuous splitting $TM=E^s\oplus E^u$ such that

- the splitting is $Df$-invariant: $D_xf(E^s_x)=E^s_{fx}$ and $D_xf(E^u_x)=E^u_{fx}$,
- $E^s$ is uniformly contracting: $\|D_xf^n(v)\|_{f^nx}\le C\lambda^n_s\|v\|_x$ for every $v\in E^s_x$,
- $E^u$ is uniformly expanding: $\|D_xf^n(v)\|_{f^nx}\ge C^{-1}\lambda^n_u\|v\|_x$ for every $v\in E^u_x$,

for some uniform constant $C\ge1$ and $0<\lambda_s<1<\lambda_u$.

So the hyperbolicity of the map $f$ does depend on the choice of Riemannian metric.

But being hyperbolic is not sensitive to the choice of metric. For example if $f$ is hyperbolic with respect to $(M,g)$ and we have another Riemannian metric $h$ with $C_1\cdot g_x(v,v)\le h_x(v,v)\le C_2\cdot g_x(v,v)$, then $f$ is also hyperbolic with respect to $(M,h)$ (with a possible different $C$).

The upper bound of the curvatures of stable and unstable manifolds (if exists) may depend on the choice of metrics. But the property of having bounded curvature should be independent of the choice of metrics.