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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.

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You can't talk about curvatures until there's at least some kind of connection, and it sounds as though you want there to be some kind of metric on $M$. Are you going to assume that this metric has any properties relative to $f$? Without that, it's unlikely that you'll get any local conditions whatsoever. – Robert Bryant Feb 21 '12 at 15:49
@Robert-Bryant Yes you are right. I do need a Riemannian metric and then the Levi-Civita connection to evaluate the curvature, and the actual bound (if exists) depends on the metric. I will modify the formulation. Thank you! – Pengfei Feb 22 '12 at 0:57
@Pengfei: I do not understand what relation you assume between the map and the metric. This is part of what Robert Bryant was asking. Without such a relation, it is unlikely that much can be said. – Benoît Kloeckner Feb 22 '12 at 6:45
@Benoît Kloeckner Thank you! I have added the definition of hyperbolicity of a map $f$. Hope it get better now :) – Pengfei Feb 24 '12 at 11:06
up vote 4 down vote accepted

Did you look at this reference ? Maybe it can help? (in the case of the geodesic flow on the unit tangent bundle of a negatively curved manifold) Ernst Heintze and Hans-Christoph Im Hof. Geometry of horospheres, Source: J. Differential Geom. Volume 12, Number 4 (1977), 481-491

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Thank you for the reference! – Pengfei Mar 11 '12 at 2:48

Maybe late, but I believe the following reference is relevant. Is from a paper from Bonatti and Viana which works even for partially hyperbolic attractors (of course the $C^2$-hypothesis is crutial):

See Lemma 3.1 here

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