Let $f:M\to M$ be a diffeomorphism on a compact riemannian manifold $M$.In the definition of a hyperbolic set we know that for all $x\in M$ there is a splitting of tangent space 


$T\_xM=E^s(x)\oplus E^u(x)$


I want to know what is the definition of angle between $E^s(x)$ and $E^u(x)$?



Angle between two subspaces
I suppose you're referring to statements in hyperbolic dynamics of the type "the angle between $E^s(x)$ and $E^u(x)$ is bounded away from zero".


I haven't seen a formal definition, but I think the typical implicit assumption is that a Riemannian metric is chosen, and under this metric the minimum angle between tangent vectors $v \in E^s(x)$ and $w \in E^u(x)$ defines the angle between $E^s(x)$ and $E^u(x)$.


This depends on the choice of metric on $M$, but if for a metric $g$ this angle is bounded away from zero, uniformly for all $x \in M$, then the same holds true for any other metric $g'$, since any two Riemannian metrics are equivalent on a compact manifold in the sense that there exists a constant $C > 1$ such that
$$
 C^{-1} g'\_x(v,w) \le g\_x(v,w) \le C g'\_x(v,w)
$$
for all $x \in M$ and $v,w \in T\_x M$.


