# Angle between two subspaces

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 $E^u(x)$?

• It seems to me that this is principally a question about the geometry of the finite-dimensional vector space $T_xM$, so I have taken the liberty of adding some tags to reflect this so as to bring the question to a more geometrically-minded audience. Since I'm not very familiar with the nomenclature of geometry my choice of tags is probably only approximately correct. Jan 22 '14 at 11:21

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 $v,w \in T_x M$.

• in particular, you can in this case modify the metric to make the stable and unstable bundles orthogonal... Jan 22 '14 at 21:51
• Jaap Eldering: Thank you for your response. I have always been thinking as you do. Maybe its better to consider $v\in E^s(x), w\in E^u(x)$ with $\Vert v\Vert=\Vert w\Vert=1$. But i am somehow confused after studying the paper"On C^1-persistently of homoclinic classes" By "Martin Sambarinho and Jose vieitez". in this paper they have defined the angle between two subspaces in a different way. Is this definition equivalent to yours?
– mac
Jan 23 '14 at 6:12
• you can find the paper here: cmat.edu.uy/cmat.eventos.16cla/docentes/samba/publicaciones
– mac
Jan 23 '14 at 6:31
• @mac: ah, if $E \oplus F$ is the splitting, they use linear map $L$ for a graph description of $F$ wrt. $E^\perp$. Draw this in 2D and you'll see that the norm of $L$ is the cotangent of their angles. In general, $\|L\|^{-1}$ is the tangent of the "usual" angle between $E$ and $F$. Jan 23 '14 at 10:20