MathOverflow is a question and answer site for professional mathematicians. It's 100% free, no registration required.

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

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 $E^u(x)$?

share|cite|improve this question
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. – Ian Morris Jan 22 '14 at 11:21
up vote 6 down vote accepted

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 $v,w \in T_x M$.

share|cite|improve this answer
in particular, you can in this case modify the metric to make the stable and unstable bundles orthogonal... – Barbara Schapira 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: – 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 $F$. – Jaap Eldering Jan 23 '14 at 10:20

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.