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I'm reading Bridson & Haefliger's book on non-positively curved spaces. Specifically the parts in II.10 on $P(n,\mathbb{R})$. These are positive definite matrices, viewed as the image of symmetric matrices under matrix exponentiation, with a metric given by $d(e^A, e^B) = |\log(e^{-A/2} e^B e^{-A/2})|$, the usual trace norm after this isometry moving $e^A$ to the identity and going back to the symmetric matrices.

I'd like to understand this situation:

Let $e^D, e^A \in P(n,\mathbb R)$ with $D$ diagonal. Consider the pair of geodesics $e^{tD}$ and $e^{tA}$ going through the origin and these points. If $A$ commutes with $D$, then these geodesics diverge linearly, as in euclidean space.

However, when $A,D$ do not commute, the geodesics diverge faster than linearly. I would like to know how this divergence depends on the distance of $e^A$ to the set $C(e^D)$ of matrices commuting with $e^D$. My intuition is that the divergence should continuously increase with this distance. E.g., I imagine $d(e^{2D},e^{2A}) - 2\cdot d(e^{D}, e^{A})$ increases continuously as $d(e^A, C(e^D))$ increases.

Is this true? Better yet if it is, is it described somewhere, or clear from the definition of this metric and some relation between this distance and commutators? Bridson & Haefliger show that the differential of $e^{-X/2}e^{X+tY}e^{-X/2}$ is a matrix series $\sum_{k=0}^{\infty} \frac{ad^{2k}_{X/2}}{(2k+1)!}(Y)$ in order to prove $P(n,\mathbb R)$ is CAT(0). But I don't know enough to know if this can be pulled together to get what I'm asking.

Any pointers or pointing would be appreciated.

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  • $\begingroup$ Actually, with the usual notion of divergence in metric spaces, see mathoverflow.net/questions/137076, geodesics in a symmetric space of rank $\ge 2$ diverge only linearly. $\endgroup$
    – Misha
    Nov 20, 2013 at 7:58
  • $\begingroup$ I see, thanks. I'm using the wrong terminology then. Do you know what this is called? I mean the distance between corresponding points of a pair of geodesic rays. This is clearly nonlinear if the geodesics are not in a common flat subspace. $\endgroup$
    – Matth
    Nov 20, 2013 at 15:57
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    $\begingroup$ Matth: If you simply compute the distance $d(\gamma_1(t), \gamma_2(t))$ then it is at most a linear function of $t$, this is just the triangle inequality. There is a classical quantity one uses instead, namely, Jacobi fields along geodesic rays: They describe infinitesimal variations of geodesics and they indeed can grow exponentially in the setting of symmetric spaces. Look for instance in Eberlein's book on manifolds of nonpositive curvature, where everything is worked out in great detail. Along flats you have no growth (parallel JF); otherwise, you get exponential growth. $\endgroup$
    – Misha
    Nov 20, 2013 at 16:46
  • $\begingroup$ @Misha Thanks for the pointer toward Jacobi fields & the Eberlein reference. It's clear to me that I'm bad at formulating this question to people with knowledge about it. I think a quantity like $d(e^{2D},e^{2A}) - 2d(e^D, e^A)$ varies as $e^A$ moves away from a flat containing $e^D$ (I have a particular metric on geodesics in mind). I want to know that a bound on this quantity yields a bound on the distance. JFs sound like they do this just as well, so I'm hopeful that this is worked out in Eberlein's book. In any case, thank you very much for your time and for responding at all! $\endgroup$
    – Matth
    Nov 22, 2013 at 18:45

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