Sectional curvature and injectivity radius of natural metric in Cotangent bundles In the following paper by Cielibeak, Ginzburg and Kerman https://arxiv.org/pdf/math/0210468.pdf they claim in page $3$ that the natural metric $\tilde g$ on $T^*M$ the sectional curvature is bounded from above na the injectivity radius is bounded away from zero. They also give a reference where this is mentioned in the book by Lafontaine and Audin "Holomorphic curves in Symplectic geometry ".
I have tried to see check why this fact is true but I got nowhere useful. I tried following their proof of Proposition $2.2$, where they prove that a twisted cotangent bundle is geometrically bounded , i.e., I tried to see what happens to the pullback of the sasaki metric under the flow $\phi_t$ given by fiberwise dilation of factor $e^t$. Afer some computations, I found out that if we consider the decomposition of $T_{(x,v)}TM= H(\theta)\oplus V(\theta)\cong T_xM\oplus T_xM$ then $d_{(x,v)}\phi_t((u,w))= (u,e^tw)$. And with this we will obtain that $\phi_t^*\tilde g((u_1,w_1),(u_2,w_2))= g_x(u_1,u_2)+e^{2t}g_x(w_1,w_2)$.  With this I am not sure how to proceed to obtain the desired claim .
Any insight ,or if anyone knows a reference where this is proved, is apprecited. Thanks in advance.
 A: I am sorry my short answer elsewhere confuses you.
I try to be more explicite. First they 'define' (or extend)
metric so that $\phi_t^*g=e^t g$ holds from the
original metric on $\Sigma$.
Specifically, any point $T^*M$ outside $\Sigma$ can be written
uniquely in the form $(x,e^a w_0)$ where $(x,w_0)\in \Sigma$.
And 'define' the metric at that point
$g_{(x,e^a w_0)}(X,Y)=e^ag_{(x,w_0)}(d\phi_a^{-1}(X), d\phi_a^{-1}(Y))$. (this nothing to do with the explicit computation of the
metric decomposing the vertical and horizontal component)
This definition of the metric implies the pullback property $\phi_t^*g_p=e^tg_p$ (this time the equality holds for any t and any point p outside of $\Sigma$. It is a consequence of above definition, and $\phi_t\phi_s=\phi_{t+s}$).
This relates the metric around p and the metric around $\phi_t(p)$
and the latter is expanded by factor $e^t$.
It is direct from the general
definition of the curvature, the sectional curvature $K$ reduces by factor $e^{-t}$ when the metric is expanded by factor $e^t$.
Hence $K(\phi_t(p))=e^{-t}K(p)$. And of course the injectivity radius is expanded when metric is expanded.
