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In Cieliebak, Ginzburg and Kerman's paper Symplectic homology and periodic orbits near symplectic submanifolds, the authors claim and give a proof that a twisted cotangent bundle will be geometrically bounded. This is Proposition 2.2. To do so they pick a fiberwise convex hypersurface $\Sigma$, enclosing the zero section and consider the flow $\phi_t$ which is dilation by $e^t$ along the fibers. Then they consider $U=\bigcup_{t\geq 0}\phi_t(\Sigma)$. Then they pick a metric $g$ on $TW\big|_{\Sigma}$ which is compatible and they extend it in $U$ by $\phi_{t}^*(g)=e^tg$. Then the author's claim that by this dilation condition on the metric $g$ along $U$ we will have that the sectional curvature of the metric will be going to zero as we approach $\infty$ along the fibers. This is the part I do not follow. Why will this be the case , how can we conclude such a result for the sectional curvature?

Any insight is appreciated, thanks in advance.

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  • $\begingroup$ Probably because $\phi_t$ dilates the metric by factor $e^t$ (by pull-back property). This implies for example the metric around the point $p$ near $\Sigma$ is expanded by $\phi_t$ by factor $e^t$ around $\phi_t(p)$ (near infinity for large $t$), hence the curvature is reduced by factor $e^{-t}$ $\endgroup$
    – anything
    Commented Jan 13, 2022 at 8:02
  • $\begingroup$ @anything: If my understanding is correct, to approach $\infty$ along a fiber means that the vectors in that fiber go to $\infty$, not that $t \to \infty$. Therefore, I don't see how your comment addresses the question. $\endgroup$
    – Alex M.
    Commented Jan 14, 2022 at 16:18

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