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I search for some properties $P$ for complete Riemannian manifolds which satisfy a kind of Sandwich property.

More precisely I search for those properties $P$ for complete Riemannian manifolds which satisfy the following:

For every manifold $M$ with $3$ complete Riemannian metrics $g_1\leq h \leq g_2$, if $(M,g_1)$ and $(M,g_2)$ satisfy $P$ then $(M,h)$ satisfies $P$, too.

What properties $P$ are some examples of this situation?

In particular, is "Flatness" an example of such squeeze property?

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    $\begingroup$ Flatness certainly isn't; consider a conformally euclidian metric $h:=e^f g_{\rm eucl}$ for some bounded function $f$ on $\mathbb R^n$, then $h$ isn't flat if $f$ is general. $\endgroup$
    – Henri
    Commented Jun 27, 2018 at 21:15
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    $\begingroup$ On type of property one can thing of though is being of prescribed (e.g. linear, quadratic, maximal) volume growth. $\endgroup$
    – Henri
    Commented Jun 27, 2018 at 21:17
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    $\begingroup$ Yes, two metrics that are quasi-isometric are either both complete or both incomplete $\endgroup$
    – Henri
    Commented Jun 27, 2018 at 21:21
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    $\begingroup$ (boundedness of $f$ is the key, not the boundedness of $e^f$. You need that $e^f$ is uniformly bounded away from zero also) $\endgroup$
    – Willie Wong
    Commented Jun 27, 2018 at 21:22
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    $\begingroup$ The intuitive idea should be that curvature is a quantity at the level of two derivatives of the metric, and we know that if three functions $f_1 \leq h \leq f_2$, there are no guarantee that $|\nabla f_1| \leq |\nabla h| \leq |\nabla f_2|$. ($h$ can be much more oscillatory than either $f_1$ and $f_2$). So realistically you only really expect to control things of regularity level at the level of the metric or its integrals. Volume being certainly one of them as Henri said. $\endgroup$
    – Willie Wong
    Commented Jun 27, 2018 at 21:25

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