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 $(M,h)$ satisfy $P$, too.

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

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

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?