Almost flat manifold

In mathematics, a smooth compact manifold M is called almost flat if for any ${\displaystyle \varepsilon >0}$ there is a Riemannian metric ${\displaystyle g_{\varepsilon }}$ on M such that ${\displaystyle {\mbox{diam}}(M,g_{\varepsilon })\leq 1}$ and ${\displaystyle g_{\varepsilon }}$ is ${\displaystyle \varepsilon }$-flat, i.e. for the sectional curvature of ${\displaystyle K_{g_{\varepsilon }}}$ we have ${\displaystyle |K_{g_{\epsilon }}|<\varepsilon }$.

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