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Any smooth manifold $M$ admits a metric with quadratic curvature decay. In fact there is a complete Riemannian metric $g$ on $M$ such that $$|K_x|=o(|x_0x|^{-2}),$$ here $|K_x|$ is absolute bound for sectional curvature of $g$ at point $x$, $x_0$ is a fixed point and and $|x_0x|$ denotes distance induced by $g$.

In particular there is no examples which you are looking for.

See p. 96 in Gromov M., Volume and bounded cohomology, Publ. Math. IHES 56 (1982) 5–99.

Any smooth manifold $M$ admits a metric with quadratic curvature decay. In fact there is a complete Riemannian metric $g$ on $M$ such that $$|K_x|=o(|x_0-x|_g^{-2}),$$ here $|K_x|$ is absolute bound for sectional curvature of $g$ at point $x$, $x_0$ is a fixed point and and $|x_0-x|_g$ denotes distance induced by $g$.

In particular there is no examples which you are looking for.

See p. 96 in Gromov M., Volume and bounded cohomology, Publ. Math. IHES 56 (1982) 5–99.

Any smooth manifold $M$ admits a metric with quadratic curvature decay. In fact there is a complete Riemannian metric $g$ on $M$ such that $$|K_x|=o(|x_0x|^{-2}),$$ here $|K_x|$ is absolute bound for sectional curvature of $g$ at point $x$, $x_0$ is a fixed point and and $|x_0x|$ denotes distance induced by $g$.

In particular there is no examples which you are looking for.

See Abresch, Lower curvature bounds, Toponogov's theorem, and bounded topology.

Any smooth manifold $M$ admits a metric with quadratic curvature decay. In fact there is a complete Riemannian metric $g$ on $M$ such that $$|K_x|=o(|x_0x|^{-2}),$$ here $|K_x|$ is absolute bound for sectional curvature of $g$ at point $x$, $x_0$ is a fixed point and and $|x_0x|$ denotes distance induced by $g$.

In particular there is no examples which you are looking for.

See p. 96 in Gromov M., Volume and bounded cohomology, Publ. Math. IHES 56 (1982) 5–99.

Any smooth manifold admits a metric with quadratic curvature decay. In fact there is a metric such that $$|K_x|=o(|x_0x|^{-2}).$$ In particular there is no examples which you are looking for.

See Abresch, Lower curvature bounds, Toponogov's theorem, and bounded topology.

Any smooth manifold $M$ admits a metric with quadratic curvature decay. In fact there is a complete Riemannian metric $g$ on $M$ such that $$|K_x|=o(|x_0x|^{-2}),$$ here $|K_x|$ is absolute bound for sectional curvature of $g$ at point $x$, $x_0$ is a fixed point and and $|x_0x|$ denotes distance induced by $g$.

In particular there is no examples which you are looking for.

See Abresch, Lower curvature bounds, Toponogov's theorem, and bounded topology.