Let $M$ be a finite-dimensional compact smooth manifold and $$\mathcal{M}et(M) = \{ g : g \text{ is a Riemannian metric on }M \}.$$

Q1-a: What metrics $g$ are very close to the given metric $g_0$? I.e. Is it possible $g\in B_\varepsilon(g_0,M)$ and $g$ has completely different curvature for sufficiently small $\varepsilon>0$?

Q1-b: For example, Is that true that there exists $\varepsilon>0$, such that all metrics in $B_\varepsilon(g_0,\Bbb S^n)\subset \mathcal{M}et(\Bbb S^n)$ are of positive curvature?

Any references about understanding the structure of $\mathcal{M}et(M)$ would be helpful.

Let $M$ be a finite-dimensional compact smooth manifold and $$\mathcal{M}et(M) = \{ g : g \text{ is a Riemannian metric on }M \}.$$

Q1-a: What metrics $g$ are very close to the given metric $g_0$? I.e. Is it possible $g\in B_\varepsilon(g_0,M)$ and $g$ has completely different curvature for sufficiently small $\varepsilon>0$?

Q1-b: For example, Is that true that there exists $\varepsilon>0$, such that all metrics in $B_\varepsilon(g_{can},\Bbb S^n)\subset \mathcal{M}et(\Bbb S^n)$ are of positive curvature?

Any references about understanding the structure of $\mathcal{M}et(M)$ would be helpful.

Let $M$ be a finite-dimensional compact smooth manifold and $$\mathcal{M}et(M) = \{ g : g \text{ is a Riemannian metric on }M \}.$$

Q1-a: What metrics $g$ are very close to the given metric $g_0$? I.e. $g\in B_\varepsilon(g_0,M)$?

Q1-b: For example, Is that true that there exists $\varepsilon>0$, such that all metrics in $B_\varepsilon(g_0,\Bbb S^n)\subset \mathcal{M}et(\Bbb S^n)$ are of positive curvature?

Any references about understanding the structure of $\mathcal{M}et(M)$ would be helpful.

Let $M$ be a finite-dimensional compact smooth manifold and $$\mathcal{M}et(M) = \{ g : g \text{ is a Riemannian metric on }M \}.$$

Q1-a: What metrics $g$ are very close to the given metric $g_0$? I.e. Is it possible $g\in B_\varepsilon(g_0,M)$ and $g$ has completely different curvature for sufficiently small $\varepsilon>0$?

Q1-b: For example, Is that true that there exists $\varepsilon>0$, such that all metrics in $B_\varepsilon(g_0,\Bbb S^n)\subset \mathcal{M}et(\Bbb S^n)$ are of positive curvature?

Any references about understanding the structure of $\mathcal{M}et(M)$ would be helpful.