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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.

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  • $\begingroup$ Q1-b is false because $g_0$ could be a metric which does not have positive curvature. $\endgroup$ Aug 7 '20 at 20:01
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    $\begingroup$ The answer is "it depends". You have to decide what topology you want to use on the space of metrics, It is completely analogous to the different possible topologies for the space of functions on $M$. Possible topologies include $C^k$, Sobolev, Holder, and $C^\infty$. You get to choose, depending on the specific requirements needed. $\endgroup$
    – Deane Yang
    Aug 7 '20 at 20:01
  • $\begingroup$ @MichaelAlbanese, if the metric $g$ is $C^2$-close to the standard metric on the sphere, then it has positive curvature. $\endgroup$
    – Deane Yang
    Aug 7 '20 at 20:02
  • $\begingroup$ @DeaneYang: As far as I can tell, $g_0$ is an arbitrary metric on $S^n$. Maybe C.F.G intended for $g_0$ to be the round metric. They should make this clear. $\endgroup$ Aug 7 '20 at 20:04
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$\mathcal{M}et(M)$ carries many natural (= invariant under the action of the group of diffeomorphisms of $M$) Riemannian metrics. See the following papers (and references therein):

  • Martin Bauer, Philipp Harms, Peter W. Michor: Sobolev metrics on the manifold of all Riemannian metrics. Journal of Differential Geometry 94, 2 (2013), 187-208. (pdf)

In particular, for the Sobolev order $\ge 2+\frac{\dim(M)}2$ metric the curvature is continuous so Q2 has a positive answer.

A more quite recent paper concentrating on the well-posedness of the geodesic equations for Sobolev metrics is this one.

More details:

In the $C^\infty$-topology, where $\mathcal{M}et(M)$ is an open subset of a Frechet space, Q2 is has always a positive answer. See here for a detailed description of this topology. Namely, the mapping $g\mapsto R^g$ which maps a metric $g$ to its curvature, is smooth $C^\infty$, and thus continuous since all is Frechet. Then, choose a finite open atlas $U_\alpha$ for $M$ a compact $K_\alpha \subset U_\alpha$ such that $\bigcup_\alpha K_\alpha = M$, and a frame $(s^i_\alpha)$ on each chart. Then, if for all $i < j$ sectional curvature $$ k(\text{span}(s^i_\alpha,s^j_\alpha)) = -\frac{g_0(R^{g_0}(s^i_\alpha,s^j_\alpha)s^i_\alpha,s^j_\alpha)}{g_0(s^i_\alpha,s^i_\alpha) g_0(s^j_\alpha,s^j_\alpha) - g_0(s^i_\alpha,s^j_\alpha)^2} \ge \epsilon_\alpha $$ for each $\alpha$, then we have $>\epsilon_\alpha/2$ for each $g$ near $g_0$. This also holds for the $C^2$-topology, or for the Sobolev metric of the order given above, involving the Sobolev lemma.

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