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.



Open neighbourhood of a point of space of Riemannian metrics
$\mathcal{M}et(M)$ 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)](https://www.mat.univie.ac.at/%7Emichor/rie-met2.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](https://www.mat.univie.ac.at/%7Emichor/rie-met3.pdf).


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](https://www.mat.univie.ac.at/%7Emichor/rie-met.pdf) 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.


