@RobertBryant What can be said about the topology of space of all semi positive matrix of size $d$ whose matrix norm is $\leq$ 1? It is compact and convex so is it homeomorphic to a disk?

@YemonChoi it is a little strange:for small disk the positive solutions of the inequality span a finite dimension but in big disk thy produce infinite dim space(according to Gabe comment on my recently deleted question). Any way the set of solutions is a lattice invariant under $\lambda$ rescalling $\lambda\geq 1$

@YemonChoi the domination of norm infinity by norm-2. A result attributed to Grothendieck. I found it in Royden real analysis.

So for a compact Riemannian manifold a natural question is to assign an upper bound for the maximum number of independent positive functions $f_1, f_2,\ldots,f_n$ with $\Delta f_i \leq f_i^2$

on the other hand the condition $\Delta f\leq f^2 $ is preserved by "addition"

It seems that the set of functions satisfying $\Delta f\leq f^2$ can not be very big. Because your question remind me of a theorem of Grothendieck which says : if $|.|_{\infty}$ is dominated by $|.|_2 $ on a a subspace of $L^{\infty}\cap \ell^2$ then the subspace is necessarily a finite dimensional space

@DavidRoberts I appreciate very much your kind help and edit

@Holonomia The identity is harmonic because the metric is parallel: $\Delta^g=0$. So I wonder this simple argument can be modified for giving a tweetable proof of harmonicity of isometries?

@SamHopkins may be a possible mqnifold structure or orbifold structure or group(oid) structure of the quotient space can be discussed here

Thank you and +1for your post

In fact I meant the closure of unbounded open contractible sets, sorry I forget the "contractible" in previous comment"