Let $(M,g)$ be a Riemannian manifold such that for each $C>0$ there is $p\in M$ and $X,Y\in T_pM$ unitary such that $K(X,Y) > C.$ Does this imply that the diameter of $(M,g)$ is infinite?
I just have an intuition about it, for example, by the neck singularity on Ricci flow, or by looking to the Gabriel's Horn: Gabriel.
I searched a lot for a counter-example and possible known theorem's on Petersen's book and other references, but I could conclude nothing, does anyone has a clue?
If $(M,g)$ is complete, then yes. Since sectional curvature is continuous, if it is unbounded then $(M,g)$ is not compact, and by the contrapositive of the Hopf-Rinow theorem, a complete non-compact Riemannian manifold has infinite diameter.
If $(M,g)$ is not complete, then no. You can take something like $(0,1)^n$ and modify it with little dimples that become smaller but sharper as you approach the boundary, so that their sectional curvature becomes arbitrarily large but the manifold's diameter does not increase significantly.
