Naively, a small "test" plane cannot anywhere lie against the torus without loss of contact in one or both of the two local principal directions. This would seem to suggest that nowhere is the curvature zero.

However, perhaps the "sophisticated" curvature is zero along a pair of circles somewhat near / at the "flat" top
- where the local surface changes from convex to saddle-like.
ie differently-signed Ricci curvatures.

Agreed, this change occurs at the flat top, but there the curvature is surely positive ? Or is a flat sheet rolled into a right cylinder still of zero curvature ?

Basically, the naive "flat test-plane" notion of zero curvature is deemed insufficient

What of the curvature of a spherical hole within a solid sphere - is the curvature at the inner surface positive or negative ?

What if the torus shrinks to have no central hole ?
and beyond that - to just a fat doughnut with a dimple in the centre ?

"torus
The torus has regions with different curvature: on the outside of the torus curvature is positive (blue), on the inside it's negative (red), and at the top and bottom circles it's zero (grey). (Image from Mark L. Irons.)"

I think a fuller explanation is required:

Naively, a small "test" plane cannot anywhere lie against the torus without loss of contact in one or both of the two local principal directions. This would seem to suggest that nowhere is the curvature zero.

However, perhaps the "sophisticated" curvature is zero along a pair of circles somewhat near / at the "flat" top

- where the local surface changes from convex to saddle-like.

ie differently-signed Ricci curvatures.

Agreed, this change occurs at the flat top, but there the curvature is surely positive ? Or is a flat sheet rolled into a right cylinder still of zero curvature ?

Basically, the naive "flat test-plane" notion of zero curvature is deemed insufficient

What of the curvature of a spherical hole within a solid sphere - is the curvature at the inner surface positive or negative ?

What if the torus shrinks to have no central hole ?

and beyond that - to just a fat doughnut with a dimple in the centre ?

"torus

The torus has regions with different curvature: on the outside of the torus curvature is positive (blue), on the inside it's negative (red), and at the top and bottom circles it's zero (grey). (Image from Mark L. Irons.)"