This continues the previous post Curvdimension and curvature of a metric profile, I.
Definition 3. (flat space) A locally compact metric space is locally flat around if there exists such that for any we have . A locally compact metric space is flat if the metric profile at any point is eventually constant.
Question 1. Metric tangent spaces are locally flat but are they locally flat everywhere? I don’t think so, but I don’t have an example.
Definition 4. Let be a locally compact metric space and a point where the metric space admits a metric tangent space. The curvdimension of at is , where is the set of all such that
Remark that the set always contains . Also, according to this definition, if the space is locally flat around then the curvdimension at is .
Question 2. Is there any metric space with infinite curvdimension at a point where the space is not locally flat? (Most likely the answer is “yes”, a possible example would be the revolution surface obtained from a graph of a infinitely differentiable function such that and all derivatives of at are equal to . This surface is taken with the distance from the 3-dimensional space, but maybe I am wrong. )
We are going to see next that the curvdimension of a sufficiently smooth riemannian manifold at any of its points where the sectional curvature is not trivial is equal to .