I continue from the previous post “Curvdimension and curvature of a metric profile II“.
Let’s see what is happening for , a sufficiently smooth ( for example), complete, connected riemannian manifold. The letter “” denotes the metric (scalar product on the tangent space) and the letter “” will denote the riemannian distance, that is for any two points the distance between them is the infimum of the length of absolutely continuous curves which start from and end in . The length of curves is computed with the help of the metric .
Notations. In this example is a differential manifold, therefore it has tangent spaces at every point, in the differential geometric sense. Further on, when I write “tangent space” it will mean tangent space in this sense. Otherwise I shall write “metric tangent space” for the metric notion of tangent space.
Let be vectors in the tangent space at $x \in X$. When the basepoint is fixed by the context then I may renounce to mention it in the various notations. For example means the norm of the vector with respect to the scalar product on the tangent space at the point . Likewise, may be used instead of ; the riemannian curvature tensor at may be denoted by and not by , and so on …
Remark 2. The smoothness of the riemannian manifold should be just enough such that the curvature tensor is and such that for any compact subset of , possibly by rescaling , the geodesic exponential makes sense (exists and it is uniquely defined) for any and for any with .
Let us fix such a compact set and let’s take a point .
Definition 5. For any we define on the closed ball of radius centered at (with respect to the distance ) the following distance: for any with ,
(The notation used here is in line with the one used in dilation structures.)
Recall that the sectional curvature is defined for any pair of vectors which are linearly independent (i.e. non collinear).
Proposition 1. Let be greater or equal than , for any and any non-collinear pair of vectors with , . Then for any and any , with , we have
Corollary 1. For any the metric space has a metric tangent space at , which is the isometry class of the unit ball in with the distance .
Corollary 2. If the sectional curvature at is non trivial then the metric profile at has curvdimension 2 and moreover