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
.
Proofs next time, but you may figure them out by looking at the section 1.3 of these notes on comparison geometry , available from the page of Vitali Kapovitch.