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We investigate the interplay between the existence of fat triangulations,

The present note is largely motivated by our theorem below, itself a continuation and generalization of previous results of Martio and Srebro [

(i)

(ii)

(iii)

Recall that quasiconformal mappings are defined as follows:

Obviously, the linear dilatation is a measure of the eccentricity of the image of infinitesimal balls. Therefore (at least if one restricts oneself to Riemannian manifolds) quasiconformal mappings can be characterized as being precisely those maps that

map infinitesimal balls into infinitesimal ellipsoids (of bounded eccentricity);

map almost balls into almost ellipsoids;

distort infinitesimal spheres by a constant factor.

In fact, if one considers (in the Riemannian manifold setting) the linear mapping

Moreover, quasiconformal mappings

distort local distances by a fixed amount;

preserve approximative shape.

(However, in these cases the characterization is not sharp, the proper class of functions characterized by this property being the so-called

Of course, one naturally asks whether the “quasiconformal” in Definition 1.2 above implies, indeed, as the name suggests, that quasiconformal mappings “almost” preserve angles (given that conformal mappings do). The answer is, as expected, positive—see [

Generalizing the definition of quasiconformality to maps that fail to be homeomorphisms necessitates a number of additional preliminary definitions and notations:

To introduce a fitting metric definition of quasiregularity, we have to restrict somewhat the class of spaces on which they are defined. However, this class still represents a generalization of more established definitions and more than suffices for our geometric purposes here.

Note that, by [

Since the theorem above concerns the existence of quasiconformal mappings, which, as we have seen, are generalizations of quasiconformal mappings and since Proposition 3.3 below concerns the existence of quasiconformal mappings between almost Riemannian manifolds (see Definition 3.1 below), their presence in the title is elucidated.

Given that the proof of Theorem 1.1 essentially requires the construction of a “chess-board” fat triangulation (followed by the alternate quasiconformal mapping of the “black” and “white” simplices to the interior, respective exterior of the standard simplex in

Given the triangulation results of [

Recall that fat triangulations (also called thick in some of the literature) are defined (in [

The following result gives a more intuitive interpretation on the notion of fatness of simplices as a function of their dihedral angles in all dimensions:

and

Having explained our concern with fat triangulation and quasimeromorphic mappings, we still have to explain the connection with differential geometry. The inherent relation between the existence of fat triangulations and differential geometry is expressed by the essential role of curvature in the construction of such triangulations. This ingrained connection is transparent in the very basic proof of Peltonen [

The reverse direction, that is the role of fat triangulations in determining (in the

Recall that the Hausdorff metric is defined as follows:

(Note that, since

The convergence of measure considered here is the weak convergence:

Recall also that, for a Riemannian manifold

where

where

In a similar manner (but technically slightly more complicated), one can define the associated boundary curvatures (

where

and

These curvatures measures are normalized by imposing the condition that:

for any flat

The need for fat triangulations as a prerequisite for Theorem 1.13 should not be surprising, in view of the characterization of fat triangulations as being those triangulations having dihedral angles bounded from below (Proposition 1.12) and in view of the following expression of the Lipschitz–Killing curvatures in terms of dihedral angles (see [

where

The differential geometric consequence of Theorems 1.13 and 1.7, as well as Corollary 1.9 is the following:

(i) If

(ii) If

Our main result is the observation that spaces that admit “good” curvature convergence in secant approximation are

Now, from Agard’s characterization of quasiconformal mappings [

where

where

That Formula (13) holds, in general, only for the absolute values of the curvatures

(ii) By our triangulation result above, namely Corollary 1.6, there exists a fat triangulation of

where

(i) For

First, let us note that, for small enough

Note that, in particular,

where

Using this generalization of principal curvatures, one can retrieve a proper analogue of Formula 14, namely

where

where

Using the convergence properties of the generalized principal curvatures and of the Lipschitz–Killing curvature measures (again, see [

(ii) The case of smooth manifolds follows immediately following the same scheme as in the previous proof.

We also bring here a generalization of Theorem 1.1, but, before stating this next result, we need to introduce (following [

The basic—and motivational—example of an almost metric space (beyond the trivial one

The relevant result here is the following theorem that was proved in [

The theorem above and the construction technique of quasimeromorphic mappings employed in [

We do not consider here the problem of extending Proposition 2.1 to the case of general quasiregular mappings, as well as discussing other related aspects relating quasiregular mappings and curvature, but we rather postpone them for further study (see [

We conclude, therefore, this paper with the following Remark and the ensuing Question:

First author’s research supported by European Research Council under the European Community’s Seventh Framework Programme (FP7/2007-2013) / ERC grant agreement n

The authors would like to thank the referees for their many helpful corrections, comments and suggestions.