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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 [

^{n}^{N}

^{n} is of class^{r},

^{n}

^{n}

^{n}

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.

^{n}

^{n}

^{n}

_{1}≥

_{2}≥ · · · ≥

_{n}

Moreover, quasiconformal mappings

distort local distances by a fixed amount;

preserve approximative shape.

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

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

^{−1}(

_{f} _{f} = {

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.

_{0} < ∞ _{0} _{f}

^{n}

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 ^{n}), the presence in the title of the “fat triangulations” is also explained. Moreover, it follows that to prove Theorem 1.1 one has first to ensure the existence of fat triangulations on manifolds. The required result is given below:

^{n}

_{0}^{n}.

Given the triangulation results of [

^{n}

^{n} admits a fat triangulation.

Recall that

^{n}_{j}(_{j}(_{0}_{0} > 0, _{0}. ^{n}_{i}_{i∈I} _{0}_{0}_{i}_{i∈I} _{0} ≥ 0 _{0}

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:

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

^{n} ^{n}^{n} _{i}^{n}_{i}

Recall that the

_{r}(_{r}(_{a∈A}

(Note that, since ^{n} is compact and since, given that the ^{n}, all these manifolds can be considered to be embedded in the same ℝ^{N}. Thus we can employ the Hausdorff metric, instead of the more abstract Gromov–Hausdorff metric.)

The convergence of measure considered here is the

_{i}}_{i} _{i}}_{i} _{X} fdµ_{i}_{X} fdµ

Recall also that, for a Riemannian manifold ^{n}, the Lipschitz–Killing curvatures are defined as follows:
_{π(j−1)π(j)} are the _{kl} denote the _{k}} is the dual basis of {_{k}}.

^{0} ≡ volume ^{2} ≡ scalar curvature^{n} ≡

_{Mn} R^{j} is also known as the integrated mean curvature (of order j).

In a similar manner (but technically slightly more complicated), one can define the associated boundary curvatures (or ^{j} which are curvature measures on ^{n}: Let {_{k}}_{1≤k≤n} be an orthonormal frame for the tangent bundle _{Mn } of ^{n}, such that, along the boundary ^{n}, _{n} coincides with the inward normal. Then, for any 2

These curvatures measures are normalized by imposing the condition that:
^{n−j}.

^{1} ≡ ^{2} ≡ ^{j}

^{n}

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 [^{j}) denotes the (^{j}, and ∡(^{i}, ^{j}) is the internal dihedral angle of ^{i} < ^{j}; ∡(^{i}, ^{j}) = Vol(^{i}, ^{j}), where the volume is normalized such that Vol(^{n}) = 1, for any ^{k}, respectively.)

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

^{n−1} ^{n}^{n}

^{j}} _{N} = ℛ_{N} _{M} = ℛ_{M}, _{N}, ℛ_{M}

^{j}|_{∂Mn} = ^{j}

Our main result is the observation that spaces that admit “good” curvature convergence in secant approximation are ^{n}. More precisely, from the considerations above we obtain immediately the following theorem:

(i) We should first underline the fact that, in the

Now, from Agard’s characterization of quasiconformal mappings [_{j−1} denotes the (_{j}. (Alternatively, by making appeal to [^{j} of a _{1} = _{1}(

That Formula (13) holds, in general, only for the absolute values of the curvatures ^{j}, is a consequence of the fact, already noted above, that interior points—that is of 0 curvature—can be mapped into (essential) vertices—^{j}’s—that is without passing to the absolute values—can be obtained if these curvatures are bounded away from 0 and the dilatation ^{j}.

(ii) By our triangulation result above, namely Corollary 1.6, there exists a fat triangulation of ^{n}. Moreover, by repeated (almost) ^{2} smaller ones, by hyper-planes (lines, for _{0} > 0, such that _{i} = _{0}, for all ^{j} of the smooth manifold ^{n} by those of a sequence of

^{n}. We should emphasize that this is,^{j}(^{n} and of its gradient, but also of φ_{0} ^{j} at a given vertex. For this one has to take into account the error, as a function of the mesh of the simplices incident to a vertex v of the sides and angles of these simplices—see [

In this approach, one regards the relevant edges of a fine enough triangulation both as the principal vectors and as semi-axes of an “infinitesimal” ellipsoid and apply directly the definition of quasiconformality, or rather its geometric interpretations (see ^{n−1} = ^{n}, ^{n−1} denotes the (_{j} are defined by:
_{1}(_{2}(_{n−1}(

(i) For

First, let us note that, for small enough _{r} is a ^{1,1}-hypersurface. Therefore, they admit principal curvatures (in the classical sense) _{i}(^{n−1}-a.a. (

Note that, in particular, _{r}, ∃ ! _{r} denotes the

Using this generalization of principal curvatures, one can retrieve a proper analogue of Formula 14, namely
_{k}(_{d−1} | <

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.

_{j}”-s are called the mean curvatures (of order j).

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

_{0} > 0,

The basic—and motivational—example of an almost metric space (beyond the trivial one ^{N}, with ^{N}, _{Eucl},

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:

^{n} ^{r}-mapping on

First author’s research supported by European Research Council under the European Community’s Seventh Framework Programme (FP7/2007-2013) / ERC grant agreement n° [203134] and by ISF grants 221/07 and 93/11.

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

^{n}

^{n}