# Fat Triangulations, Curvature and Quasiconformal Mappings

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

**Theorem 1.1 ([4])**Let M

^{n}be a connected, oriented n-dimensional (n ≥ 2) submanifold of ℝ

^{N}(for some N sufficiently large), with boundary, having a finite number of compact boundary components, and such that one of the following condition holds:

- (i)
- M
^{n}is of class C^{r}, 1 ≤ r ≤ ∞ , n ≥ 2; - (ii)
- M
^{n}is a PL manifold and n ≤ 4; - (iii)
- M
^{n}is a topological manifold and n ≤ 3.

^{n}with spherical metric.

**Definition 1.2**

- map infinitesimal balls into infinitesimal ellipsoids (of bounded eccentricity);
- map almost balls into almost ellipsoids;
- distort infinitesimal spheres by a constant factor.

^{n}→ ℝ

^{n}, then f′(B

^{n}) = E(f′) is an ellipsoid of semi-axes a

_{1}≥ a

_{2}≥ · · · ≥ a

_{n}(and equal to the square roots of the eigenvalue of the adjoint mapping of f′) and the characterizations above follow. Not only this, but, in fact,

- distort local distances by a fixed amount;
- preserve approximative shape.

**Remark 1.3**

**Definition 1.4**

- (i)
- open iff it maps open sets onto open sets;
- (ii)
- discrete iff f
^{−1}(y) is discrete (in M ), for any y ∈ N.

**Definition 1.5**

_{f}of f (also called the critical set in some of the literature) is defined as

_{f}= {x ∈ M | fis not locally homeomorphic at x}

**Definition 1.6**

- 1.
- It is sense preserving, open and discrete;
- 2.
- H(f, x) is locally bounded;
- 3.
- There exists H
_{0}< ∞ such that H(f, x) ≤ H_{0}for a.e. x ∈ B_{f}.

^{n}(equipped with standard metric).

^{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:

**Theorem 1.7 ([4])**Let M

^{n}be a Riemannian manifold satisfying the conditions in the statement of Theorem 1.1 above.

_{0}, then there exists a global fat triangulation of M

^{n}.

**Remark 1.8**

**Corollary 1.9 ([4])**Let M

^{n}be a Riemannian manifold satisfying the conditions in the statement of Theorem 1.1 above.

^{n}admits a fat triangulation.

**Remark 1.10**

**Definition 1.11**

^{n}; 0 ≤ k ≤ n be a k-dimensional simplex. The fatness φ of τ is defined as being:

_{j}(σ) and diam σ stand for the Euclidean j-volume and the diameter of σ respectively. (If dim σ = 0, then Vol

_{j}(σ) = 1, by convention.) A simplex τ is φ

_{0}-fat, for some φ

_{0}> 0, if ϕ(τ) ≥ φ

_{0}. A triangulation (of a submanifold of $\mathbb{R}$

^{n}) T = {σ

_{i}}

_{i∈I}is φ

_{0}-fat if all its simplices are φ

_{0}-fat. A triangulation T = {σ

_{i}}

_{i∈I}is fat if there exists φ

_{0}≥ 0 such that all its simplices are φ

_{0}-fat.

**Proposition 1.12 ([10])**There exists a constant c(k) that depends solely upon the dimension k of τ such that

**Theorem 1.13 ([10])**Let M

^{n}be a compact Riemannian manifold, with or without boundary, and let ${M}_{i}^{n}$ be a sequence of fat PL (piecewise flat) manifolds, that are secant approximations of M

^{n}, converging to M

^{n}in the Hausdorff metric. Denote by $\mathcal{R}$ and $\mathcal{R}$

_{i}respectively, the Lipschitz–Killing curvatures of M

^{n}, ${M}_{i}^{n}$. Then $\mathcal{R}$

_{i}→ $\mathcal{R}$ in the sense of measures.

**Definition 1.14**

_{r}(A) denotes the r-neighborhood of A, i.e., U

_{r}(A) = ∪

_{a∈A}B(a, r). (Here, B(a, r), denotes, as usually, the open ball of center a and radius r.)

^{n}is compact and since, given that the PL manifolds ${M}_{i}^{n}$ are secant approximations of M

^{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.)

**Definition 1.15**

_{i}}

_{i}be of sequence of Borel measures of finite mass on X . The sequence {µ

_{i}}

_{i}is said to converge (weakly) to a measur e µ iff ∫

_{X}fdµ

_{i}→ ∫

_{X}fdµ, for any bounded, positive and continuous function f : X → ℝ.

^{n}, the Lipschitz–Killing curvatures are defined as follows:

_{π(j−1)π(j)}are the curvature 2-forms and ω

_{kl}denote the connection 1-forms, and they are interrelated by the structure equations:

_{k}} is the dual basis of {e

_{k}}.

**Remark 1.16**

^{0}≡ volume and R

^{2}≡ scalar curvature. Moreover, R

^{n}≡ Gauss–Bonnet–Chern form, (for n = 2k).

**Remark 1.17**

_{Mn}R

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

^{j}which are curvature measures on ∂M

^{n}: Let {e

_{k}}

_{1≤k≤n}be an orthonormal frame for the tangent bundle T

_{Mn }of M

^{n}, such that, along the boundary ∂M

^{n}, e

_{n}coincides with the inward normal. Then, for any 2k + 1 ≤ j ≤ n, we define

^{n−j}.

**Remark 1.18**

^{1}≡ area boundary, H

^{2}≡ mean curvature for inward normal (as expected given the generic names for these H

^{j}-s), etc.

**Remark 1.19**

^{j}) denotes the (spherical) link of σ

^{j}, and ∡(σ

^{i}, σ

^{j}) is the internal dihedral angle of σ

^{i}< σ

^{j}; ∡(σ

^{i}, σ

^{j}) = Vol(L(σ

^{i}, σ

^{j}), where the volume is normalized such that Vol($\mathbb{S}$

^{n}) = 1, for any n. (See [10] for further details.) (Here χ, Vol denote, as usual, the Euler characteristic and volume of σ

^{k}, respectively.)

**Theorem 1.20 ([15])**Let N = N

^{n−1}be a not necessarily connected manifold, such that N = ∂M, M = M

^{n}, where M

^{n}is, topologically, as in the statement of Theorem 1.7.

^{j}} on $\overline{M}$ = M ∪ N, such that $\mathcal{R}$|

_{N}= $\mathcal{R}$

_{N}and $\mathcal{R}$|

_{M}= $\mathcal{R}$

_{M}, except on a regular (arbitrarily small) neighbourhood of N , where $\mathcal{R}$

_{N}, $\mathcal{R}$

_{M}denote the curvature measures of N, M respectively.

**Remark 1.21**

^{j}|

_{∂Mn}= H

^{j}and, in the case of PL manifolds, it represents the contribution of the (n − j)-dimensional simplices that belong to the boundary. (For an explicit formula, see any of the formulas (3.23), (3.38) or (3.39) of [10].)

**Remark 1.22**

## 2. Quasiconformal Mappings Between Manifolds

^{n}. More precisely, from the considerations above we obtain immediately the following theorem:

**Theorem 2.1**

**Proof**

_{j−1}denotes the (j − 1)-dimensional measure (content) of the (j-dimensional dihedral) angle α

_{j}. (Alternatively, by making appeal to [5], Section 6, one can obtain the double inequality above directly in terms of K itself, albeit at the precise of loosing intuitiveness.) But, by Formula (12), the Lipschitz–Killing curvatures R

^{j}of a PL (piecewise flat) manifold are functions of the measures of the dihedral angles in dimension ≥ n − j (and, implicitly, on the dimension n of the manifold), thence there exists C

_{1}= C

_{1}(K, n, j) such that Formula (13) holds.

^{j}, is a consequence of the fact, already noted above, that interior points—that is of 0 curvature—can be mapped into (essential) vertices—i.e., carriers of positive or negative curvature. Note that the respective inequalities for the R

^{j}’s—that is without passing to the absolute values—can be obtained if these curvatures are bounded away from 0 and the dilatation K ≈ 1, where “closeness to 1” is a function of lower/upper bounds on the curvatures R

^{j}.

^{n}. Moreover, by repeated (almost) parallel (or median) subdivisions (i.e., obtained dividing any given simplices into k

^{2}smaller ones, by hyper-planes (lines, for n = 2) parallel to the faces of given the simplex), the mesh of the triangulation can be made arbitrarily small, while ensuring that the fatness of the elements of a such sequence of subdivisions {${M}_{i}^{n}$} is uniformly bounded from below, i.e., there exists φ

_{0}> 0, such that φ

_{i}= φ(${M}_{i}^{n}$) ≥ φ

_{0}, for all i ∈ ℕ. (Note that in fact, the construction requires such repeated subdivisions—see [4].) This, in conjunction with [10], Theorem 5.1 (for manifolds without boundary and the interior of manifolds with boundary) and Theorem 8.1 (for the case of the boundary curvatures) assures the existence of arbitrarily good approximations in measure of the Lipschitz–Killing curvatures R

^{j}of the smooth manifold M

^{n}by those of a sequence of PL approximations $\left\{{M}_{i}^{n}\right\}$. The desired result now follows immediately from (1).

**Remark 2.2**

^{n}. We should emphasize that this is, however, only an approximation in measure and the error |R

^{j}(U) − ${R}_{i}^{j}$(U)| is a function not only on the curvature tensor of M

^{n}and of its gradient, but also of φ

_{0}and on the mesh of the triangulation, and, of course, of the volume of the neighbourhood U (of a given vertex) where curvatures are approximated—see [10] Formula (5.4). However, in many applications (see, e.g. [17]), one wishes to estimate the error in the approximation of R

^{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 [10], Formula (1.21) and Formula (4.7) and Appendix A2, respectively. Moreover, the change of dihedral angles of these approximations is (as expected) a function of the sectional curvatures at v—see [10], Formula (5.10).

**Alternative Proof**

^{n−1}= ∂M

^{n}, d$\mathscr{H}$

^{n−1}denotes the (n − 1)-dimensional Hausdorff measure, and where the symmetric functions S

_{j}are defined by:

_{1}(x), k

_{2}(x), . . . , k

_{n−1}(x) being the principal curvatures—see e.g. [18].

_{r}is a C

^{1,1}-hypersurface. Therefore, they admit principal curvatures (in the classical sense) ${k}_{i}^{\epsilon}$(x +

**n**) at almost any point p = x +

**n**, where

**n**denotes the normal unit vector (at x). Define the generalized principal curvatures by: ${k}_{i}\left(\epsilon ,n\right)={k}_{i}^{\epsilon}\left(x+n\right)$. Then k

_{i}(ε,

**n**) exist $\mathscr{H}$

^{n−1}-a.a. (x,

**n**).

_{r}, ∃ ! x ∈ X nearest to y}

_{r}denotes the r-neighbourhood of X, and that the reach itself has to be strictly positive.

_{k}(X, B) denote the so called Lipschitz–Killing curvature measures (see [18] and the bibliography therein for details), and nor(X) denotes the (unit) normal bundle of X:

**n**∈ S

_{d−1}| <

**n**, v >≤ 0, v ∈ Tan(X, x))} is the normal cone (to X at the point x ∈ T), dual to the tangent cone (to X at the point x ∈ T).

**Remark 2.3**

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

## 3. Quasimeromorphic Mappings on Almost Riemmanian Manifolds

**Definition 3.1**

- 1.
- M is a smooth manifold;
- 2.
- There exists a (smooth) Riemannian metric g on M and a constant C
_{0}> 0, such that, for any x ∈ M , there exists a neighbourhood U(x), such that$${C}_{0}^{-1}d\left(y,z\right)\le {\text{dist}}_{g}\left(y,z\right)\le {C}_{0}d\left(y,z\right)$$

^{N}, with d being the Euclidean distance in ℝ

^{N}, d = dist

_{Eucl}, i.e., precisely the setting which we were concerned in the previous section: The secant approximation of an embedded smooth manifold, with its Euclidean (ambient) metric is an almost Riemannian manifold (relative, so to say, to the approximated smooth manifold).

**Theorem 3.2 ([15])**Let (M, d) be an almost Riemannian manifold, where M satisfies the conditions in the statement of Theorem 1.1. Then it admits a fat triangulation.

**Proposition 3.3**

## 4. Final Remarks

**Remark 4.1**

**Question 1**

^{n}admit a q

^{r}-mapping on ${\mathbb{S}}^{n}$ iff it admits “good” curvature convergence in secant approximation?

## Acknowledgements

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Saucan, E.; Katchalski, M.
Fat Triangulations, Curvature and Quasiconformal Mappings. *Axioms* **2012**, *1*, 99-110.
https://doi.org/10.3390/axioms1020099

**AMA Style**

Saucan E, Katchalski M.
Fat Triangulations, Curvature and Quasiconformal Mappings. *Axioms*. 2012; 1(2):99-110.
https://doi.org/10.3390/axioms1020099

**Chicago/Turabian Style**

Saucan, Emil, and Meir Katchalski.
2012. "Fat Triangulations, Curvature and Quasiconformal Mappings" *Axioms* 1, no. 2: 99-110.
https://doi.org/10.3390/axioms1020099