We provide an alternative, simpler proof of the existence of thick
triangulations for noncompact $\mathcal{C}^1$ manifolds. Moreover, this proof
is simpler than the original one given in \cite{pe}, since it mainly uses tools
of elementary differential topology. The role played by curvatures in this
construction is also emphasized.Comment: 7 pages Short not

Katchalski, Meir

Saucan, Emil

English

arXiv

Abstract: We provide an alternative, simpler proof of the existence of thick triangulations for noncompact C 1 manifolds. Moreover, this proof is simpler than the original one given in [1], since it mainly uses tools of elementary differential topology. The role played by curvatures in this construction is also emphasized

Emil Saucan

Meir Katchalski

CiteSeerX

8 The Open Mathematics Journal, 2009, 2, 8-11  
 
 1874-1177/09 2009 Bentham Open 
Open Access 
The Existence of Thick Triangulations – An “Elementary” Proof 
Emil Saucan
*
 and Meir Katchalski  
Mathematics Department, Technion, Technion City, Haifa 32000, Israel 
Abstract: We provide an alternative, simpler proof of the existence of thick triangulations for noncompact C
1 
manifolds. 
Moreover, this proof is simpler than the original one given in [1], since it mainly uses tools of elementary differential 
topology. The role played by curvatures in this construction is also emphasized.  
Keywords: Thick triangulation, C
1 
manifolds, cut locus.  
1. INTRODUCTION  
 The existence of the so called “thick” or “fat” 
triangulations is important both in Pure Mathematics, in 
Differential Geometry (where it plays a crucial role in the 
computation of curvatures for piecewise-flat approximations 
of smooth Riemannian manifolds, with applications to Regge 
Calculus, see [2]); and in Geometric Function Theory 
(mainly in the construction of quasimeromorphic mappings, 
see [1, 3-7]). Thick triangulations play also an important role 
in many applications, mainly in Computational Geometry, 
Computer Graphics, Image Processing and their related 
fields (see, [8-13], to name just a few).  
 Recall that “thick” or “fat” triangulations are defined as 
follows:  
Definition 1.1. Let   
n 
;0  k  n be a k-dimensional 
simplex. The thickness  of  is defined as being:  
 
= ( ) = inf
<
dim = j
Vol
j
( )
diam j
.
 
 The infimum is taken over all the faces of ,  < , and 
Volj( ) and diam  stand for the Euclidian j-volume and the 
diameter of  respectively. (If dim  = 0, then Volj( ) = 1, by 
convention). 
 A simplex  is 0-thick, for some 0 > 0, if ( )  0. A 
triangulation (of a submanifold of 
n
) T = { i}i I is 0-thick 
if all its simplices are 0-thick. A triangulation T = { i}i I is 
thick if there exists 0  0 such that all its simplices are 0-
thick.  
 The definition above is the one introduced in [14]. For 
some different, yet equivalent definitions of thickness, see 
[1, 4, 15-17].  
 For C
 
Riemannian manifolds without boundary the 
existence has been  proved by  Peltonen  [1].  This result was  
 
*Address correspondence to this author at the Mathematics Department, 
Technion, Technion City, Haifa 32000, Israel;  
E-mail: semil@tx.technion.ac.il 
 
1991 Mathematics Subject Classification. AMS Classification. 53A07, 
57R05, 57M60, A05.  
extended by the first author to include manifolds of lower 
differentiability class with boundary [18] and to a large class 
of orbifolds [7].  
 Peltonen’s proof is based on the construction of an 
exhaustion of the manifold using a delicate curvature-based 
argument, both to decide the “size” of the compact “pieces” 
(i.e. of the elements of the exhaustion) and to chose the mash 
of the triangulation (see [1] and also Section 2). The 
technique used in [18] is much more elementary, using only 
the Differential Topology apparatus (and results) of [17]. 
This discrepancy between methods creates a kind of 
“esthetic asymmetry” that naturally gives rise to the 
following question:  
Question 1.2. Is it possible to prove the existence of thick 
triangulations for non-compact manifolds using only 
techniques of Elementary Differential Topology?  
 We shall prove that the answer to the question above is 
positive by showing that the “mashing” technique of thick 
triangulations developed in [18] allows us to discard the 
curvature considerations of the original proof.  
 However, in discarding the curvature-related information, 
one also loses geometric intuition and, with it, any possibi-
lity of applying the technique in any non-trivial, concrete 
case. Hence, the next question ensues immediately:  
Question 1.3. Can one recover the Differential Geometric 
information (i.e. curvatures) from the constructed PL 
triangulation?  
 We show that, again, the answer is affirmative, and it 
follows from the results of [14]. Moreover, we indicate how 
this approach can also be simplified using tools that may be 
considered more “elementary”.  
 The remainder of the paper is organized as follows: In 
the next Section we briefly sketch, for the benefit of the 
reader and for the sake of the paper’s self-containment, the 
main steps in the proofs of the main results in [1] and [18]. 
In Section 3, we show how our result in [18] allows us to 
give a simpler, “elementary” proof of Peltonen’s theorem. 
Finally, in the last section, we discuss the role played by 
curvature in our simplified proof and indicate how, using our 
method, one still can recapture the Differential Geometric 
information encoded in Peltonen’s proof.  
The Existence of Thick Triangulations – An “Elementary” Proof The Open Mathematics Journal, 2009, Volume 2     9 
2. BACKGROUND  
 We bring below a very brief sketch of the methods used 
in proving the existence of thick triangulations. We 
concentrate mainly on those aspects that are pertinent to our 
present study.  
2.1. Open Riemannian Manifolds 
 First, a number of necessary definitions:  
 Let M
n 
denote an n-dimensional complete Riemannian 
manifold, and let M
n 
be isometrically embedded into 
 
(“ ”-
s existence is guaranteed by Nash’s Theorem – see, e.g. [1]).  
 Let B (x, r) = {y  
|
deucl < r}; B
 
(x, r) = S
1
(x, r). If 
x  M
n
, let 
n
(x, r) = M
n 
 B (x, r), 
n
(x, r) = expx(B
n
(0, r)), 
where: expx denotes the exponential map: expx : Tx(M
n
)  
M
n
, and where B
n
(0, r)  Tx(M
n
), B
n
(0,r) = {y  
n|
deucl(y, 
0) < r}.  
 The following definitions generalize in a straightforward 
manner classical ones used for surfaces in 
3
:  
Definition 2.1.  
(1)  S
1
(x, r) is tangent to M
n 
at x  M
n 
iff there exists 
S
n
(x, r)  S
1
(x, r), such that Tx(S
n
(x, r))  Tx(M
n
).  
(2)  Let l  
 
be a line, then l is secant to X  M
n 
iff|l  
X|  2.  
Definition 2.2.  
(1)  S
1
(x, ) is an osculatory sphere at x  M
n 
iff:  
(a) S
1
(x, ) is tangent at x; and  
(b) B
n
(x, )  M
n 
= .  
(2)  Let X  M
n
. The number  = X = sup{ > 0|S
1
(x, 
) osculatory at any x  X} is called the maximal 
osculatory (tubular) radius at X.  
Remark 2.3. There exists an osculatory sphere at any point of 
M
n 
(see [19] ).  
Definition 2.4. 
 Let U  M
n
,U , be a relatively compact set, and let = T 
= x  (x, U). The number U = max{r|
n
(x, r) is 
connected for all r  U, x   T }, is called the maximal 
connectivity radius at U.  
 Note that the maximal connectivity radius and the 
maximal osculatory radius are interconnected by the following 
inequality ([1], Lemma 3.1):  
 
U
3
3
U
.  
 We are now able to present the main steps of Peltone’s 
proof, which generalizes both the result and method of proof 
of Cairns [19]:  
(1)  (a) Construct an exhaustion of M
n 
by compact sets 
{Ei}, generated by the pair (Ui, i), where:  
(i)  Ui is the relatively compact set Ei \  E
 
i 1 and  
(ii)  i is a number that controls the fatness of the 
simplices of the triangulation of Ei , to be 
constructed in (2) below, such that it will not 
differ to much on adjacent simplices, i.e.:  
(ii1)  The sequence ( i)i 1 descends to 0;  
(ii2) 2 i  i 1.  
 The numbers i are chosen such that they satisfy the 
following bounds: 
 
i
1
4
min{
U
i 1
,
U
i
,
U
i+1
}.  
(b)  (i) Produce a maximal set A, |A| 0, such that 
A  Ui satisfies: 
(i1)  a density condition, namely:  
d(a, b)  i/2 , for all a, b  A, and for all i  1; 
(i2)  a “gluing” condition for Ui,Ui+1 , i.e. 
their intersection is large enough.  
 Note that according to the density 
condition (i1), the following holds: For 
any i and for any x  i, there exists a 
 A such that d(x, a)  i/2.  
(ii)  Prove that the Dirichlet complex 
 
{
i
}  defined 
by the sets Ai is a cell complex and every cell 
has a finite number of faces (so that it can be 
triangulated in a standard manner).  
(2)  Consider first the dual complex , and prove that it is 
a Euclidean simplicial complex with a “good” density. 
Project then  on M
n
 (using the normal map). Finally, 
prove that the resulting complex   can be 
triangulated by fat simplices.  
2.2. Manifolds With Boundary 
 The idea of the proof in this case is to build first two fat 
triangulations: T1 of a product neighbourhood N of M
n 
in M
n 
and T2 of int M
n 
(its existence follows from Peltonen’s 
result), and then to “mash” the two triangulations into a new 
triangulation T, while retaining their thickness (see [18]). 
While the mashing procedure of the two triangulations is 
basically the classical one of [17], the triangulation of T1 has 
been modified, in order to ensure the thickness of the 
simplices of T1 (see [18], Theorem 2.9). To thicken 
triangulations one can use either the method used in [14], or, 
alternatively, the one developed in [5]. For the technical 
details, see [18].  
3. MAIN RESULT  
 The idea of the proof is to use the basic fact that M
n 
is -
compact, i.e. it admits an exhaustion by compact submani-
folds {Kj}j (see, e.g. [21]). This is a standard fact for metri-
zable manifolds. However, it is conceivable that the “cutting 
surfaces” Nij, i=1,….kj Nij = Kj, are merely C
0
, so even the 
existence of a triangulation for these hypersurfaces is not 
always assured, hence a fortiori that of smooth triangular-
tions. (See. e.g. [20] for a brief review of the results regar-
ding the existence of triangulations).  
 To show that one can obtain (by “cutting along”) smooth 
hypersurfaces, we briefly review the main idea of the proof 
10     The Open Mathematics Journal, 2009, Volume 2 Saucan and Katchalski 
of the -compactness of M
n 
(for the full details, see, for 
example [21]): Starting from an arbitrary base point x0  M
n
, 
one considers the interval I = I(x0)= {r> 0|
n
(x0, r) is 
compact}, where 
n
(x0, r) is as in Section 2. If I = , 
 1  n
(x, n), thence -compact. If I  , one constructs the 
compacts sets Kj, K0 = {x0}, Kj+1 = y kj 
n
(y, r(y)), where 
r(y) =  sup I(y). Then it can be shown that M
n
 = j>0 Kj, i.e. 
M
n 
is -compact. The smoothness of the surfaces Nij now 
follows from Wolter’s result [22] regarding the 2-differentia-
bility of the cut locus of the exponential map.  
 By [15] (see also [16]) the sets Kj and Nij admit thick 
triangulations and, moreover, these triangulations have 
thickness 1 = 1(n) and 2 = 2(n  1), respectively. One 
can thus apply repeatedly the “mashing” technique 
developed in [18], for collars of Nij in Kj and Kj+1, j  0, 
rendering a triangulation of M
n
, of uniform thickness  =  
(n) (see [14, 18]).  
Remark 3.1. Instead of the less known and more complicated 
method of [15], we could have employed the simpler and 
more intuitive one in [16]. However, the later one still makes 
use of the principal curvatures (at each point of) the 
manifold, thus using this method the “Differential Geometric 
content”, so to speak, of our proof would still be rather 
higher. Since we strive to obtain a proof that is as elementary 
as possible, i.e. using only (or mainly) the tools of 
elementary differential topology, we prefer to adopt the 
methods of Cairns’ earlier work.  
Remark 3.2. In some special cases a “purely” Euclidean 
construction can be achieved, without resorting to embeddings 
and piecewise-flat (or just piecewise-linear) approximations. 
Such a construction is provided in [23], where a method of 
dividing a non-compact hyperbolic manifold into (canonical) 
Euclidean pieces is described. Note that, moreover, these 
pieces can be easily subdivided into thick simplices using a 
number of the methods mentioned above (and in the 
bibliography).  
Remark 3.3. The opposite problem, that of extending a thick 
triangulation from the interior of a manifold to its boundary, 
is also worth considering, in itself and for its importance in 
Geometric Function Theory (see [7]).  
4. CURVATURES  
 We conclude with a few brief remarks regarding the role 
of curvature:, its possible “recovery” in the PL context, and 
further directions of study.  
 First, let us note that, manifestly enough, our simplified 
proof is not “Differential Geometry free”, since we used both 
the exponential map and Wolter’s result. Regarding an 
“elementary” proof, this is evidently a weakness, as is the 
appeal we have made to Nash’s Embedding Theorem (with 
its complicated Differential Geometric apparatus). One 
would be tempted to substitute for the balls 
n
(x, j), 
Euclidean balls B (x, j), and replace, if necessary, the 
surfaces Ljk = B (x, j)  M
n 
by C
1 
surfaces 
 
L
j
*
 that are -
close to Lj. Such an approach would allow us to consider 
only C
1 
em-beddings, thus permitting us to dispense with the 
use of Nash’s Theorem. Unfortunately, in general, one 
cannot discard the use of the exponential map (and, 
consequently, neither can one ignore Wolter’s work). Indeed, 
for wildly embedded manifolds, the simple method above is 
not applicable. However, if only tame embeddings are 
considered, than the method above can be employed.  
 Secondly, let us note that Wolter’s method is – obviously 
– not independent of curvature. Therefore, the curvature 
considerations do play a decisive role, even if in a more 
“soft” manner. Moreover, even if applying the very simple 
and direct method described in the previous paragraph, the 
curvature plays an intrinsic role in determining the meshes of 
the triangulations of two adjacent “pieces” Kj and Kj+1, (Kj = 
B
 
(x, j)  M
n
), and their common boundary 
 
L
jk
*
.
 
For this 
reason, a “naive” exhaustion of the manifold using Euclidean 
balls, hence without control of curvature, may prove, in 
general, to be counterproductive, rendering not a monotone 
(and even highly oscillating) series of meshes for the 
members of the exhaustion.  
 Finally, as noted in the Introduction, the (Lipschitz-
Killing) curvatures have “good” convergence properties, 
under piecewise-flat (secant) approximations of M
n
. This 
result of Cheeger et al. (see [14]) is the goal for which they 
developed the “mashing of triangulations” technique 
mentioned above. However, a simpler and more direct 
approach to curvatures computation in PL approximation, 
along the lines indicated in [24] and based on metric 
curvatures, is currently in preparation [25].  
ACKNOWLEDGMENTS 
 The first author wishes to express his gratitude to 
Professor Shahar Mendelson – his warm support is gratefully 
acknowledged. He would also like to thank Professor Klaus-
Dieter Semmler for bringing to his attention Wolter’s work. 
Second author’s research partially supported by the Japan 
Technion Society Research Fund.  
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Received: December 01, 2008 Revised: March 25, 2009  Accepted: March 27, 2009 
 
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The existence of thick triangulations – an “elementary” proof

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The existence of thick triangulations -- an "elementary" proof

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