Annales de la Faculté des Sciences de Toulouse

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A dual Moser–Onofri inequality and its extensions to higher dimensional spheres

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Large scale ocean models beyond the traditional approximation

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ANNALES
DE LA FACULTÉ
DES SCIENCES
Mathématiques
TERUAKI KITANO AND TAKAYUKI MORIFUJI
A note on Riley polynomials of 2-bridge knots
Tome XXVI, no 5 (2017), p. 1211-1217.
<http://afst.cedram.org/item?id=AFST_2017_6_26_5_1211_0>
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Annales de la faculté des sciences de Toulouse Volume XXVI, no 5, 2017
pp. 1211-1217
A note on Riley polynomials of 2-bridge knots (∗)
Teruaki Kitano (1) and Takayuki Morifuji (2)
ABSTRACT. — In this short note we show the existence of an epi-
morphism between groups of 2-bridge knots by means of an elementary
argument using the Riley polynomial. As a corollary, we give a classifica-
tion of 2-bridge knots by Riley polynomials.
RÉSUMÉ. — Dans cette note nous montrons l’éxistence d’un epimor-
phism entre les groupes des noeuds à deux ponts par un argument élémen-
taire en utilisant le polynôme de Riley. Comme corollaire, nous donnons
une classification des noeuds à deux ponts par polynômes de Riley.
1. Introduction
Let K = S(α, β) be a 2-bridge knot given by the Schubert normal form
(see [1, Chapter 12]). Here α > 0 and β are relatively prime odd integers
satisfying −α < β < α. We denote the set of all such pairs (α, β) by S.
Two knots S(α, β) and S(α′, β′) are equivalent if and only if α = α′ and
β′ ≡ β±1 (mod α). In this note, we will identify S(α, β) with its mirror
image S(α, β)∗ = S(α,−β). Namely we will consider two 2-bridge knots
S(α, β) and S(α′, β′) to be equivalent if and only if α = α′ and either
β′ ≡ β±1 (mod α) or β′ ≡ −β±1 (mod α).
The knot group G(K) = π1(S3 \K) of K = S(α, β) has a presentation
G(K) = 〈x, y |wx = yw〉
(*) Reçu le 1er décembre 2015, accepté le 22 septembre 2016.
Keywords: Riley polynomial, 2-bridge knot, epimorphism.
2010 Mathematics Subject Classification: 57M25.
(1) Department of Information Systems Science, Faculty of Science and Engineering,
Soka University, Tangi-cho 1-236, Hachioji, Tokyo 192-8577, Japan — kitano@soka.ac.jp
(2) Department of Mathematics, Hiyoshi Campus, Keio University, Yokohama
223-8521, Japan — morifuji@z8.keio.jp
Article proposé par Stepan Orevkov.
– 1211 –
Teruaki Kitano and Takayuki Morifuji
where w = xε1yε2 · · ·xεα−2yεα−1 and εi = (−1)[
β
α i]. Here we write [r] for
the greatest integer less than or equal to r ∈ R. It is easily checked that
εi = εα−i holds. The above presentation is not unique for a 2-bridge knot
K itself, but the existence of at least one such presentation follows from
Wirtinger’s algorithm applied to the Schubert normal form of S(α, β). The
generators x and y come from the two overpasses and present the meridian
of S(α, β) up to conjugation.
A representation ρ : G(K) → SL(2;C) is called parabolic if tr ρ(x) =
tr ρ(y) = 2 holds and ρ is nonabelian (i.e. Im ρ is a nonabelian subgroup of
SL(2;C)). We consider a map ρ from {x, y} to SL(2;C) given by
x 7→ X =
(
1 1
0 1
)
and y 7→ Y =
(
1 0
−u 1
)
,
where u 6= 0 ∈ C. Moreover we define a matrix W = (wij) by W =∏(α−1)/2
l=1 X
ε2l−1Y ε2l . Riley proved in [6, Theorem 2] that the above map
ρ gives a parabolic representation of G(K) if and only if w11 = 0 holds. It
is easy to see that any parabolic representation of G(K) can be realized in
the above form up to conjugation. We call this polynomial w11 ∈ Z[u] the
Riley polynomial of K = S(α, β) and denote it by φS(α, β)(u). Then we have
a map R : S → Z[u] defined by the correspondence (α, β) 7→ φS(α, β)(u).
The purpose of the present note is to show the following result.
Theorem 1.1. — Let K1 = S(α1, β1), K2 = S(α2, β2) be 2-bridge
knots. If φS(α1, β1)(u) = φS(α2, β2)(u), then there exists an isomorphism
G(K1)→ G(K2) and thus K1 and K2 are equivalent.
Remark 1.2. — As we will see in the next section, the Riley polynomial
is not determined by the knot group itself (see Example 2.1). Hence the
converse statement of Theorem 1.1 does not hold. In other words, the map
R : S → Z[u] does not descend to the quotient S/ ∼, the set of equivalence
classes of 2-bridge knots.
The above theorem directly follows from Theorem 3.1 which states the
existence of an epimorphism between groups of 2-bridge knots. In the next
section, we quickly review some properties of the Riley polynomial. We will
prove Theorem 1.1 in Section 3. In the last section, we give a proof of a
special case of Hartley–Murasugi’s result to make this note as self-contained
as possible.
– 1212 –
A note on Riley polynomials of 2-bridge knots
2. Properties of Riley polynomials
The definition of the Riley polynomial φS(α, β)(u) depends on a choice
of a presentation of the knot group G(K). Namely G(K) itself does not
determine φS(α, β)(u).
Example 2.1. — Let us consider a pair of equivalent 2-bridge knots K =
S(7, 3) and K ′ = S(7, 5). They have the isomorphic knot groups G(K) ∼=
G(K ′) but different ε-sequences
(εi) = (1, 1,−1,−1, 1, 1) and (ε′i) = (1,−1, 1, 1,−1, 1).
By a straightforward calculation, we have
φS(7,3)(u) = 1− 2u+ u2 − u3 and φS(7,5)(u) = 1− 2u− 3u2 − u3.
Namely it shows that φS(7,3)(u) 6= φS(7,5)(u) although K = S(7, 3) and
K ′ = S(7, 5) are equivalent.
On the other hand, a 2-bridge knot S(α, β) and its mirror image
S(α, β)∗ = S(α,−β) share the Riley polynomial.
Claim 2.2. — Let K = S(α, β) and K∗ = S(α,−β) its mirror image.
Then we have φS(α,−β)(u) = φS(α, β)(u).
Proof. — Since [r] + [−r] = −1 holds for r ∈ R \ Z, ε-sequences for K
and K∗ satisfy ε∗i = −εi for 1 6 i 6 α − 1. By the inductive argument on
the length of ε-sequence, we can show that
w∗11 = w11, w∗12 = −w12, w∗21 = −w21, w∗22 = w22
for
W = (wij) =
(α−1)/2∏
l=1
Xε2l−1Y ε2l
and W ∗ = (w∗ij) =
(α−1)/2∏
i=1
Xε
∗
2l−1Y ε
∗
2l .
Therefore we have φS(α,−β)(u) = φS(α, β)(u). 
Let S+ be the subset of S satisfying β > 0 and further
S∗+ =
{
(α, β) ∈ S+
∣∣ ∃ (α, β′) ∈ S+ s. t. ββ′ ≡ 1 (mod α) and β′ < β}.
We define S to be S+ \ S∗+. As an immediate corollary of Theorem 1.1 and
Claim 2.2, we have the following.
Corollary 2.3. — The restriction of R, namely R|S : S → Z[u] is
injective.
– 1213 –
Teruaki Kitano and Takayuki Morifuji
3. Proof of Theorem 1.1
Let Ki = S(αi, βi) (i = 1, 2) be 2-bridge knots and φS(αi, βi)(u) (i = 1, 2)
their Riley polynomials. To prove Theorem 1.1, we first show the following.
Theorem 3.1. — If φS(α2, β2)(u) is a factor of φS(α1, β1)(u), then there
exists an epimorphism from G(K1) to G(K2).
Proof. — Now we fix the presentations of G(K1) and G(K2) as in Sec-
tion 1:
G(K1) = 〈x1, y1 |w1x1 = y1w1〉, G(K2) = 〈x2, y2 |w2x2 = y2w2〉.
Assume φS(α1, β1)(u) = φS(α2, β2)(u)ψ(u) for some ψ(u) ∈ Z[u], where the
degree of φS(α1, β1)(u) is n + k and deg φS(α2, β2)(u) = n. Let u1, u2, . . . , un
be the zeros of φS(α2, β2)(u) counting multiplicity. They are also zeros of
φS(α1, β1)(u) by the assumption.
Since each zero of the Riley polynomial corresponds to a parabolic rep-
resentation of the knot group, we can consider all parabolic representations
ρ11, ρ
1
2, . . . , ρ
1
n+k for G(K1) and ρ21, ρ22, . . . , ρ2n for G(K2), where
ρ1j (x1) = ρ2j (x2) =
(
1 1
0 1
)
and ρ1j (y1) = ρ2j (y2) =
(
1 0
−uj 1
)
hold for 1 6 j 6 n. We take the direct products of these representations as
Φ1 : G(K1) 3 g 7→
(
ρ1j (g)
)
j=1,...,n+k ∈ SL(2;C)
n+k,
Φ2 : G(K2) 3 g 7→
(
ρ2j (g)
)
j=1,...,n ∈ SL(2;C)
n
and put Γi = ImΦi which is a subgroup of SL(2;C)n+k or SL(2;C)n gen-
erated by Φi(xi) and Φi(yi). The natural projection to the first n factors
p : SL(2;C)n+k → SL(2;C)n induces a homomorphism p : Γ1 → SL(2;C)n,
and clearly p(Φ1(x1)) = Φ2(x2) and p(Φ1(y1)) = Φ2(y2) hold. Hence we
obtain an epimorphism p : Γ1 → Γ2.
By Thurston’s hyperbolization theorem for Haken 3-manifolds, any knot
in the 3-sphere S3 is either a torus knot T (p, q), or a satellite knot, or a
hyperbolic knot, i.e. its complement admits a complete hyperbolic metric
with finite volume (see [10]). By [8] a knot with bridge number 2 cannot
be a satellite, hence a 2-bridge knot is a torus knot T (2, q) where q is odd
(because the bridge number of T (p, q) is min{|p|, |q|} by [8]), or a hyperbolic
knot. So let us consider the following two cases.
(1) If K2 is hyperbolic, there exists a parabolic and faithful representa-
tion of G(K2) into SL(2;C) (see [10]). Hence Φ2 is injective. Apply-
ing the inverse of Φ2, we obtain an epimorphism
Φ−12 |Γ2 ◦ p ◦ Φ1 : G(K1)→ G(K2) .
– 1214 –
A note on Riley polynomials of 2-bridge knots
This completes the proof in this case.
(2) If K2 = T (2, q), a torus knot with an odd integer q, there is a
parabolic faithful representation G(K2)/Z(G(K2)) → PSL(2;R) ⊂
PSL(2;C), where Z(G(K2)) ∼= Z is the center of G(K2) (see [9]). In
fact, the image of G(K2)/Z(G(K2)) is isomorphic to the (2, q,∞)-
triangle Fuchsian group in PSL(2;R). This homomorphism lifts to
SL(2;C) and the lift is also faithful, so we can apply the same
argument as in (1). Namely we have an epimorphism G(K1) →
G(K2)/Z(G(K2)). Finally by a result of Hartley–Murasugi [3] (see
also Section 4), this epimorphism lifts to G(K2).

Proof of Theorem 1.1. — If φS(α1, β1)(u) = φS(α2, β2)(u), we obtain two
epimorphisms ϕ12 : G(K1) → G(K2) and ϕ21 : G(K2) → G(K1) by Theo-
rem 3.1. Hence we have the epimorphism ϕ21 ◦ϕ12 : G(K1)→ G(K1). Since
any knot group G(K) is known to be Hopfian (this follows from the facts
that any knot group is residually finite [4] and a finitely generated, residually
finite group is Hopfian [5]), namely, every epimorphism of G(K) onto itself
is an isomorphism, we see that ϕ12 and ϕ21 are injective. Therefore we can
conclude G(K1) ∼= G(K2). Every 2-bridge knot is prime, so K1 is equivalent
to K2 (see [2, Corollary 2.1]). This completes the proof. 
4. Lifting problems
In this section, we prove that any epimorphism from any knot group onto
the quotient of a torus knot group by its center can lift to the torus knot
group. This is a special case of the result of Hartley and Murasugi (see [3]),
but here we give a proof to make this note self-contained.
Let K be a knot and G(K) its knot group with the abelianization α :
G(K)→ Z ∼= 〈m〉. We simply write G(p, q) to G(T (p, q)) for the (p, q)-torus
knot T (p, q). It is known that G(p, q) has the following presentation
G(p, q) = 〈x, y |xp = yq〉
and its center is the infinite cyclic group generated by z = xp = yq. We write
π : G(p, q)→ G(p, q) = G(p, q)/〈z〉 = 〈x, y |xp = yq = 1〉
and γ : Z = 〈m〉 3 m 7→ m ∈ Z/pq = 〈m |mpq = 1〉. Further we take the
abelianization β : G(p, q) 3 x 7→ mq, y 7→ mp ∈ Z/pq.
Proposition 4.1 (Hartley–Murasugi [3]). — If ϕ : G(K) → G(p, q) is
an epimorphism such that β ◦ ϕ = γ ◦ α, then there exists a lift ϕ : G(K)→
G(p, q) of ϕ : G(K)→ G(p, q) such that π ◦ ϕ = ϕ.
– 1215 –
Teruaki Kitano and Takayuki Morifuji
Proof. — The torus knot group G(p, q) = γ∗(G(p, q)) can be described
as the fiber product
γ∗(G(p, q)) γ̃ = π−−−−→ G(p, q)
β̃
y yβ
Z −−−−→
γ
Z/pq.
More precisely, the fiber product γ∗(G(p, q)) is a subgroup of G(p, q)×Z as
γ∗
(
G(p, q)
)
=
{
(g,ms) ∈ G(p, q)× Z |β (g) = γ(ms)
}
.
Since there exists an exact sequence
1→ Ker(π) = 〈(1,mpq)〉 → γ∗(G(p, q))→ G(p, q)→ 1,
we can see G(p, q) ∼= γ∗(G(p, q)).
By the assumption, we have an epimorphism ϕ : G(K) → G(p, q) such
that β ◦ ϕ = γ ◦ α. For any g ∈ G(K) we then define
ϕ(g) = (ϕ(g), α(g)) ∈ G(p, q)× Z ,
and it gives an epimorphism ϕ : G(K)→ G(p, q) satisfying π ◦ ϕ = ϕ. 
Acknowledgments
After finishing this work, Makoto Sakuma kindly told us the paper [7]
which shows the existence of an epimorphism between groups of 2-bridge
links from the view point of Markoff trace maps. This paper was written
while the authors were visiting Aix-Marseille University. They would like to
express their sincere thanks for the hospitality. The authors would like to
thank Michel Boileau and Michael Heusener for helpful comments. They also
would like to thank an anonymous referee for his/her useful comments. This
research was supported in part by JSPS KAKENHI 25400101 and 26400096.
Bibliography
[1] G. Burde, H. Zieschang & M. Heusener, Knots, 3 ed., De Gruyter Studies in
Mathematics, vol. 5, Walter de Gruyter, 2014, xiii+417 pages.
[2] C. M. Gordon & J. Luecke, “Knots are determined by their complements”, J. Am.
Math. Soc. 2 (1989), no. 2, p. 371-415.
[3] R. Hartley & K. Murasugi, “Homology invariants”, Can. J. Math. 30 (1978),
p. 655-670.
– 1216 –
A note on Riley polynomials of 2-bridge knots
[4] J. Hempel, “Residual finiteness for 3-manifolds”, in Combinatorial group theory and
topology (Alta, Utah, 1984), Annals of Mathematics Studies, vol. 111, Princeton
University Press, 1987, p. 379-396.
[5] A. I. Mal′tsev, “On isomorphic matrix representations of infinite groups”, Rec.
Math. Moscou 8 (1940), p. 405-422.
[6] R. Riley, “Parabolic representations of knot groups, I”, Proc. Lond. Math. Soc. 24
(1972), p. 217-242.
[7] M. Sakuma, “Epimorphisms between 2-bridge knot groups from the view point of
Markoff maps”, in Intelligence of low dimensional topology 2006 (Hiroshima, Japan),
Series on Knots and Everything, vol. 40, World Scientific, 2007, p. 279-286.
[8] H. Schubert, “Über eine numerische Knoteninvariante”, Math. Z. 61 (1954), p. 245-
288.
[9] P. Scott, “The geometries of 3-manifolds”, Bull. Lond. Math. Soc. 15 (1983), p. 401-
487.
[10] W. P. Thurston, Three-dimensional geometry and topology. Vol 1., Princeton Math-
ematical Series, vol. 35, Princeton University Press, 1997, x+311 pages.
– 1217 –


A note on Riley polynomials of 2-bridge knots

Matthew Paddick

Transverse nonlinear instability of Euler–Korteweg solitons

Leonard Gross

Hypercontractivity for local states of the quantized electromagnetic field.

Lev Soukhanov

ANNALES
DE LA FACULTÉ
DES SCIENCES
Mathématiques
LEV SOUKHANOV
Diffusion-orthogonal polynomial systems of maximal weighted degree
Tome XXVI, no 2 (2017), p. 511-518.
<http://afst.cedram.org/item?id=AFST_2017_6_26_2_511_0>
© Université Paul Sabatier, Toulouse, 2017, tous droits réservés.
L’accès aux articles de la revue « Annales de la faculté des sci-
ences de Toulouse Mathématiques » (http://afst.cedram.org/), implique
l’accord avec les conditions générales d’utilisation (http://afst.cedram.
org/legal/). Toute reproduction en tout ou partie de cet article sous quelque
forme que ce soit pour tout usage autre que l’utilisation à fin strictement
personnelle du copiste est constitutive d’une infraction pénale. Toute copie
ou impression de ce fichier doit contenir la présente mention de copyright.
cedram
Article mis en ligne dans le cadre du
Centre de diffusion des revues académiques de mathématiques
http://www.cedram.org/
Annales de la faculté des sciences de Toulouse Volume XXVI, no 2, 2017
pp. 511-518
Diffusion-orthogonal polynomial systems of maximal
weighted degree
Lev Soukhanov (1)
ABSTRACT. — A diffusion-orthogonal polynomial system is a bounded
domain Ω in Rd endowed with the measure µ and the second-order elliptic
differential operator L, self adjoint w.r.t L2(Ω, µ), preserving the space
of polynomials of degree 6 n for any n. This notion was initially defined
in [2], and 2-dimensional models were classified.
It turns out that the boundary of Ω is always an algebraic hypersurface
of degree 6 2d. It was pointed out in [2] that in dimension 2, when the
degree is maximal (so, equals 4), the symbol of L (denoted by gij) is a
cometric of constant curvature.
We present the self-contained classification-free proof of this property,
and its multidimensional generalisation.
RÉSUMÉ. — Un système de diffusion polynomial est la donnée d’un
domaine borné Ω ⊂ Rd équipé d’une mesure µ et d’un opérateur différen-
tiel elliptique d’ordre deux L, autoadjoint sur L2(Ω, µ), tels que l’espace
de polynômes de degré 6 n est invariant par L pour tout n. Cette notion
est introduite dans [2] et les modèles en dimension 2 y sont classifiés.
Une conséquence est que la frontière de Ω est toujours une hypersur-
face algébrique de degré 6 2d. D’après [2], en dimension 2 et lorsque le
degré est maximal (égal à 4) le symbole gij de L est une cométrique de
courbure constante.
Nous présentons une démonstration indépendante de cette propriété
valable en toute dimension.
1. Introduction
Definition 1.1. — A model is a triple (Ω, µ, L), such that Ω is a
bounded domain in Rd, µ is a measure on Ω, L is a second-order differential
elliptic operator, self-adjoint in L2(Ω, µ), preserving the space of polynomials
of degree 6 n for any n:
L := gij∂i∂j + hi∂i
(1) — amligart@yandex.ru
The study has been partially supported by the Russian Academic Excellence Project
‘5-100’, National Research University Higher School of Economics, Simons Foundation .
– 511 –
Lev Soukhanov
This notion naturally extends the notion of classical orthogonal polyno-
mial systems (we restrict ourselves only to the bounded case). It was initially
defined in [2], and full classification of these systems in dimension 2 was ob-
tained.
Also, there is a weighted case: let us consider the natural numbers w1, . . . ,
wd > 1, called “weights” of the variables x1, . . . , xd.
The space Wk of polynomials of weighted degree 6 k is the span of
monomials of the form xs11 . . . x
sd
d such that
∑
siwi 6 k. We will say that
the polynomial has a weighted degree 6 k if it is contained in the space Wk.
Then one can introduce the following definition:
Definition 1.2. — A weighted model (for the given weights) is defined
in a same way as a model, but L preserves the spaces Wk instead.
At the moment, there is no known classification of weighted models even
in degree 2.
The following important property holds: the boundary of Ω is a (part
of) algebraic hypersurface and has a weighted degree at most 2(
∑
wi) (it is
proved in [2] and will be explained throughout the text).
Definition 1.3. — The weighted model is called maximal if the degree
of boundary of Ω is exactly 2(
∑
wi).
Remark 1.4. — In [2] it has been noted as a consequence of the classifica-
tion that in the non-weighted case, the curvature of the metric gij = (gij)−1
is constant and non-negative for all maximal models.
We explain this fact and prove it in a much broader scope.
Theorem (Theorem 2.9). — For a maximal weighted model of any di-
mension the riemannian metric gij = (gij)−1 is of constant scalar curvature.
We are currently unable to prove (either in the weighted case, or even in
non-weighted, but without using classification) that the curvature should be
non-negative.
In dimension 2, the constant scalar curvature property implies that the
metric is locally modelled on the round sphere, the euclidean plane or the
hyperbolic plane. Then we make use of other invariants (covariant derivatives
of the curvature tensor) and extend this property to higher dimensions:
Theorem (Corollary 3.10). — The riemannian metric of the maximal
model is locally homogeneous (in any dimension).
– 512 –
Diffusion-orthogonal polynomial systems of maximal weighted degree
The author wants to thank Dominique Bakry, Stepan Orevkov, Stefan
Nemirovsky, Vadim Schechtman, Misha Verbitsky and Dmitry Korb for use-
ful discussions on the subject and also Jean-Pierre Otal and the anonymous
referee for pointing out multiple inaccuracies and grammar mistakes in the
early versions of the paper. The author also thanks the university of Toulouse
for their hospitality.
2. Scalar curvature of g
2.1. Necessary facts about diffusion-orthogonal models
Now we are going to present the list of some useful results of [1] and [2]
about (weighted) models.
Property 2.1. — Measure µ is always a smooth function multiple of
the Lebesgue measure on Ω. We will denote this function as µ by abuse of
notation.
Property 2.2. — Hence, by self-adjointness, L is of the form 1µ∂iµg
ij∂j.
Property 2.3. — It is quite easy to see that gij is a symbol of L, and
then gij must be a polynomial of weighted degree 6 (wi+wj) (if not, L won’t
preserve Wk’s).
Property 2.4. — The boundary of Ω is always contained in the hyper-
surface det(gij) = 0, hence has a weighted degree at most 2(
∑
wi).
Denote det(gij) by D, G = D−1.
Property 2.5. — For a maximal model the polynomial gradg(D) =
gij∂iD is divisible by D.
These properties are contained in Theorem 5.1 of [1].
2.2. Strategy of the proof
In the two following subsections we prove two facts about the scalar
curvature Sc(g). In the first one we obtain by inspection of the formula that
Sc(g) is a rational function of weighted degree 6 0. In the second we prove
that it is, in fact, polynomial, hence, constant.
In what follows the degree of a rational function P/Q is defined as
deg(P )− deg(Q).
– 513 –
Lev Soukhanov
2.3. Formula for Sc(g)
The metric is defined from the cometric as
gij = (gij)−1
gij = ĝijD−1
By the maximality of the model degree
deg(D−1) = −2
∑
wi
and so
deg(ĝij) 6 (2
∑
wi)− (wi + wj)
deg(gij) 6 −wi − wj
The Christoffel symbols are defined as follows:
Γijk = ∂kgij − ∂jgki − ∂igjk
Γkij = gklΓkij
The degrees are as follows: (partial along i derivation lowers degree by wi)
deg(Γijk) 6 −wi − wj − wk
deg(Γkij) 6 wk − wi − wj
The formula of the Riemann curvature tensor is:
Rlijk = ∂iΓljk − ∂jΓlik + ΓsikΓljs − ΓlisΓsjk
The degree is
deg(Rlijk) 6 wl − wi − wj − wk
The scalar curvature is the following contraction:
Sc = Rjijkg
ik
So, its degree is
deg(Sc) 6 0
2.4. Scalar curvature is a polynomial
Claim 2.6. — D vanishes on the boundary of Ω with multiplicity 1.
Proof. — Otherwise, the degree of the boundary would be strictly less
than the degree of D, which contradicts maximality of Ω. 
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Diffusion-orthogonal polynomial systems of maximal weighted degree
Recall also that, by the Property 2.5, gij∂jD is divisible by D.
Now we are going to exploit this fact to substitute the coordinates in
such a way that makes the metric g invertible. For the rest of this part we
consider all equations to be defined over C. We denote the complexification
of the symmetric 2-form g by the same letter.
Consider a variety S (“double cover”, defined by the equation z2 = D in
the space Cd+1 with coordinates (x1, . . . , xd, z)). It is obviously smooth at
the points where D 6= 0, and it is also smooth at the smooth points of the
set {z = 0, D = 0} (because D vanishes with multiplicity 1). Let us denote
by SR the set (Ω×R)∩ S - set of real points of S projected into Ω. Also we
denote the projection from S to Cd as π, the set of smooth points of S as
SSm, the set {p ∈ Cd|D(p) = 0} as ∆.
Lemma 2.7. — The pullback g̃ = π?g is an invertible (as a section of
S2(TSSm)) regular symmetric 2-form on SSm. Its restriction on the smooth
points of SR is a smooth riemannian metric.
Proof. — At first, notice that the pullback of g (either metric or cometric)
is well defined on the preimage of Cd\{D = 0} because π is a covering
away from ∆. So, g̃ is continued from an open dense subset on SSm as a
symmetric 2-form with rational coefficients on SSm, possibly having poles
in the ramification divisor π−1∆.
The rest is to check the absence of poles, which will be a local analysis
in a neighborhood of the smooth point of π−1∆.
Pick local coordinates y0, . . . , yd−1 in such a way that the local equation
of the boundary is y0 = D = 0 (this is possible by the implicit function
theorem), and the point p is (0, . . . , 0). Denote by J the Jacobian of the
change of coordinates x→ y.
By the Property 2.5, gradg(y0)/y0 is regular, which means that the coef-
ficients gy0y∗ are divisible by y0. Also, gy0y0 is not divisible by y20 (otherwise,
det(g) = DJ−1, and, hence, D would vanish with multiplicity > 1).
For the neighborhood of S around p pick the local coordinates
ỹ20 = z2 = y0
ỹi = yi.
By the chain rule:
∂ỹi = ∂yi
∂ỹ0 = 2ỹ0∂y0
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Lev Soukhanov
Hence, in the ỹ-coordinates,
gỹ∗ỹ∗ = gy∗y∗ , ∗ 6= 0
gỹ0ỹ∗ = gy0y∗/(2ỹ0), ∗ 6= 0
gỹ0ỹ0 = gy0y0/(4ỹ20)
Notice that, as gy0y∗ and gy∗y∗ are divisible by y0, these coefficients do
not blow up. Also notice that the determinant of the matrix g is divided
by ỹ20 = y0 totally (first row and first column are divided by ỹ0 each), so it
becomes a non-vanishing function (in the y-coordinates it was yJ). Hence,
the form g is regular and invertible in the ỹ-coordinates.
Moreover, the form g is positive-definite at smooth points of SR (as it is
invertible and positive definite in an open dense subset), so it is a smooth
riemannian metric on the smooth part of SR. 
Theorem 2.8. — The scalar curvature is a polynomial.
Proof. — Let us denote the irreducible components of the polynomial D
as D = D1 . . . Dk. There are no multiple components, because the degree
of the boundary is maximal. Also, for each Di there is smooth point on the
boundary such that Di vanishes in it (otherwise we could exclude this Di
from the equation of the boundary).
By the formula of the scalar curvature, it is a rational function of the
irreducible form P/Da11 . . . Dann (because all the divisions performed are di-
visions on the determinant of gij in the inverse matrix formula).
For each Di choose the smooth point pi on the boundary at which
Di(pi) = 0 but P (pi) 6= 0 (it can be done by a general position argument).
It suffices to check that the scalar curvature is finite at pi’s.
Scalar curvature is an invariant of the metric, so it can be calculated on
the preimages of the points pi’s on S instead (in the ỹ-coordinates). There, by
the previous lemma, we know that the metric is invertible smooth symmetric
2-form, hence the scalar curvature is finite. 
Theorem 2.9. — For a maximal degree model (gij) has constant scalar
curvature.
Proof. — Scalar curvature is a polynomial of degree 0 or less, so it is
constant. 
For the dimension 2, this theorem guarantees that the metric is locally
modelled on one of the three classical spaces: sphere, euclidean plane or
hyperbolic plane. In the higher dimensions the various possible refinements
of this theorem are possible.
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Diffusion-orthogonal polynomial systems of maximal weighted degree
3. Local homogeneity of g
The scalar curvature is important, but it is only the first in the big list
of local metric invariants.
Definition 3.1. — A function of g is a rational function, depending on
∂s1 . . . ∂skg
ij with only (possibly) det(g)n in a denominator.
Remark 3.2. — For example, the coefficients of (gij) and all of their
derivatives are functions of g (as the inverse matrix is ĝ/det(g), and ĝ is
polynomial in g).
Definition 3.3. — A local metric invariant I(g) is a function of g in-
variant under change of coordinates.
Remark 3.4. — For example, the scalar curvature, or, generally, any
contraction of the Riemann curvature tensor and the metric tensor is a local
metric invariant.
Definition 3.5. — The metric is called locally homogeneous if any two
points have isometric neighborhoods.
Definition 3.6. — An invariant is called a scalar Weyl invariant if it is
a full tensor contraction of gij’s, gij’s and covariant derivatives of Riemann
curvature tensor ∇s1 . . .∇srRlijk’s.
Theorem 3.7 (Prufer, Tricerri, Vanhecke). — A metric is locally homo-
geneous if and only if all its scalar Weyl invariants are constant.
This is a version of a famous result of Singer. It is proved in [3].
Lemma 3.8. — Any scalar Weyl invariant on a maximal model is a ra-
tional function of degree 0.
Proof. — It is by the same inspection of the formula as for scalar curva-
ture:
deg(Rlijk) = wl − wi − wj − wk
deg(gij) 6 wi + wj
deg(gij) 6 −wi − wj
deg(Γkij) 6 wk − wi − wj
deg(∂i) = −wi
So, the degree of any formula with these symbols 6 sum of weights of upper
indices minus sum of weights of lower indices. But, as the scalar invariant
is a full contraction, each lower index is paired with an upper index, so the
degree is 6 0 
– 517 –
Lev Soukhanov
Lemma 3.9. — Any local invariant on a maximal model is polynomial.
Proof. — The same proof as for scalar curvature, literally: it is a rational
function with only possibly powers of D in the denominator, and it does not
tend to infinity at smooth points of boundary. 
Corollary 3.10 (of the theorems above). — For a maximal model the
metric is locally homogeneous.
Bibliography
[1] D. Bakry, “Symmetric diffusions with polynomial eigenvectors”, https://arxiv.org/
abs/1403.7468v1, 2014.
[2] D. Bakry, S. Orevkov &M. Zani, “Orthogonal polynomials and diffusion operators”,
https://arxiv.org/abs/1309.5632v2, 2014.
[3] F. Prüfer, F. Tricerri & L. Vanhecke, “Curvature invariants, differential operators
and local homogeneity”, Trans. Am. Math. Soc. 348 (1996), no. 11, p. 4643-4652.
– 518 –


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