Information geometry in a global setting
Atsuhide MORI

Abstract. We begin a global study of information geometry. In this article, we
describe the geometry of normal distributions by means of positive and negative
contact structures associated to the suspension Anosov flows on Sol3 -manifolds.

1.

Introduction

Information geometry is basically a local study of a parameter space of afamily of probability distributions by means of differential geometry (see [1]).
We begin a global study of this object. In this article, we restrict ourselvesto the case of normal distributions. Then the parameter space is the upper
half-plane H = {(m, s) ∈ R2 | s > 0} with a certain non-Kähler metric and a
non-metric connection, where m denotes the mean and s the standard deviation.
The background of the metric and the connection is a certain potential function
D on H × H. On the other hand, we know that the product H × H is forparametrizing abelian surfaces. Precisely, the quotient of the product H × H
under the action of a Hilbert modular group is a non-compact singular complex
surface which parametrizes the isomorphism classes of complex abelian surfaces
with a real multiplication structure. Its topology was studied by Hirzebruch [4].
He took the quotient M of H×H under the action of a semi-direct product ZnZ2of lattices of R and R2 which acts on H × H freely and properly discontinuously.
Then the neighborhoods of ∞ form the end model for a Hilbert modular surface.
From contact topological point of view, M ∪ {∞} is the cone with positive
and negative symplectic structures whose base M0 is a Sol3 -manifold with thepositive and negative contact structures associated to the suspension Anosov
flow (see [6] and §4 below for the precise meaning). Now the result of this article
can be summarized as follows.
i) (The model in §3.) We describe the information geometry of H byintroducing a self-correspondence F ⊂ H × H which identifies the convolution operation for the probability densities in each factor of H × H
to the Bayesian learning for those in the other factor.
ii) (The propositions in §5.) The action of Z n Z2 preserves the metricand the connection on each factor of H × H. The sum (resp. thedifference) of the natural area forms descends to the positive (resp.
negative) symplectic structure on the quotient M . The surface F is2010 Mathematics Subject Classification. 57R17, 57R30, 62B10.
Key words and phrases. Information geometry, contact structure, foliation.

2

Atsuhide MORIa Lagrangian correspondence which descends to a densely immersed
Lagrangian submanifold L(⊂ M0 ) ⊂ M with respect to the negative
symplectic structure. Further L is decomposed into Legendrian submanifolds of M0 with respect to the positive contact structure.
iii) (Theorem 1.) We can take a contact Hamiltonian flow on the preimageof M0 such thata) the surface F is an invariant submanifold,b) the push-forward of the induced flow on F to each of the factorsof H × H is tangent to a foliation whose leaves are geodesics forthe non-metric connection, andc) the iterations of convolutions along the foliation of the first factor
correspond to those of Bayesian learnings along the foliation of
the second factor under F .
Note that we could take a similar contact flow just partially on the quotient. Indeed the contact Hamiltonian function for the flow in c) is the inverse coefficientof variance m/s on the first factor which is not preserved under the Z2 -action(but preserved under the Z-action). We raise open problems concerning the geometric characterization of the potential function D on H × H and the relation
between abelian varieties and pairs of normal distributions (§6).

2.

Information geometry

In a smooth setting of parametric statistics, we consider an open set U ⊂ Rn
and a positive function p on R × U with the normality condition
Z ∞p(x, X)dx ≡ 1.
−∞

Each point X ∈ U presents a random variable with probability density pX (x) =p(x, X). We consider the relative entropy
Z ∞D(X, Y ) =−∞pX (x) log

1
1
− log
pY (x)pX (x)

dx

for (X, Y ) ∈ U × U as a fundamental potential. It defines a separating premetric1
on U , which is called the Kullback-Leibler divergence in information geometry.
Using the tensor notation only in this section, we define the Fisher metric g and

1A premetric or prametric on a set U is a non-negative function on U × U which vanishes

along the diagonal set. If it is positive elsewhere, we say that it is separating. Each value of apremetric is called a distance.

Information geometry in a global setting3

the e(xponential)-connection ∇ by the cubic approximation
D(X, X + ∆X) ≈1 X
gij ∆X i ∆X j
2! i,j+

1 X(∂i gjk + Γjk,i )∆X i ∆X j ∆X k
3!
i,j,k



1 X
or D(X + ∆X, X) ≈gij ∆X i ∆X j
2! i,j+
1 X(∂i gjk + ∂j gki − Γij,k )∆X i ∆X j ∆X k
3!
i,j,k

of a small distance, where the coefficients in the expression(s) are supposed tobe symmetric, eg., ∂i gjk + Γjk,i = ∂j gki + Γki,j . The coefficients Γij,k of thee-connection all vanish for an exponential family!
nXipX (x) = exp
X fi (x) + f0 (x) − ν(X) ,i=1hence the name. Here fi (x) are any functions and ν(X) the normalizer. We also
consider the linear combinations of the e-connection and the metric connection.
For example, the m(ixture)-connection Γ∗jk,i = ∂j gki − Γij,k satisfies Γ∗ij,k = 0nXfor any mixture pX (x) =X l ql (x) of n + 1 probability densities ql (x), wherel=0

X 1 , . . . , X n > 0 and X 0 = 1 − X 1 − · · · − X n > 0. See the book [1] for more on
information geometry.
Hereafter we restrict ourselves to the case of the normal distributions,
namely we put1(x − m)2p(m,s) (x) = √
exp −
2s2
2πs
for (m, s) ∈ H = R × R>0 . Then the Kullback-Leibler divergence is expressed as
D((m, s), (m0 , s0 ))



Z ∞
(x − m)2(x − m)2(x − m0 )2
s
1
√
exp −
−
+
−
log
dx
=
2s2
2s2
2(s0 )2
s0
2πs
−∞
s2
(m0 − m)2
s
1
=− +
+
0
+
− log 0 .
2 2(s0 )22(s0 )2s
It is easy to read the Fisher metric g and the e-connection ∇ from the cubic
approximation
D((m + ∆m, s + ∆s), (m, s)) ≈1 ∆m2 + 2∆s2
1 ∆s3
−
.
2
2
s
3 s3

4

Atsuhide MORI

Namely,−2−6dm2 + 2ds2
, Γ12,1 = Γ21,1 = 3 , Γ22,2 = 3 ,s2ssand the other coefficients Γij,k vanish. Notethatthenormaldistributionsm −1
form an exponential family parametrized by,. This implies that thes2 2s2
geodesics for the e-connection are the horizontal lines, the vertical half-lines, andthe upper semi-parabolas. On the other hand, the geodesics for the m-connection
are vertical half-lines and upper semi-circles, which coincide with those for thedm2 + 1ds2
(6= g).
Poincaré metric
s2
g=

3.

The basic model

Let (m, s, M, S) denote the coordinate system of H × H. Take the correspondencem
1
F : M = − 2 and S =
s
s


m M
⇔
+
= 0 and sS = 1sS
and regard it as a submanifold of H × H. For any point (m, s, M, S) in H × H,
we put2mM+ exp(−h)+ exp(−2h) − 1 + 2hSs,f (s, m, S, M ) =2
where h = − log(sS).
Proposition 1. Suppose that under the correspondence F a point (m0 , s0 )of the first factor of H × H is identified with the point (M, S) of the second
factor, i.e., (m0 , s0 , M, S) ∈ F . Then the function f (m, s, M, S) presents the
Kullback-Leibler divergence D((m, s), (m0 s0 )).
Proof.

 
2
1 −m0
−m0
s m
s
s2
2f s, m, 0 , 0 2 =+·− 1 − 2 log 0+s (s )s0s0 s(s0 )2s= 2D((m, s), (m0 , s0 ))
We define the product (m, s)∗(m0 , s0 ) on the first factor of H×H by putting
p(m, s) ∗ (m0 s0 ) = m + m0 , s2 + (s0 )2 .
It represents the convolution of probability density functionsp(m,s)∗(m0 ,s0 ) (x) = (p(m,s) ∗ p(m,s) )(x).

Information geometry in a global setting

5

We define another product (M, S) · (M 0 , S 0 ) on the second factor by puttings!
M (S 0 )2 + (M 0 )S 2
S 2 (S 0 )20
0,(M, S) · (M , S ) =.
S 2 + (S 0 )2
S 2 + (S 0 )2This represents the normalized pointwise product.Z ∞p(M,S)·(M 0 ,S 0 ) = p(M,S) (x) · p(M 0 ,S 0 ) (x)p(M,S) (x) · p(M 0 ,S 0 ) (x)dx.
−∞

We see that a normalized pointwise product presents a Bayesian learning asfollows. Suppose that we have a prior probability density p(x) and a likelihoodq(x) of a new data. The likelihood q(x) is the conditional probability density ofthe data under p(x). Then we obtain the posterior probability density p(x)q(x)/rfrom Bayes’ rule, where r is the probabilityof the data. Since r does not dependZ∞

on x, we see that it is the normalizer

p(x)q(x)dx.
−∞

Proposition 2. The above correspondence F on H×H identifies the product ∗ on the first factor (resp. the second factor) with the product · on the second
factor (resp. the first factor).
Proof. Let (m, s, M, S) and (m0 , s0 , M 0 , S 0 ) be two points on F . Then wehaves

S 2 (S 0 )2
1
=p
2
0
2
2
S + (S )s + (s0 )2

and
m0 1
m 1
−
−
M (S 0 )2 + (M 0 )S 2
s2 (s0 )2(s0 )2 s2
=
2
0
2
1
1
S + (S )
+ 0 2
s2
(s )
−(m + m0 )= 2.
s + (s0 )2
Further F is invariant under the interchange of the order of the product H ×
H.

Thus the correspondence F defines a Fourier-like transformation on H, hence
the notation.
4.

Contact/foliation topology of Sol3 -manifolds

We recall contact/foliation topology of certain T 2 -bundles. First we should
notice that contact structure is soft enough to interest topologists since Gray’s
stability [3] says that any smooth homotopy of a contact structure on a closed(i.e., compact borderless) manifold can be realized as an isotopy of the manifold. Further the symplectization of a contact structure is well-defined up to6

Atsuhide MORI

isotopy which preserves the R-fibers. Note also that many foliators imagines contact topology as a “discretization” of foliation theory at least in 3-dimensionalcase. Honda [5] completed the isotopy classification of contact structures on T 2 bundles essentially in this spirit (precisely, by using convex surfaces instead ofholonomy of leaves). Let A ∈ SL(2, Z) be the monodromy map of a T 2 -bundle.
In the case where trA > 2, the T 2 -bundle possesses a canonical pair (ξ+ , ξ− )of positive and negative contact structures associated to the suspension Anosov
flow. Several authors studied the isotopy classes of these contact structures interms of Honda’s classification. Kasuya [6], without appealing to the classification, related the contact structures ξ± to Hirzebruch’s construction [4] of a cusp.
Hereafter we use relevant part of his description of the contact structures ξ± .
We notice that he also related it to Ryo Furukawa’s observation concerning thetoric construction of contact submanifold of S 5 in [9].
It is well-known thatanyelement A of SL(2, Z) with trA > 2 is conjugate
1 1
1 0
to a positive word ofand. Thus we can take a map
0 1
1 1
Z/rZ 3 k 7→ bk ∈ Z≥2
not identically 2 such that A is conjugate to−1 

sk tk
bk+r−1 1
bk+r−2
=
uk vk
−1
0
−1
Indeed we have

−1
1 0
1 1



1
0

1
1



1
1

0
1

bk −2 ! 

1
0

1
1




···

bk
−1





0
1

=

bk
−1

1
0

−1
.

1
0


.

Of course, one may start with a given recurring sequence bk ≥ 2 since the traceof the above composition map is greater than 2 unless identically bk = 2. The
continued fraction1
wk = bk −1
bk+1 −
bk+2 + · · ·is a quadratic irrational number satisfying
tk wk 2 − (sk − vk )wk − uk = 0.
We have 0 < wk < 1 < wk , where wk denotes the other solution of the quadratic
equation (i.e., the irrational conjugate of ω). Put c = log(w1 · · · wr ). Take any(large) positive constant K > 0 to fix the vectors

 
−1 

x1
1
−1
K
=
x2
w1 −w1
0
and


y1
y2




=

1
w1

−1
−w1

−1 

0
K


.

Information geometry in a global setting

7

Then we define the quotient manifold M (x, y, c) = H × H/ ∼ by the equivalences(m, s, M, S) ∼ (ec m, ec s, e−c s, e−c S),(m, s, M, S) ∼ (m + x1 , s, M + x2 , S),and(m, s, M, S) ∼ (m + y1 , s, M + y2 , S)
for any (m, s, M, S) ∈ H × H. Note that the map(m, s, M, S) 7→ (ec m, ec s, e−c s, e−c S)
preserves the lattice on the mM -plane generated by x = (x1 , x2 ) and y = (y1 , y2 ).
Thus M (x, y, c) is a 4-manifold diffeomorphic to a T 2 -bundle over the open
annulus S 1 × R.
Precisely, we take the function h = − log(sS) on H ×H ⊂ C × C. It is1
1,
is clearly positive
strictly plurisubharmonic since the Hessian diag4s2 4S 2
definite. Now we shift our ground to consider, instead of the complex structure
Jstd , the exact symplectic structure dλ+ with fixed primitive




h+t
h−t
dm dM
∗
+
= exp
dm + expdM,
λ+ = −dJstd
dh =
s
S
2
2
S
where t = log . The above strong pseudo-convexity is expressed as dh(X) > 0sfor the vector field X which is defined by ιX dλ+ = λ+ . The flow foliation of X
carries the holonomy invariant transverse contact structure which is defined by
the 1-form λ+ . Conversely the symplectic manifold H × H is the symplectizationof the section {h = 0} whose R-fibers are the flow lines of X. The function hdescends to the quotient M (x, y, c), so that the 0-level set is a closed contact
3-manifold M0 (x, y, c). Then M (x, y, c) is the symplectization of M0 (x, y, c).
The function t also descends to M (x, y, c), so that its restriction to the section
M0 (x, y, c) defines a T 2 -bundle projection to the circle R/2cZ. The T 2 -fiber is thequotient cof the mM
 -plane by the lattice generated by x and y. The monodromye0mapwith respect to the fundamental basis of the mM -plane can
0 e−c

−1
s1 t1
also be written as∈ SL(2, Z) with respect to the T 2 -basis {x, y}.
u1 v1This finally leads us to shift our ground to consider the contact structure ξ+ on
the T 2 -bundle with monodromy A.
We can take the global frame (e1 , e2 , e3 ) of T M0 (x, y, c) by putting
e1 = e−t/2 ∂m ,e2 = et/2 ∂M ,and e3 = 2∂t .

3

It satisfies the sol -relations[e1 , e2 ] = 0,[e3 , e1 ] = −e1 ,and[e3 , e2 ] = e2 .

The dual coframe (e∗1 , e∗2 , e∗3 ) satisfies the corresponding relationsde∗3 = 0, de∗1 = −e∗1 ∧ e3 , and de∗2 = e∗2 ∧ e∗3 ,8Atsuhide MORI

wheree∗1 = et/2 dm, e∗2 = e−t/2 dM, and e∗3 = dt/2.
We call M0 (x, y, c) the Sol3 -manifold associated to A. The suspension Anosov
flow is the flow generated by e3 . Since it contracts e1 and expands e2 , the 1forms α± = e∗1 ± e∗2 define a pair of positive and negative contact structures
ξ± , which we call the bi-contact structure associated to the flow. (This ξ+
coincides with the above one, hence we use the same notation.) The suspension
Anosov flow drifts the positive and negative contact structures toward the stable
Anosov foliation defined by the equation e∗1 = 0. Further we have the negative
symplectic form dλ− on M (x, y, c) which is the symplectization of the negative
contact structure ξ− , where




h−t
h+t
h/2 ∗
∗
dm − expdM
λ− = e (e1 − e2 ) = exp
2
2
We have the following contact 5-manifold which is crucial in contemporary contact topology since it shows how to generalize the classical propeller constructionof 3-dimensional contact structure to higher dimensions so that the propeller
keeps its relation to convex (hyper)surface theory especially to Giroux torsion(see [10] and [7]). Take η = e∗1 + cos θe∗2 + sin θdϕ on M0 (x, y, c) × S 1 × R, whereθ ∈ S 1 and ϕ ∈ R are the coordinates. It defines a positive contact structuresinceη ∧ (dη)2 = (e∗1 + cos θe∗2 + sin θdϕ)∧ (e∗3 ∧ e∗1 − sin θdθ ∧ e∗2 + cos θe∗2 ∧ e∗3 + cos θdθ ∧ dϕ)2= (4 cos4 θ + 2 sin2 θ)e∗1 ∧ e∗2 ∧ e∗3 ∧ dθ ∧ dϕ > 0.
We divide the hypersurface {ϕ = 0} into two regions Σ± according to the sign ofsin θ. Then the restrictions of the 2-form dη on ±intΣ± are symplectic structures,
where −intΣ− denotes intΣ− with reversed orientation. We can see that ±intΣ±are symplectomorphic to each other by switching the sign of θ-coordinate. We+
further divide the region Σ+ into two regions Σ++ and Σ− according to the signof cos θ. The following observation fits the scope of [8].
+Proposition 3. intΣ++ × R (resp. intΣ− × R) is contactomorphic toker(λ+ + dϕ) (resp. ker(λ− + dϕ)) on M (x, y, c) × R (resp. −M (x, y, c) × R).

Proof. We haveη= e(H+T )/2 dm ± e(H−T )/2 dM + dϕ,sin θ| cos θ|where H = log
and T = t − log | cos θ|.
sin2 θ



Namely, we consider the symplectic manifold intΣ+ as the result of thenext procedure expressing that scope. The 4-manifold M (x, y, c) with the conformal class of the positive symplectic structure dλ+ and its orientation-reversion−M (x, y, c) with the conformal class of the positive symplectic structure dλ− areInformation geometry in a global setting

9

pasted together so as to be realized as the symplectic region intΣ+ along theLevi-flat ends each of which is expressed as {h = +∞} ≈ M0 (x, y, c) and carries
the stable Anosov foliation e∗1 = 0. This can be viewed as the interchange ofthe sign of the bi-contact structure at the limit of the Anosov flow via the stable foliation. That is why we consider the positive symplectic structure dλ+ as
paired with the negative one dλ− on M (x, y, c). (This idea also comes from the
topology of Hirzebruch-Inoue or hyperbolic Inoue surfaces.)
5.

Results
Now we investigate the model in §3 in the light of the description in §4.

Proposition 4. The correspondence F and the function f are invariant
under the monodromy map (m, s, M, S) 7→ (ec m, ec s, e−c s, e−c S).
Proof. The monodromy map preserves the inverse coefficients of varianceMmand, and the strictly plurisubharmonic function h = − log(sS).
sS
Proposition 5. The symplectic structuredλ+ =dm ∧ ds dM ∧ dS
+
,
s2
S2

the sum of Fisher metrics
dm2 + 2ds2
dM 2 + 2dS 2
+
,
s2
S2
and the almost complex structure11
J : ∂m 7→ √ ∂s , ∂M 7→ √ ∂S2
2√
on H × H(3 (m, s, M, S)) satisfy g(·, ·) = 2dλ+ (·, J·) and descend to the quotient M (x, y, c).
g=

Proof. We have

and

√

√

1
1
2dλ+ (∂m , √ ∂s ) = 2s2√22dλ+ (∂s , − 2∂m ) = 2 .
s

The rest is clear.

Proposition 6. The “Fourier” correspondence F ⊂ (H × H, dλ+ ) is asmooth surface contained in the contact-type hypersurface H = {sS = 1} withpositive contact structure ker α+ . It is the union of Legendrian lines{eT /2 m + e−T /2 M = 0

andt = T}(T ∈ R).

These lines descend to the quotient M0 (x, y, c) of H as Legendrian curves which
form a dense immersion of the surface F .

10Atsuhide MORI

Proof. We see that
m M
+
= 0.
s
S
Together with sS = 1, this defines F . The Legendrian lines for T = T0 + 2c and
T = T0 descend to the same Legendrian immersed curve on (M0 (x, y, c), ker α+ ),which is either closed or dense in the toral fiber {t = T } ⊂ M0 (x, y, c) dependingon whether the slope −eT is rational or irrational with respect to the basis {x, y}.

et/2 m + e−t/2 M = 0⇔

Proposition 7. We have a negative symplectic structuredm ∧ ds dM ∧ dS
−
dλ− =
s2
S2
on H × H, with respect to which the surface F is a Lagrangian correspondence.
Proof. The tangent space of F is expressed as
T F = hs∂m − S∂M , m∂m + s∂s − M ∂M − S∂S i.
Then we havedm ∧ ds(s∂m , m∂m + s∂s ) dM ∧ dS(−S∂M , −M ∂M − S∂S )−= 1 − 1 = 0.
s2
S2
The correspondence F also satisfies the next interesting property.
Proposition 8. We call a geodesic for the e-connection an e-geodesic.
Any e-geodesic on the first factor of H × H corresponds to an e-geodesic on thesecond factor via F , and vice-versa.
Proof. We haveS(k1 s2 + k2 m + k3 ) = s(k3 S 2 + k2 M + k1 ) = 0,
where ki are any constants.



The next theorem shows that a certain contact flow describes Bayesianlearning processes.
Theorem 1. The contact Hamiltonian vector field Y of the restriction ofmthe inverse coefficient of variation
to the hypersurface {sS = 1} ⊂ H × Hswith respect to α+ is expressed asm
1
1
Y = e1 − e3 = m∂m − (S∂S − s∂s )|{sS=1} .
s2
2
It is tangent to the surface F . Let Yi denote the push-forward of the restriction
Y |F to the i-th factor of H × H (i = 1, 2). Then the flow of Y1 comes out of theorigin (0, 0) ∈ H along the half-parabolic (or half-liner) e-geodesics, and the flowof Y2 goes into the M -axis {S = 0} ⊂ H along the vertical half-liner e-geodesics.
The former can be discretized into the iterations of convolutions and therefore
the latter into those of Bayesian learnings.

Information geometry in a global setting





11





m M
1 m M
+
=
+
vanishes
s
S
2 s
S
1
along F . The Lie derivative of α+ is LY α+ = α+ . The Hamiltonian function
2
mis ιY α+ = . The rest is easy.
sProof. The interior product ιY d

Remark 1. Let α be a positive contact form on an oriented 3-manifold
M0 , i.e., a 1-form satisfying α ∧ dα > 0. Then the 2-form d(eh α) on the product
R × M0 = M (h ∈ R) defines the symplectic structure as the symplectizationof the contact structure. Given a function H on M0 , we can define the contact
Hamiltonian vector field Y for H as the well-defined push-forward of the usuale = eh H to M0 . Then we have
Hamiltonian vector field Ye for the function Hα(Y ) = H. Note that Y is contact, i.e., it preserves the contact structure ker αsince LYe eh α = ιYe d(eh α) + dιYe (eh α) = −d(eh H) + d(eh H) = 0. From thenon-integrability of the contact structure ker α, we see that a contact vector fieldY 0 with α(Y 0 ) = 0 must be zero. Thus any contact vector field is a contact
e and change the section of the R-fibration ofHamiltonian vector field. If we fix H
M , we obtain another pair of contact form ef α and contact Hamiltonian function
e−f H on the base manifold M0 which defines the same contact Hamiltonian
vector field Y . This means that, for a fixed contact structure, each contact form
presents an isomorphism between the space of functions and the space of contact
vector fields as well as a section of the symplectization.
6.

Problems

Since information geometry concerns parameter spaces with geometric structures, it would have some relation to moduli theory. This was one of the starting
points of this research. In the present, we have no intrinsic relation between thepairs of normal distributions and abelian surfaces.
Problem 1. Is there any number theoretical relation between two normal
distributions which enables us to relate the pair to an abelian surface with realmultiplication ?
The other starting point was the following splitting result proved in [8] (seealso [2]): A non-singular flow on a closed 3-manifold is projectively Anosov if
and only if it is simultaneously tangent to a mutually transverse pair of positive
and negative contact structures, i.e., to a bi-contact structure. In order to consider the topology of the mixed parameter space, as a closed manifold or as aspace with non-trivial topology, we have to know how to split it into individual
parameter spaces. This motivates us to consider Anosov flows. However, thiskind of hyperbolicity usually makes the dynamics chaotic under the compactnessof the space (like Arnold’s cat map). Thus it is reasonable to consider quotientmanifolds of an already split space. The author is trying to find another example
than the pair of normal distributions from this point of view.

12

Atsuhide MORI

There are still questionable points on our own model in this article. For
example,Problem 2. Is there any geometrical reason for taking the particular function f ?
The author consider them as future tasks and raises the above one as anopen problem.
References[1] Shun-ichi Amari, Information geometry and its applications, App. Math. Sci. , Springer,
Tokyo, 2016.
[2] Yakov Eliashberg and William Thurston, Confoliations, A. M. S. University Lecture Series
13 (1997).
[3] John Gray, Some global properties of contact structures, Ann. of Math. 69(2) (1959),
421450[4] Friedrich Hirzebruch, Hilbert modular surfaces, L’Ensignement Math. 19 (1973), 183-28.
[5] Ko Honda, On the classification of tight contact structures II, J. Diff. Geom. 55 (2000),
83-143.
[6] Naohiko Kasuya, The canonical contact structure on the link of a cusp singularity, Tokyo
J. Math. 37 (2014), 1-20.
[7] Patrick Massot, Klaus Niederkrüger, and Chris Wendl, Weak and strong fillability of higher
dimensional contact manifolds, Invent. Math. 192(2)(2013), 287-373.
[8] Yoshihiko Mitsumatsu, Anosov flows and non-Stein symplectic manifolds, Ann. Inst.
Fourier 45 (1995), 1407-1421.
[9] Atsuhide Mori, The Reeb foliation arises as a family of Legendrian submanifolds at theend of a deformation of the standard S 3 in S 5 , C. R. Acad. Sci. Paris, Ser. I 350 (2012),
67-70.
[10] Atsuhide Mori, On higher dimensional violation of Thurston-Bennequin inequality,
preprint (2015). This includes “Reeb foliations on S 5 and contact 5-manifolds violating
the Thurston-Bennequin inequality” (arXiv:0906.3237) and “On the violation of ThurstonBennequin inequality for a certain non-convex hypersurface” (arXiv:1111.0383).