Diﬀerential
geometry of
projective or
centroaﬃne
surfaces

Diﬀerential geometry of projective or
centroaﬃne surfaces

AtsushiFujioka.

Contents

Atsushi Fujioka

Introduction
Projective
surfaces
Projective
minimalsurfaces
Isothermallyasymptotic
surfaces
Centroaﬃne
surfaces
Centroaﬃne
minimalsurfaces..

.
Faculty of Engineering Science
Kansai University

March 2, 2013
Hokkaido University
Workshop on Geometry of Statistical Manifolds
and Related Topics
1 / 40Contents
Diﬀerential
geometry of
projective or
centroaﬃne
surfaces
AtsushiFujioka..

1

Introduction

2

Projective surfaces

3

Projective minimal surfaces

4

Isothermally asymptotic surfaces

5

Centroaﬃne surfaces

6

Centroaﬃne minimal surfaces

Contents
Introduction
Projective
surfaces..

Projective
minimalsurfaces
Isothermallyasymptotic
surfaces..

Centroaﬃne
surfaces
Centroaﬃne
minimalsurfaces..

2 / 40OverviewDiﬀerential
geometry of
projective or
centroaﬃne
surfaces
AtsushiFujiokaContents
Introduction
Projective
surfaces
Projective
minimalsurfaces
Isothermallyasymptotic
surfaces
Centroaﬃne
surfaces
Centroaﬃne
minimalsurfaces

Pn : the n-dimensional real projective space
Projective diﬀerential geometry: diﬀerential geometry ofsubmanifolds in Pn
Many classes of surfaces in projective diﬀerential geometry
related to integrable systems:◦ Projective minimal surfaces◦ Isothermally asymptotic surfaces..
.
Centroaﬃne minimal surfaces: A class of surfaces in centroaﬃne diﬀerential geometry(1994 C. P. Wang)
Integrable systems(2000 W. K. Schief)
Can be considered as projective
surfaces3 / 40References
Diﬀerential
geometry of
projective or
centroaﬃne
surfaces
AtsushiFujiokaContents
Introduction
Projective
surfaces
Projective
minimalsurfaces
Isothermallyasymptotic
surfaces
Centroaﬃne
surfaces◦ E. V. Ferapontov, Integrable systems in projective diﬀerential
geometry, Kyushu J. Math., 54 (2000), 183–215◦ C. Rogers and W. K. Schief, Bäcklund and Darboux
transformations. Geometry and modern applications insoliton theory. Cambridge Texts in Applied Mathematics,
2002◦ T. Sasaki, Projective diﬀerential geometry and linear
homogeneous diﬀerential equations, Rokko Lectures inMathematics, 5, 1999
◦ T. Sasaki, Line congruence and transformation of projective
surfaces, Kyushu J. Math., 60 (2006), 101–243..
.

Centroaﬃne
minimalsurfaces4 / 40Projective surfaces and surfaces in the Euclideanspace
Diﬀerential
geometry of
projective or
centroaﬃne
surfaces
AtsushiFujiokaContents
Introduction
Projective
surfaces
Projective
minimalsurfaces
Isothermallyasymptotic
surfacesz : D → P3 : a projective surface(x, y ): local coordinatesz(x, y ) = [z 1 (x, y ), z 2 (x, y ), z 3 (x, y ), z 4 (x, y )]z corresponds to a surface in R3 .
If z 1 6= 0,( 2 3 4)
z z z
ẑ :=
, ,
.
z1 z1 z1

Centroaﬃne
surfaces
Centroaﬃne
minimalsurfaces
5 / 40Symmetric 2-form
Diﬀerential
geometry of
projective or
centroaﬃne
surfaces
AtsushiFujiokaContents
Introduction
Projective
surfaces
Projective
minimalsurfacesz : D → P3 : a projective surface(x, y ): local coordinates
Assume zxy , zx , zy , z are linearly independent on D.
{
zxx = lzxy + azx + bzy + pz,
zyy = mzxy + czx + dzy + qz

(1)

Deﬁne a symmetric 2-form ϕ by
ϕ = ldx 2 + 2dxdy + mdy 2 .

Isothermallyasymptotic
surfaces
Centroaﬃne
surfaces
Centroaﬃne
minimalsurfaces

Proposition
ϕ is conformal to the second fundamental form of ẑ.
6 / 40Proof of Proposition
Diﬀerential
geometry of
projective or
centroaﬃne
surfaces.

Proof
Consider the caseAtsushiFujiokaz = [λ, λf ], λ : D → R \ {0}, f : D → R3

Contents
Introduction
Projective
surfaces
Projective
minimalsurfaces
Isothermallyasymptotic
surfaces
Centroaﬃne
surfaces
Centroaﬃne
minimalsurfaces

From the ﬁrst equation of (1)(λxx , λxx f + 2λx fx + λfxx )= l(λxy , λxy f + λx fy + λy fx + λfxy ) + a(λx , λx f + λfx )+b(λy , λy f + λfy ) + p(λ, λf )Hence
2λx fx + λfxx = l(λx fy + λy fx + λfxy ) + aλfx + bλfy.
7 / 40Proof of Proposition (continued)
Diﬀerential
geometry of
projective or
centroaﬃne
surfaces
AtsushiFujiokaContents

Proof (continued).

Since λ 6= 0






fxy
fxx
det  fx  = l det  fx fyfy

Introduction
Projective
surfaces
Projective
minimalsurfaces
Isothermallyasymptotic
surfaces
Centroaﬃne
surfaces
Centroaﬃne
minimalsurfaces

Similar computation can be done from the second equation of(1).
On the other hand, multiplying the second fundamental form ofẑ by kfx × fy k, we have






fyy
fxy
fxx
det  fx  dx 2 + 2 det  fx  dxdy + det  fx  dy 2 .
fy
fy
fy
.
8 / 40Asymptotic line coordinatesDiﬀerential
geometry of
projective or
centroaﬃne
surfaces
AtsushiFujiokaContents
Introduction
Projective
surfacesz : D → P3 : a projective surface(x, y ): local coordinates
zxy , zx , zy , z: linearly independent on D
Moreover, we assume z is indeﬁnite, i.e., the symmetric 2-form
ϕ is indeﬁnite.
By the above proposition, we can choose asymptotic line
coordinates as (x, y ), so thatl = m = 0.

Projective
minimalsurfaces
Isothermallyasymptotic
surfaces
Centroaﬃne
surfaces
Centroaﬃne
minimalsurfaces

Put


zxy
 zx 
Φ= zy  .
z

9 / 40A system of linear partial diﬀerential equations
Diﬀerential
geometry of
projective or
centroaﬃne
surfaces
AtsushiFujiokaContents
Introduction
Projective
surfaces
Projective
minimalsurfaces
Isothermallyasymptotic
surfaces
Centroaﬃne
surfaces
Centroaﬃne
minimalsurfaces

Proposition.

Φx = AΦ, Φy = BΦ,

(2)

where



a ay + bc by + bd + p bq + py
 0

a
b
p
,
A=
 1

0
0
0
0
1
0
0



d cx + ac + q dx + bc cp + qx
 1

0
0
0
.
B =
 0

c
d
q
0
0
1
0

.
10 / 40

Integrability condition
Diﬀerential
geometry of
projective or
centroaﬃne
surfaces
AtsushiFujiokaContents
Introduction
Projective
surfaces
Projective
minimalsurfaces
Isothermallyasymptotic
surfaces
Centroaﬃne
surfaces
Centroaﬃne
minimalsurfaces.

Proposition
The integrability condition for the system of linear partial
diﬀerential equations (2) can be written byLy = −2bcx − cbx ,Mx = −2cby − bcy ,
bMy + 2Mby + byyy = cLx + 2Lcx + cxxx ,

(3)

where
a = θx , d = θy ,
1L = θxx − θx2 − bθy − by − 2p,2M = θ − 1 θ2 − cθ − c − 2q.
yy
x
x
2 y

(4)

.
11 / 40

Proof of Proposition
Diﬀerential
geometry of
projective or
centroaﬃne
surfaces
AtsushiFujioka.

Proof
The integrability condition is
Ay − Bx + [A, B] = O.

Contents
Introduction
Projective
surfaces
Projective
minimalsurfaces
Isothermallyasymptotic
surfaces
Centroaﬃne
surfaces
Centroaﬃne
minimalsurfaces

From the second, third and fourth rows, we have identities.
From the (1, 1)-entry,
ay = dx .
Hence ∃ θ satisfying (4).
From the (1, 2)-entry, we have the second equation of (3).
From the (1, 3)-entry, we have the ﬁrst equation of (3).
From the (1, 4)-entry and the ﬁrst and second equations of (3),
we have the third equation of (3).
.
12 / 40

Canonical system
Diﬀerential
geometry of
projective or
centroaﬃne
surfaces
AtsushiFujiokaz : D → P3 : an indeﬁnite projective surface(x, y ): asymptotic line coordinates{
zxx = θx zx + bzy + pz,
zyy = czx + θy zy + qz

Contents
Introduction
Projective
surfaces
Projective
minimalsurfaces
Isothermallyasymptotic
surfaces
Centroaﬃne
surfaces
Centroaﬃne
minimalsurfaces

λ : D → R \ {0}If we put z = λw ,zx = λx w + λwx , zxx = λxx w + 2λx wx + λwxx .
θPutting λ = e 2 , we may assume{
zxx = bzy + pz,
zyy = czx + qz.

(5)(canonical system)
13 / 40

Coordinate transformationDiﬀerential
geometry of
projective or
centroaﬃne
surfaces
AtsushiFujiokaContents

For the canonical system (5), consider coordinatetransformation:u = f (x), v = g (y ).
λ : D → R \ {0}If we put z = λw ,Introduction
Projective
surfaces
Projective
minimalsurfaces
Isothermallyasymptotic
surfaces
Centroaﬃne
surfaceszx = λx w +λwu f 0 , zxx = λxx w +2λx wu f 0 +λwuu (f 0 )2 +λwu f 00 .
C
If λ = √ 0 0 (C ∈ R \ {0}), we have another canonical system:f g{
wuu = b̄wv + p̄w ,wvv = c̄wu + q̄w .

Centroaﬃne
minimalsurfaces
14 / 40

Transformation ruleDiﬀerential
geometry of
projective or
centroaﬃne
surfaces
AtsushiFujiokaContents
Introduction
Projective
surfaces
Projective
minimalsurfaces
Isothermallyasymptotic
surfaces
Centroaﬃne
surfaces
Centroaﬃne
minimalsurfaces

Proposition.

)
1 g 00p − b 0 + {f ; x} ,
2 g
(
)
1
1 f 00f0c̄ = c 0 2 , q̄ = 0 2 q − c 0 + {g ; y } ,
(g )
(g )
2 f
g0
1
b̄ = b 0 2 , p̄ = 0 2
(f )
(f )

(

where {f ; x} and {g ; y } are the Schwarzian derivatives of f
and g respectively:
( )
( )
( )
1 f 00 0 1 f 00 2 1 f 000 3 f 00 2−=−.
{f ; x} :=
2 f0
4 f0
2 f0
4 f0
.
15 / 40

Schwarzian derivative and linear ordinary
diﬀerential equation of second order
Diﬀerential
geometry of
projective or
centroaﬃne
surfaces

Proposition.

z1 (x), z2 (x): linearly independent R-valued solutions to

AtsushiFujiokazxx + α(x)z = 0

Contentsz1
z2=⇒ α = {f ; x}f :=Introduction
Projective
surfaces
Projective
minimalsurfaces.

ProofDiﬀerentiating fz2 = z1 twice and using (6), we have

Isothermallyasymptotic
surfaces

f 00
2z20
=
−
.
f0
z2

Centroaﬃne
surfaces
Centroaﬃne
minimalsurfaces

(6)

..

.

16 / 40

Projective metric and Darboux cubic form
Diﬀerential
geometry of
projective or
centroaﬃne
surfaces

The transformation rule for b, c in the canonical system (5):b̄ = bAtsushiFujiokaContents
Introduction
Projective
surfaces
Projective
minimalsurfaces
Isothermallyasymptotic
surfaces
Centroaﬃne
surfaces
Centroaﬃne
minimalsurfaces

g0
f0
,
c̄
=
c
(f 0 )2(g 0 )2Henceb̄c̄dudv = bcdxdy=⇒ bcdxdy is invariant for the projective surface z.
(projective metric)
Moreoverb̄du 3 + c̄dv 3 = f 0 g 0 (bdx 3 + cdy 3 )=⇒ The conformal class of bdx 3 + cdy 3 is invariant for the
projective surface z.
(Darboux cubic form)
17 / 40

The case that Darboux cubic form vanishes
Diﬀerential
geometry of
projective or
centroaﬃne
surfaces
AtsushiFujioka

Consider the case b = c = 0.
The integrability condition (3) becomespy = 0, qx = 0.
u1 (x), u2 (x): linearly independent R-valued solutions toContents

zxx = p(x)z

Introduction
Projective
surfaces
Projective
minimalsurfaces
Isothermallyasymptotic
surfaces
Centroaﬃne
surfaces
Centroaﬃne
minimalsurfacesv1 (y ), v2 (y ): linearly independent R-valued solutions tozyy = q(y )z
If we putz = [u1 v1 , u1 v2 , u2 v1 , u2 v2 ],z is a projective surface with b = c = 0, which is a quadratic:(u1 v1 )(u2 v2 ) = (u1 v2 )(u2 v1 ).
18 / 40

The case that projective metric vanishes
Diﬀerential
geometry of
projective or
centroaﬃne
surfaces
AtsushiFujioka

Consider the case c = 0.
The integrability condition (3) becomes(−by − 2p)y = 0, (−2q)x = 0, bMy + 2Mby + byyy = 0.
By the transformation rule,Contents
Introduction
Projective
surfaces
Projective
minimalsurfaces
Isothermallyasymptotic
surfaces
Centroaﬃne
surfaces
Centroaﬃne
minimalsurfacesq̄ =1(q(y ) + {g ; y }) .
(g 0 )2

May assume q = 0, so that M = 0 andzxx = (α(x)y 2 + β(x)y + γ(x))zy + (−α(x)y + δ(x))z, zyy = 0.
By a further computation, we have a ruled surface:z = A(x) + yB(x).
19 / 40Deﬁnition of projective minimal surfaces
Diﬀerential
geometry of
projective or
centroaﬃne
surfaces
AtsushiFujiokaContents
Introduction
Projective
surfaces
Projective
minimalsurfaces
Isothermallyasymptotic
surfaces

Projective minimal surfaces: Extremals for the integral of the
projective metricz : D → P3 : an indeﬁnite projective surface(x, y ): asymptotic line coordinates
D: boundedΦ(x, y , t): a 1-parameter family of indeﬁnite projective surfacess.t.
Φ(x, y , 0) = z(x, y ), Φ|∂D = z|∂D
Note that (x, y ) are asymptotic line coordinates at t = 0.

Centroaﬃne
surfaces
Centroaﬃne
minimalsurfaces
20 / 40

Review of equiaﬃne diﬀerential geometryDiﬀerential
geometry of
projective or
centroaﬃne
surfaces
AtsushiFujiokaContents
Introduction
Projective
surfaces
Projective
minimalsurfaces
Isothermallyasymptotic
surfaces
Centroaﬃne
surfacesf : D → R3 : an aﬃne surface with transversal vector ﬁeld ξ(x1 , x2 ): local coordinates
Gauss formula:fxi xj = Γ1ij fx1 + Γ2ij fx2 + h(∂xi , ∂xj )ξ

(i, j = 1, 2)

h: the aﬃne metric
Two kinds of area elements:

fx1◦ θ(∂x1 , ∂x2 ) := det  fx2 ξ1◦ ω(X1 , X2 ) := |det(h(Xi , Xj ))| 2(θ(X1 , X2 ) = 1)

Centroaﬃne
minimalsurfaces
21 / 40

Blaschke surfaces
Diﬀerential
geometry of
projective or
centroaﬃne
surfaces

Consider nondegenerate aﬃne surfaces, i.e., the aﬃne metric is
nondegenerate, which is independent of choice of transversal
vector ﬁeld.

AtsushiFujioka

Proposition
Contents
Introduction
Projective
surfaces
Projective
minimalsurfaces
Isothermallyasymptotic
surfaces
Centroaﬃne
surfaces
Centroaﬃne
minimalsurfaces.

f : D → R3 : a nondegenerate aﬃne surface∇: the induced connection=⇒ Changing the transversal vector ﬁeld, if necessary, we have∇θ = 0, θ = ω.
(7)
Blaschke normal: the transversal vector ﬁeld satisfying (7)Blaschke surfaces: aﬃne surfaces with Blaschke normal astransversal vector ﬁeld
Blaschke metric: the aﬃne metric of Blaschke surfaces.

22 / 40

Fubini-Pick invariant
Diﬀerential
geometry of
projective or
centroaﬃne
surfaces
AtsushiFujiokaContents
Introduction
Projective
surfaces
Projective
minimalsurfaces
Isothermallyasymptotic
surfaces
Centroaﬃne
surfaces
Centroaﬃne
minimalsurfacesf : D → R3 : a Blaschke surface
Deﬁne a cubic form C by
C (X , Y , Z ) = X (h(Y , Z )) − h(∇X Y , Z ) − h(Y , ∇X Z )for vector ﬁelds X , Y , Z on the surface f .
(x1 , x2 ): local coordinates
hij := h(∂xi , ∂xj ), (hij ) := (hij )−1
Cijk := C (∂xi , ∂xj , ∂xk )Fubini-Pick invariant:
1
1
kC k2h = hip hjq hkr Cijk Cpqr
8823 / 40

Method of equiaﬃne diﬀerential geometryDiﬀerential
geometry of
projective or
centroaﬃne
surfaces
AtsushiFujiokaContents
Introductionz : D → P3 : an indeﬁnite projective surface(x, y ): local coordinatesz = [1, f ]=⇒ f becomes a Blaschke surface s.t.
{
fxx = afx + bfy + lfxy ,fyy = cfx + dfy + mfxy .

Projective
surfaces
Projective
minimalsurfaces
Isothermallyasymptotic
surfaces
Centroaﬃne
surfaces
Centroaﬃne
minimalsurfaces

Proposition
If (x, y ) are asymptotic line coordinates,
1
kC k2h θ = ±bcdx ∧ dy .
8
In particular, kC k2h θ is invariant for the projective surface z.
24 / 40

Blaschke normal and Blaschke metricDiﬀerential
geometry of
projective or
centroaﬃne
surfaces
AtsushiFujiokaContents

f : D → R3 : an indeﬁnite Blaschke surface, i.e., the Blaschke
metric is indeﬁnite, s.t.
{
fxx = āfx + b̄fy + l̄fxy ,fyy = c̄fx + d̄fy + m̄fxy
ξ: the Blaschke normal

Introduction
Projective
surfaces
Projective
minimalsurfaces
Isothermallyasymptotic
surfaces
Centroaﬃne
surfaces
Centroaﬃne
minimalsurfacesξ = ζfx + ηfy + λfxy
If (x, y ) are asymptotic line coordinates,ζ = η = 0, λ2 = ±1.
|fx , fy , fxy |h: the Blaschke metric
(
)
(
)
1
l̄ 1
h(∂x , ∂x ) h(∂x , ∂y )(hij ) :==h(∂y , ∂x ) h(∂y , ∂y )λ 1 m̄
25 / 40

Area elements and cubic form
Diﬀerential
geometry of
projective or
centroaﬃne
surfaces
AtsushiFujiokaθ, ω: the area elements
θ=ω=±

√

1 − l̄ m̄
dx ∧ dyλC : the cubic form
Contents
Introduction
Projective
surfaces
Projective
minimalsurfaces
Isothermallyasymptotic
surfaces
Centroaﬃne
surfaces
Centroaﬃne
minimalsurfaces

C111 : = C (∂x , ∂x , ∂x )
( )
(
)
(
)
l̄
l̄
l̄
l̄
1
=
− 2 ā − ζ− 2 b̄ − η
λ x
λ
λ
λ
λ
C112 = C121 = C211 := C (∂x , ∂x , ∂y ), . . .
If (x, y ) are asymptotic line coordinates,
C112 = C121 = C211 = C122 = C212 = C221 = 0.
In particular, the above proposition holds.
26 / 40

Canonical system via equiaﬃne diﬀerential
geometryDiﬀerential
geometry of
projective or
centroaﬃne
surfaces
AtsushiFujiokaContents
Introduction
Projective
surfaces
Projective
minimalsurfaces
Isothermallyasymptotic
surfaces
Centroaﬃne
surfaces
Centroaﬃne
minimalsurfacesz : D → P3 : an indeﬁnite projective surface(x, y ): asymptotic line coordinates
Canonical system:{
zxx = bzy + pz,
zyy = czx + qz
[ ϕ]
ϕz = e− 2 , e− 2 f=⇒ f is a Blaschke surface s.t.
{
fxx = ϕx fx + bfy ,fyy = cfx + ϕy fy .
and
1
1
1
1
p = − ϕxx + ϕ2x + ϕy b, q = − ϕyy + · · · .
242227 / 40

First variation formula
D: boundedΦ(x, y , t): a 1-parameter family of indeﬁnite Blaschke surfacess.t.

Diﬀerential
geometry of
projective or
centroaﬃne
surfaces
AtsushiFujioka

Φ(x, y , t) = f + t(αfx + βfy + γfxy ) + o(t)(t → 0)α, β, γ : D → R, α|∂D = β|∂D = γ|∂D = 0

Contents
Introduction
Projective
surfaces

Proposition
∫
∫
d
1
1
kC k2h θ = ∓{(bMy + 2Mby + byyy )dt t=0 D 8
2 D

Projective
minimalsurfaces
Isothermallyasymptotic
surfaces+(cLx + 2Lcx + cxxx )} γdxdy ,Centroaﬃne
surfaces
Centroaﬃne
minimalsurfaceswhere
L = −by − 2p, M = −cx − 2q.
.

28 / 40

Aﬃne spheres
Diﬀerential
geometry of
projective or
centroaﬃne
surfaces
AtsushiFujiokaContents
Introduction
Projective
surfaces
Projective
minimalsurfaces
Isothermallyasymptotic
surfaces
Centroaﬃne
surfaces
Centroaﬃne
minimalsurfaces[ ϕ]
ϕz = e − 2 , e − 2 f : D → P3 : an indeﬁnite projective surface
given by canonical system
ξ: the Blaschke normal of f
(
)
(
)(
)
ξx
ϕxy + bc by + ϕy b
fx
=⇒
=λ
ξy
cx + ϕx c ϕxy + bc
fy
A direct computation shows thatby + ϕy b = cx + ϕx c = 0
implies the projective minimality:bMy + 2Mby + byyy = cLx + 2Lcx + cxxx = 0.
Proposition
Aﬃne spheres are projective minimal, i.e., if f is an aﬃnesphere, z is projective minimal.
29 / 40

Demoulin surfaces and Godeaux-Rozet surfaces
Diﬀerential
geometry of
projective or
centroaﬃne
surfaces
AtsushiFujiokaContents
Introduction
Projective
surfaces
Projective
minimalsurfaces
Isothermallyasymptotic
surfaces
Centroaﬃne
surfaces
Centroaﬃne
minimalsurfaces

Assume b, c 6= 0.
By the transformation rule, we can deﬁne projective invariant
quadratic forms Pdx 2 , Qdy 2 by
1
1 cxx
1 c2
1 byy
1 by2
1
P := p + by −+ x2 , Q := q + cx −
+
.
2
2 c
4c
2
2 b
4 b2
Demoulin surfaces: P = Q = 0
Godeaux-Rozet surfaces: P = 0 or Q = 0
Proposition{Aﬃne spheres} ⊂ {Demoulin surfaces}⊂ {Godeaux-Rozet surfaces}⊂ {Projective minimal surfaces}
30 / 40Deﬁnition of isothermally asymptotic surfaces
Diﬀerential
geometry of
projective or
centroaﬃne
surfaces
AtsushiFujiokaContents
Introduction
Projective
surfaces
Projective
minimalsurfaces
Isothermallyasymptotic
surfaces
Centroaﬃne
surfaces
Centroaﬃne
minimalsurfacesz : D → P3 : an indeﬁnite projective surface
Assume b, c 6= 0.
()
bIsothermally asymptotic surfaces: log=0c xy◦ Changing the coordinates, if necessary, the above condition isequivalent to b = c.
◦ Aﬃne spheres are isothermally asymptotic.
◦ The surface z deﬁnes 3 families of curves (3-web):· asymptotic curves· zero curves of the Darboux cubic form (Darboux’s curves)
For isothermally asymptotic surfaces, the above 3-web ishexagonal, i.e., locally diﬀeomorphic to 3 families of parallellines.
31 / 40Web curvatureDiﬀerential
geometry of
projective or
centroaﬃne
surfaces
AtsushiFujiokaContents
Introduction
Projective
surfaces
Projective
minimalsurfaces

Consider a 3-web given by 1-forms ω1 , ω2 , ω3 .
Each curve is a zero curve of one of ωi ’s.
Normalization: ω1 + ω2 + ω3 = 0=⇒ ∃ γ: a 1-form s.t.
dω1 = ω1 ∧ γ,(web structure equations)dω2 = ω2 ∧ γ,dω3 = ω3 ∧ γγ: the Chern connection form
The web curvature K is deﬁned by

Isothermallyasymptotic
surfaces
Centroaﬃne
surfaces
Centroaﬃne
minimalsurfacesdγ = K ω1 ∧ ω2 .
Proposition
Hexagonality ⇐⇒ K = 0
32 / 40

Proof of Proposition and the case of the 3-web for
projective surfaces
Diﬀerential
geometry of
projective or
centroaﬃne
surfaces
AtsushiFujiokaContents
Introduction
Projective
surfaces
Projective
minimalsurfaces
Isothermallyasymptotic
surfaces
Centroaﬃne
surfaces
Centroaﬃne
minimalsurfaces.

Proof
K = 0 =⇒ γ = d log f(∃ f : locally)=⇒ d(f ωi ) = 0(i = 1, 2, 3)=⇒ f ωi = dui(∃ ui : locally)=⇒ u1 + u2 + u3 : constant(hexagonality).

The above 3-web for the projective surface is given by
1

1

1

1

ω1 = −b 3 dx, ω2 = −c 3 dy , ω3 = b 3 dx + c 3 dy
(
)
by
1 cx
=⇒ γ = −dx + dy
3 c
b
..
)
(
1
1
b
=⇒ K = − (bc)− 3 log
3
c xy

.

33 / 40Deﬁnition of centroaﬃne surfaces
Diﬀerential
geometry of
projective or
centroaﬃne
surfaces
AtsushiFujioka

Centroaﬃne surfaces: aﬃne surfaces with position vector astransversal vector ﬁeld
Gauss formula:fxi xj = Γ̃1ij fx1 + Γ̃2ij fx2 − h̃(∂xi , ∂xj )f(i, j = 1, 2)Contents
Introduction
Projective
surfaces
Projective
minimalsurfaces
Isothermallyasymptotic
surfaces
Centroaﬃne
surfaces
Centroaﬃne
minimalsurfacesh̃: the centroaﬃne metric
Consider indeﬁnite case.
(x, y ): asymptotic line coordinatesψ := h̃(∂x , ∂y )K : the Euclidean Gaussian curvature of fd: the signed distance from the origin to the tangent plane(the Euclidean support function))(
1Kρ := − log − 44d34 / 40

Gauss formula in asymptotic line coordinatesDiﬀerential
geometry of
projective or
centroaﬃne
surfaces.

Gauss formula

AtsushiFujiokaContents
Introduction
Projective
surfaces
Projective
minimalsurfaces
Isothermallyasymptotic
surfaces
Centroaﬃne
surfaces
Centroaﬃne
minimalsurfaces

where

)
(

ψx
α


+ ρx fx + fy ,
fxx =ψψfxy = −ψf + ρy fx + ρx fy ,

(
)


ψy
β


fyy =+ ρy fy + fx ,
ψ
ψ

/ 

f
f
α = ψ det  fx  det  fx  ,
fxx
fy

/ 

f
f
β = ψ det  fy  det  fy  .
fx
fyy

(8)



.
35 / 40

Integrability condition
Diﬀerential
geometry of
projective or
centroaﬃne
surfaces
AtsushiFujiokaContents
Introduction
Projective
surfaces
Projective
minimalsurfaces
Isothermallyasymptotic
surfaces
Centroaﬃne
surfaces
Centroaﬃne
minimalsurfaces

Proposition.

The integrability condition for the Gauss formula (8) can be
written byαβ(log |ψ|)xy = −ψ − ψ 2 + ρx ρy ,αy + ρx ψx = ρxx ψ,
βx + ρy ψy = ρyy ψ.

.

If ρ is constant and α, β 6= 0, changing the coordinates, if
necessary, we obtain Tzitzéica equation:(log ψ)xy = −ψ −1.
ψ2
36 / 40Deﬁnition of centroaﬃne minimal surfaces
Diﬀerential
geometry of
projective or
centroaﬃne
surfaces
AtsushiFujiokaContents
Introduction
Projective
surfaces
Projective
minimalsurfaces
Isothermallyasymptotic
surfaces
Centroaﬃne
surfaces
Centroaﬃne
minimalsurfaces

Centroaﬃne minimal surfaces: Extremals for the area integral of
the centroaﬃne metric3
f : D → R : an indeﬁnite centroaﬃne surface(x, y ): asymptotic line coordinates
˜ the induced connection∇:∇h̃ : the Levi-Civita connection for the centroaﬃne metric h̃˜ − ∇h̃ : the diﬀerence tensor
C̃ := ∇
1
T := trh̃ C̃ : the Tchebychev vector ﬁeld
2∇h̃ T : the Tchebychev operator
Propositionf : centroaﬃne minimal ⇐⇒ ρxy = 0⇐⇒ tr ∇h̃ T = 0
37 / 40

Fundamental examples
Diﬀerential
geometry of
projective or
centroaﬃne
surfaces

Proper aﬃne spheres: Blaschke surfaces whose aﬃne shapeoperator is a non-zero scalar operator
The center: The point where the Blaschke normals of properaﬃne spheres meet

AtsushiFujioka

Propositionf : a proper aﬃne sphere centered at the origin ⇐⇒ ρ: constantContents
Introduction
Projective
surfaces⇐⇒ T = 0

Projective
minimalsurfaces◦ Ellipsoids, hyperboloids centered at the origin =⇒ T = 0◦ Elliptic paraboloids removing the vertex which is the origin,
hyperbolic paraboloids removing the saddle point which is.
the origin =⇒ T 6= 0, ∇h̃ T = 0◦ In 1995, H. L. Liu and C. P. Wang classiﬁed centroaﬃne

Isothermally .
asymptotic
surfaces
Centroaﬃne
surfaces
Centroaﬃne
minimalsurfaces..

minimal surfaces with ∇h̃ T = 0.

.
38 / 40

Examples with non-vanishing Tchebychev operator
Diﬀerential
geometry of
projective or
centroaﬃne
surfaces
AtsushiFujiokaContents
Introduction
Projective
surfaces
Projective
minimalsurfaces
Isothermallyasymptotic
surfaces
Centroaﬃne
surfaces
Centroaﬃne
minimalsurfaces

( u
)
e
eu
1
cos v ,
sin v , 1 +(2006 F)uuu
· The centroaﬃne curvature is 1.
· α = β, ρ: linear w.r.t. u, v

◦f =· T : an eigenvector of ∇h̃ T (cf. 2004 L. Vrancken)· Projective minimal and isothermally asymptotic◦ f = A0 (u) + vA(u) (2009 F)A
A: an R3 -valued function s.t. det  A0  6= 0
A00
· The centroaﬃne curvature is 1.
· The Pick invariant vanishes.
· ∇h̃ T is not diagonalizable.
· Projective minimal
39 / 40

Diﬀerential
geometry of
projective or
centroaﬃne
surfaces
AtsushiFujiokaContents
Introduction

Thank you for your attention.

Projective
surfaces
Projective
minimalsurfaces
Isothermallyasymptotic
surfaces
Centroaﬃne
surfaces
Centroaﬃne
minimalsurfaces
40 / 40