Centroaﬃne
surfaces ofcohomogeneityone
AtsushiFujioka
Contents
Introduction
Aﬃne
hypersurfaces
Blaschkehypersurfaces
Centroaﬃne
hypersurfaces
Centroaﬃnesurfaces

Centroaﬃne surfaces of cohomogeneity one
Atsushi Fujioka
Faculty of Engineering Science
Kansai University

September 3, 2019
Ritsumeikan University
The 5th Japan-China Geometry Conference
Joint work with H. Furuhata

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Contents
Centroaﬃne
surfaces ofcohomogeneityone

1

Introduction

2

Aﬃne hypersurfaces

3

Blaschke hypersurfaces

4

Centroaﬃne hypersurfaces

5

Centroaﬃne surfacesAtsushi
FujiokaContents
Introduction
Aﬃne
hypersurfaces
Blaschkehypersurfaces
Centroaﬃne
hypersurfaces
Centroaﬃne
surfaces2 / 27

Backgrounds
Centroaﬃne
surfaces ofcohomogeneityone
AtsushiFujiokaContents
Introduction
Aﬃne
hypersurfaces
Blaschkehypersurfaces
Centroaﬃne
hypersurfaces
Centroaﬃne
surfaces

Problem
Consider an analogue of surfaces of revolution in aﬃnediﬀerential geometry.
Hypersurfaces in the aﬃne space: various choices ofa transversal vector field◦ Blaschke normal vector field ⇝ Blaschke hypersurfaces◦ Restriction of the radial vector field ⇝ centroaﬃne
hypersurfaces
Equiaﬃne rotation surfaces:
an analogue of surfaces of revolution for Blaschke surfaces
Blaschke normal vector fields meet a fixed line or are parallel
to a fixed plane.
Centroaﬃne surfaces: Consider surfaces of cohomogeneity one.
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Centroaﬃne diﬀerential geometry and centroaﬃne
surfaces of cohomogeneity one
Centroaﬃne
surfaces ofcohomogeneityone
AtsushiFujiokaContents

Centroaﬃne diﬀerential geometry:
Study properties of submanifolds in the aﬃne space which
are invariant by centroaﬃne transformations.
Centroaﬃne transformation: aﬃne transformation fixing
the origin

Aﬃne
hypersurfaces
Blaschkehypersurfaces
Centroaﬃne
hypersurfaces⇝Introduction

Group of centroaﬃne transformations of Rn ∼= GL(n, R)
A centroaﬃne surface of cohomogeneity one is expressed asf (x, y ) = γ(x)e yA ,Centroaﬃne
surfaces

whereγ: a curve in R3 ,A ∈ gl(3, R).
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Example 1
Flat proper aﬃne spheres centered at the origin
Centroaﬃnesurfaces ofcohomogeneityone
AtsushiFujiokaContents
Introduction
Aﬃne
hypersurfaces
Blaschkehypersurfaces
Centroaﬃne
hypersurfaces
Centroaﬃne
surfaces

Example 1 (cf. 1990 Magid-Ryan)f = (X , Y , Z ) : M 2 → R3 : a flat proper aﬃne sphere centeredat the origin=⇒ Up to centroaﬃne congruence,
XYZ = 1or (X 2 + Y 2 )Z = 1.

These are of cohomogeneity one.
For example, in the former case,
γ(x) = (e x , 1, e −x ),0 0 0A =  0 1 0 .
0 0 −1

The above surfaces are included in the next example.
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Example 2
Nondegenerate flat centroaﬃne surfaces with vanishing Tchebychev operator
Centroaﬃne
surfaces ofcohomogeneityone
AtsushiFujiokaContents
Introduction
Aﬃne
hypersurfaces
Blaschkehypersurfaces
Centroaﬃne
hypersurfaces
Centroaﬃne
surfaces

Example 2 (1995 Liu-Wang)f = (X , Y , Z ) : M 2 → R3 : a nondegenerate flat centroaﬃne
surface, Tchebychev operator = 0=⇒ Up to centroaﬃne congruence, f is one of the following:(1) X p Y q Z r = 1 (p, q, r ∈ R, pqr (p + q + r ) ̸= 0),{()} 2X(2) exp −p tan−1 Y(X + Y 2 )q Z r = 1(p, q, r ∈ R, r (2q + r )(p 2 + q 2 ) ̸= 0),(3) Z = −X (p log X + q log Y ) (p, q ∈ R, q(p + q) ̸= 0),2(4) Z = ±X log X + YX ,(5) f = (e x , ψ1 (x)e y , ψ2 (x)e y ), ψ1 , ψ2 are linearly independent
solutions to the diﬀerential equation ψ ′′ − ψ ′ − λψ = 0 foran arbitrary function λ = λ(x).
These are of cohomogeneity one.
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The Gauss-Weingarten formulaCentroaﬃne
surfaces ofcohomogeneityone

f : M n → Rn+1 : a hypersurface (immersion)
ξ: a vector field along f(f , ξ) or f : an aﬃne hypersurface

AtsushiFujiokaContents
Introduction
Aﬃne
hypersurfaces
Blaschkehypersurfaces
Centroaﬃne
hypersurfaces
Centroaﬃne
surfaces⇕ def.
∀x ∈ M, ξ(x) ⋔ f∗ Tx M

ξ is called a transversal vector field.
Let f be an aﬃne hypersurface.
D: the standard flat connection on Rn+1=⇒ The Gauss-Weingarten formula:{DX f∗ Y = f∗ ∇X Y + h(X , Y )ξ(X , Y ∈ X(M))
DX ξ = −f∗ SX + τ (X )ξ
∇: the induced connection, h: the aﬃne fundamental form
S: the aﬃne shape operator, τ : the transversal connection form
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Integrability conditions
The fundamental equations or the Gauss-Codazzi-Ricci equations
Centroaﬃne
surfaces ofcohomogeneityone
AtsushiFujiokaContentsf : M n → Rn+1 : an aﬃne hypersurface
X , Y , Z ∈ X(M) =⇒The Gauss equation:
R(X , Y )Z = h(Y , Z )SX − h(X , Z )SY(R: the curvature tensor for the induced connection ∇)Introduction

The Codazzi equation for h:Aﬃne
hypersurfaces(∇X h)(Y , Z ) + τ (X )h(Y , Z ) = (∇Y h)(X , Z ) + τ (Y )h(X , Z )Blaschkehypersurfaces

The Codazzi equation for S:

Centroaﬃne
hypersurfaces
Centroaﬃne
surfaces(∇X S)(Y ) − τ (X )SY = (∇Y S)(X ) − τ (Y )SXThe Ricci equation:h(X , SY ) − h(SX , Y ) = dτ (X , Y )These are the integrability conditions for the Gauss-Weingarten
formula.
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Equiaﬃne hypersurfaces
Relative normalization
Centroaﬃne
surfaces ofcohomogeneityone
AtsushiFujiokaContents
Introduction
Aﬃne
hypersurfaces
Blaschkehypersurfaces
Centroaﬃne
hypersurfaces
Centroaﬃne
surfacesf : M n → Rn+1 : an aﬃne hypersurface
ξ: a transversal vector fieldFix a volume element ω on Rn+1 which is parallel w.r.t.
the standard flat connection D.
⇝ θ: the volume element induced by (f , ξ)θ(X1 , . . . , Xn ) = ω(f∗ X1 , . . . , f∗ Xn , ξ) (X1 , . . . , Xn ∈ X(M))
Proposition∇X θ = τ (X )θ

(X ∈ X(M))

Definition
ξ or f : equiaﬃne ⇐⇒ ∇θ = 0 ⇐⇒ τ = 0
def.
When the Euclidean metric is given, the unit normal vector
field is equiaﬃne.
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Fundamental facts about equiaﬃne hypersurfaces
1/2
Centroaﬃne
surfaces ofcohomogeneityone
AtsushiFujiokaContents

Propositionf : M n → Rn+1 : an equiaﬃne hypersurfacen≥2S = λ id. (∃ λ : M → R)=⇒ λ: constantIntroduction
Aﬃne
hypersurfaces
Blaschkehypersurfaces
Centroaﬃne
hypersurfaces
Centroaﬃne
surfaces

Proof
By equiaﬃneness, the Codazzi equation for S becomes(∇X S)(Y ) = (∇Y S)(X ).
Substitute S = λ id. into the above equation, we have(X λ)Y = (Y λ)X .
In the following, assume n ≥ 2.
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Fundamental facts about equiaﬃne hypersurfaces
2/2
Centroaﬃne
surfaces ofcohomogeneityone
AtsushiFujiokaContents
Introduction
Aﬃnehypersurfaces
Blaschkehypersurfaces

Propositionf : M n → Rn+1 : an equiaﬃne hypersurface(1) S = 0 ⇐⇒ Transversal vector fields are parallel to eachother.
(2) S = λ id. (λ ∈ R \ {0}) ⇐⇒ Transversal vector fields meeta fixed point (the center).
Proof of ⇒ in (2)By equiaﬃneness, the Weingarten formula becomes

Centroaﬃne
hypersurfaces

DX ξ = −f∗ SX .

Centroaﬃne
surfacesg (x) := f (x) +1ξ(x)λ(x ∈ M)=⇒ DX g = 011 / 27

Nondegenerate, definite, indefinite aﬃne
hypersurfaces
Centroaﬃne
surfaces ofcohomogeneity
oneAtsushi
FujiokaContentsDefinitionf : M n → Rn+1 : an aﬃne hypersurface
h: the aﬃne fundamental formf : nondegenerate (resp. definite, indefinite)⇕ def.

Introduction
Aﬃne
hypersurfaces
Blaschkehypersurfaces
Centroaﬃne
hypersurfaces
Centroaﬃne
surfaces

h: nondegenerate (resp. definite, indefinite)
Proposition
The above definition is independent of a choice of a transversal
vector filed ξ.
Proofξ¯ := λξ + (tangential part) (λ : M → R \ {0}) =⇒ λh̄ = h
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Definition of Blaschke hypersurfaces
Centroaﬃne
surfaces ofcohomogeneityone
AtsushiFujiokaContents
Introduction
Aﬃne
hypersurfaces
Blaschkehypersurfaces
Centroaﬃne
hypersurfaces
Centroaﬃne
surfacesf : M n → Rn+1 : a nondegenerate aﬃne hypersurface
ξ: a transversal vector fieldτ : the transversal connection form
θ: the volume form induced by (f , ξ)
The aﬃne fundamental form h is nondegenerate.
⇝ ωh : the volume form w.r.t. hProposition
There exists a transversal vector field ξ with τ = 0, θ = ωh ,which is unique up to sign (the Blaschke normal vector field).
In particular, the Blaschke normal vector field is equiaﬃne.
In this case, f is called a Blaschke hypersurface.
h is called the Blaschke metric.
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Aﬃne hyperspheres
Centroaﬃne
surfaces ofcohomogeneityone
AtsushiFujiokaContents
Introduction
Aﬃne
hypersurfaces
Blaschkehypersurfaces
Centroaﬃne
hypersurfaces
Centroaﬃne
surfaces

Definitionf : M n → Rn+1 : a Blaschke hypersurface
S: the aﬃne shape operator
f : an aﬃne hypersphere ⇐⇒ S: a scalar operator
def.
Improper aﬃne hyperspheres: aﬃne hyperspheres with S = 0
Blaschke normal vector fields areparallel to each other.
Proper aﬃne hyperspheres: aﬃne hyperspheres with S ̸= 0
Blaschke normal vector fields meeta fixed point (the center).

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Examples of aﬃne hyperspheres
Centroaﬃne
surfaces ofcohomogeneityone
AtsushiFujioka

Example (proper quadrics)◦ Proper quadrics without center are improper aﬃnehyperspheres.
◦ Proper quadrics with center are proper aﬃne hyperspheres.

Contents
Introduction
Aﬃne
hypersurfaces
Blaschkehypersurfaces
Centroaﬃne
hypersurfaces
Centroaﬃne
surfaces

Theorem (1990 Magid-Ryan, cf. Example 1)
f = (X , Y , Z ) : M 2 → R3 : an aﬃne sphere, the curvature of
the Blaschke metric = 0 (flat)=⇒ Up to aﬃne congruence, f is one of the following:◦ Z = X 2 + Y 2 (definite, improper),◦ Z = XY + λ(X ), λ: an arbitrary function of X
(indefinite, improper),◦ XYZ = 1 (definite, proper),◦ (X 2 + Y 2 )Z = 1 (indefinite, proper).
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Definition of centroaﬃne hypersurfaces and thediﬀerence tensor
Centroaﬃne
surfaces ofcohomogeneityone
AtsushiFujiokaContents
Introduction
Aﬃne
hypersurfaces
Blaschkehypersurfaces
Centroaﬃne
hypersurfaces
Centroaﬃne
surfaces

Definitionf : M n → Rn+1 : a hypersurface(n+1)∑∂
ξ := −xi∂xi fi=1f : a centroaﬃne hypersurface⇕ def.
(f , ξ): an aﬃne hypersurfacef : M n → Rn+1 : a nondegenerate centroaﬃne hypersurface∇: the induced connection
h: the aﬃne fundamental form (the centroaﬃne metric)
ˆ the Levi-Civita connection for h∇:ˆ the diﬀerence tensor
C := ∇ − ∇:16 / 27

The Tchebychev vector field and the Tchebychev
operator
Centroaﬃne
surfaces ofcohomogeneityone
AtsushiFujiokaContents

T :=1
trh C : the Tchebychev vector fieldnProposition
T =0⇕

Introduction
Aﬃne
hypersurfaces
Blaschkehypersurfaces
Centroaﬃne
hypersurfaces
Centroaﬃnesurfaces

f : As a Blaschke hypersurface, the aﬃne shape operator isa nonzero scalar operator.
Blaschke normal vector fields meet the origin.
(a proper aﬃne hypersphere centered at the origin)ˆ : the Tchebychev operator
T := ∇T
1995 Liu-Wang: Classified nondegenerate centroaﬃne surfaces
with T ̸= 0, T = 0 (cf. Example 2).
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Centroaﬃne minimal hypersurfaces
Centroaﬃne
surfaces ofcohomogeneityone
AtsushiFujiokaContents
Introduction
Aﬃnehypersurfaces
Blaschkehypersurfaces
Centroaﬃne
hypersurfaces

ft : M n → Rn+1 : a one parameter family of nondegenerate
centroaﬃne hypersurfaces s.t.
f0 = f and ft = f , Im (ft )∗ = Im f∗ on the boundary of M
∂ft= λf + (tangential part) (λ : M → R)∂t t=0ωht : the volume form w.r.t. the centroaﬃne metric ht of ft
The first variation formula (1994 Wang)∫∫dnλ (tr T ) ωhωh = ±dt t=0 M t
2 M

Centroaﬃne
surfaces

Definition (1994 Wang)f : centroaﬃne minimal ⇐⇒ tr T = 0def.
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Definite, indefinite aﬃne surfaces and theEuclidean Gaussian curvature
Centroaﬃne
surfaces ofcohomogeneityone
AtsushiFujiokaContentsPropositionf : M 2 → R3 : a nondegenerate aﬃne surface
K : the Euclidean Gaussian curvaturef : definite (resp. indefinite) ⇐⇒ K > 0 (resp. K < 0)

Introduction
Aﬃne
hypersurfaces
Blaschkehypersurfaces
Centroaﬃne
hypersurfaces
Centroaﬃne
surfaces

Proof(x1 , x2 ): a local coordinate
The Gauss formula becomes
fxi xj = Γ1ij fx1 + Γ2ij fx2 + h(∂xi , ∂xj )ξ

(i = 1, 2).

n: the unit normal vector field
Taking the inner product of the above equation and n, we have⟨fxi xj , n⟩ = h(∂xi , ∂xj )⟨ξ, n⟩.
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The normal form for nondegenerate centroaﬃne
surfaces of cohomogeneity one
Centroaﬃne
surfaces ofcohomogeneityone
AtsushiFujiokaContents

For simplicity, consider the indefinite case.
Lemmaf : M 2 → R3 : an indefinite centroaﬃne surface
cohomogeneity one=⇒ f can be expressed asIntroduction
Aﬃne
hypersurfacesBlaschke
hypersurfaces
Centroaﬃne
hypersurfaces

f (u, v ) = γ(u + v )e (u−v )A = γ(s)e tA(the normal form), where(u, v ): an asymptotic line coordinate,

Centroaﬃne
surfaces

γ: a curve in R3 ,s = u + v,

A ∈ gl(3, R),t = u − v.
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Notations
Centroaﬃne
surfaces ofcohomogeneityone
AtsushiFujiokaContents
Introduction
Aﬃne
hypersurfaces
Blaschkehypersurfaces
Centroaﬃne
hypersurfaces
Centroaﬃne
surfacesf : M 2 → R3 : an indefinite centroaﬃne surface(u, v ): an asymptotic line coordinate(∂ ∂)φ := h ∂u, ∂v
d: the Euclidean support function from the origin()K1
ρ := − log − 44d(= log(±r ))(r : the equiaﬃne support function from the origin)
/ 
ffa := φ det  fu  det  fu fuu
fv
/ 
ffb := φ det  fv  det  fv fvv
fu
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The Gauss formula for the normal form
Centroaﬃne
surfaces ofcohomogeneityone
AtsushiFujiokaContents
Introduction
Aﬃnehypersurfaces
Blaschkehypersurfaces
Centroaﬃne
hypersurfaces
Centroaﬃnesurfaces

f : M 2 → R3 : an indefinite centroaﬃne surface
cohomogeneity one
expressed in the normal form=⇒ The Gauss( formula:)φ′1a′
fuu =+ α + C1 f u + f v ,
φ2φ)( ′
1bφ+ α′ − C1 fv + fu ,
fvv =
φ2φ
()
()

′ − 1C′ + 1C
f=−φf+αf+α uv1u1 fv ,22
whereφ, α, a, b: functions of s,ρ(s, t) = α(s) +C1t,2C1 = tr A.
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The integrability conditions for the normal form
Centroaﬃne
surfaces ofcohomogeneityone
AtsushiFujiokaContents
Introduction
Aﬃne
hypersurfaces
Blaschkehypersurfaces
Centroaﬃne
hypersurfaces
Centroaﬃne
surfaces

The integrability conditions for the normal form:
( )′φ′ab1= −φ − 2 + (α′ )2 − C12

φφ4
()
1a′ + α′ + C1 φ′ = α′′ φ
2
()

1′′b + α − C1 φ′ = α′′ φ2From the second and the third equations, ∃ λ = λ(s), ∃ C2 ∈ R
s.t.
λ′ = α′′ φ − α′ φ′ ,1a = − C1 φ + λ,2

1b = C1 φ + λ − C2 .
2

By a further computation, we can construct nondegenerate
centroaﬃne surfaces of cohomogeneity one.
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Special cases
Centroaﬃne
surfaces ofcohomogeneityone
AtsushiFujiokaContents
Introduction
Aﬃne
hypersurfaces
Blaschkehypersurfaces
Centroaﬃne
hypersurfaces
Centroaﬃne
surfaces

Propositionf : M 2 → R3 : an indefinite centroaﬃne surface
cohomogeneity one
expressed in the normal form
(1) C1 = C2 = 0=⇒ Up to centroaﬃne congruence, f is one of
the followingequiaﬃne rotation surfaces:(µ(x) cos y , µ(x) sin y , x),f (x, y ) = (µ(x) cosh y , µ(x) sinh y , x),)(x, xy , 12 xy 2 + µ(x) .
(2) f : a proper aﬃne sphere centered at the origin⇐⇒ α: constant, C1 = 0(3) f : centroaﬃne minimal ⇐⇒ α: a linear function
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The case of nonflat proper aﬃne spheres
Centroaﬃne
surfaces ofcohomogeneityone
AtsushiFujiokaContents
Introduction
Aﬃne
hypersurfaces
Blaschkehypersurfaces
Centroaﬃne
hypersurfaces
Centroaﬃne
surfaces

Theorem (F.-Furuhata)f : M 2 → R3 : an indefinite nonflat proper aﬃne spherecentered at the origin
cohomogeneity one=⇒ In the normal form, γ and A are given by the following:ca−b◦ The minimal polynomial of A is t 3 − t +, where48a, b, c ∈ R.
◦ γ is a solution to the diﬀerential equation(2φ′ + a + b)γ ′ = γ{4φA2 + (a − b)A − 2φ2 E },where φ ̸= 0, φ′ ̸= 0, − a+b2 and(φ′ )2 = −2φ3 + cφ2 + ab.
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The case of centroaﬃne miminal surfaces withconstant curvature
Centroaﬃne
surfaces ofcohomogeneityone
AtsushiFujiokaContents
Introduction
Aﬃne
hypersurfaces
Blaschkehypersurfaces
Centroaﬃne
hypersurfaces
Centroaﬃne
surfaces

Theorem (F.-Furuhata)f : M 2 → R3 : a centroaﬃne minimal surface
cohomogeneity one
κ := the curvature of the centroaﬃne metric: constant=⇒ κ = 0, 1
Devided into theree cases.
Expressed explicitly.
◦ κ = 0, T = 0 (cf. 1995 Liu-Wang)◦ κ = 1 and ruled (cf. 2009 F: without assumption ofcohomogeneity one) or a part of an ellipsoid (definite)◦ Other surfaces with κ = 1 (cf. 2006 F: special cases)

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Centroaﬃne
surfaces ofcohomogeneityone
AtsushiFujiokaContents
Introduction

Thank you for your attention.

Aﬃne
hypersurfaces
BlaschkehypersurfacesCentroaﬃne
hypersurfaces
Centroaﬃne
surfaces

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