中心アファイン曲面に対する３次形式の平行性と再帰性
藤岡敦

中心アファイン曲面に対する３次形式の平行性と再帰性

Contents
Introduction
Aﬃne
hypersurfaces
Blaschkehypersurfaces

藤岡敦
関西大学システム理工学部数学科

Centroaﬃnehypersurfaces
Centroaﬃnesurfaces

２０１６年１１月４日（金）福岡大学, ２０１６年度福岡大学微分幾何研究集会(濱本久二雄氏 (関西大学), 中井康允氏 (西宮北高等学校) との共同研究)

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Contents
中心アファイン曲面に対する３次形式の平行性と再帰性

1

Introduction

2

Aﬃne hypersurfaces

3

Blaschke hypersurfaces

4

Centroaﬃne hypersurfaces

5

Centroaﬃne surfaces

藤岡敦

Contents
Introduction
Aﬃne
hypersurfaces
Blaschkehypersurfaces
Centroaﬃne
hypersurfaces
Centroaﬃne
surfaces2 / 22

Backgrounds
中心アファイン曲面に対する３次形式の平行性と再帰性
藤岡敦Contents
Introduction
Aﬃne
hypersurfaces
Blaschkehypersurfaces
Centroaﬃne
hypersurfaces
Centroaﬃne
surfaces◦ Cubic form: A fundamental invariant for aﬃne hypersurfaces◦ Equiaﬃne diﬀerential geometry• Maschke-Pick-Berwald’s Theorem:
A Blaschke hypersurface with vanishing cubic form is apiece of a quadric.
• 1989 Nomizu-Pinkall:
If the cubic form of a Blaschke surface does not vanish andis parallel relative to the induced connection, the surface is1
a piece of a Cayley surface: z = xy − x 3 .
3· A graph of a cubic polynomial· A ruled surface· Equiaﬃnely homogeneous· An improper aﬃne sphere3 / 22

Problem
中心アファイン曲面に対する３次形式の平行性と再帰性

Problem
Characterize fundamental centroaﬃne surfaces by the cubicform.

藤岡敦

Contents
Introduction
Aﬃne
hypersurfaces
Blaschkehypersurfaces
Centroaﬃne
hypersurfaces
Centroaﬃnesurfaces◦ Examples of fundamental centroaﬃne surfaces• Quadrics• A ruled surface given byf (x, y ) = A′ (x) + yA(x)A
A is an R3 -valued function s.t. det  A′  ̸= 0.
A′′· The curvature of the centroaﬃne metric is 1.
· The Pick function vanishes.
· Centroaﬃne minimal· Projective minimal
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Euclidean diﬀerential geometry and aﬃnediﬀerential geometry
中心アファイン曲面に対する３次形式の平行性と再帰性
藤岡敦

Contents
Introduction
Aﬃne
hypersurfaces
Blaschkehypersurfaces
Centroaﬃne
hypersurfacesCentroaﬃne
surfaces

Consider hypersurfaces in the Euclidean space.
• Euclidean diﬀerential geometry· ∃ a unit normal vector field· Study properties invariant under the Euclideantransformation.
• Aﬃne diﬀerential geometry· Choose a transversal vector field.
◦ Equiaﬃne diﬀerential geometry· Take a Blaschke normal vector field.
· Study properties invariant under the equiaﬃnetransformation.
◦ Centroaﬃne diﬀerential geometry· Take a radial vector field.
· Study properties invariant under the aﬃnetransformation fixing the origin.
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Gauss-Weingarten formula
Euclidean diﬀerential geometry
中心アファイン曲面に対する３次形式の平行性と再帰性
藤岡敦

Contents
Introduction
Aﬃne
hypersurfaces
Blaschkehypersurfaces
Centroaﬃne
hypersurfaces
Centroaﬃne
surfacesf : M n → Rn+1 : a hypersurface
D: the standard flat connection on Rn+1
X , Y ∈ X(M)◦ Euclidean diﬀerential geometry
n: a unit normal vector field=⇒ Gauss-Weingarten formula:{DX f∗ Y = f∗ ∇X Y + h(X , Y )n (Gauss)
DX n = −f∗ SX(Weingarten)∇: the Levi-Civita connection
h: the second fundamental form
S: the shape operator

6 / 22

Gauss-Weingarten formulaAﬃne diﬀerential geometry
中心アファイン曲面に対する３次形式の平行性と再帰性
藤岡敦Contents
Introduction
Aﬃne
hypersurfaces
Blaschkehypersurfaces
Centroaﬃne
hypersurfaces
Centroaﬃne
surfacesf : M n → Rn+1 : a hypersurface
D: the standard flat connection on Rn+1
X , Y ∈ X(M)◦ Aﬃne diﬀerential geometry
ξ: a transversal vector field=⇒ Gauss-Weingarten formula:{DX f∗ Y = f∗ ∇X Y + h(X , Y )ξ (Gauss)DX ξ = −f∗ SX + τ (X )ξ
(Weingarten)
f is called an aﬃne hypersurface.
∇: the induced connection
h: the aﬃne fundamental form
Considered as a metric.
S: the aﬃne shape operator
τ : the transversal connection form
7 / 22

Nondegenerate, definite or indefinite aﬃne
hypersurfaces
中心アファイン曲面に対する３次形式の平行性と再帰性
藤岡敦

Contents
Introduction
Aﬃne
hypersurfaces
Blaschkehypersurfaces
Centroaﬃne
hypersurfaces
Centroaﬃne
surfaces

Definitionf : M n → Rn+1 : an aﬃne hypersurface
h: the aﬃne fundamental formf : nondegenerate (resp. definite, indefinite)⇕ def.
h: nondegenerate (resp. definite, indefinite)
Proposition
The above definition is independent of the choice of thetransversal vector field ξ.
Proofξ¯ := φξ + f∗ Z (φ : M → R \ {0}, Z ∈ X(M)) =⇒ φh̄ = h
8 / 22

Cubic form
中心アファイン曲面に対する３次形式の平行性と再帰性
藤岡敦

Contents
Introduction
Aﬃne
hypersurfaces
Blaschkehypersurfaces
Centroaﬃne
hypersurfaces
Centroaﬃnesurfaces

f : M n → Rn+1 : an aﬃne hypersurface∇: the induced connection
h: the aﬃne fundamental form
τ : the transversal connection form
X , Y , Z ∈ X(M)
Codazzi equation(∇X h)(Y , Z ) + τ (X )h(Y , Z ) = (∇Y h)(X , Z ) + τ (Y )h(X , Z )The cubic form:
C (X , Y , Z ) := (∇X h)(Y , Z ) + τ (X )h(Y , Z )Defines a symmetric (0, 3)-tensor.

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Definition of Blaschke hypersurfaces
中心アファイン曲面に対する３次形式の平行性と再帰性
藤岡敦

ContentsIntroduction
Aﬃnehypersurfaces
Blaschkehypersurfaces

f : M n → Rn+1 : a nondegenerate aﬃne hypersurface
θ: the volume form induced by the immersion fθ(X1 , . . . , Xn ) := ω(f∗ X1 , . . . , f∗ Xn , ξ) (X1 , . . . , Xn ∈ X(M))ω: a parallel volume form with respect to the flat connection Dωh : the volume form with respect to the aﬃne fundamental
form h
X1 , . . . , Xn ∈ X(M) s.t. θ(X1 , . . . , Xn ) = 11ωh (X1 , . . . , Xn ) := |det(h(Xi , Xj ))| 2

Centroaﬃne
hypersurfaces

Proposition

Centroaﬃne
surfaces∃ ξ (unique up to the sign) s.t. τ = 0 & θ = ωhThe above ξ is called a Blaschke normal vector field.
f is called a Blaschke hypersurface.
h is called a Blaschke metric.
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Characterizations by the cubic form
中心アファイン曲面に対する３次形式の平行性と再帰性
藤岡敦

Contents
Introduction
Aﬃne
hypersurfaces
Blaschkehypersurfaces
Centroaﬃne
hypersurfaces
Centroaﬃne
surfacesf : M n → Rn+1 : a Blaschke hypersurfaceC : the cubic form
Maschke-Pick-Berwald’s theorem
C = 0 =⇒ f : a piece of a quadric∇: the induced connection
Theorem (Nomizu-Pinkall 1989)
n = 2, ∇C = 0, C ̸= 0 =⇒ f : a piece of a Cayley surface
1988 Vrancken: n = 3, ∇C = 0, C ̸= 0
ˆ the Levi-Civita connection for the aﬃne fundamental form h∇:ˆ = 0, C ̸= 0
1989 Magid-Nomizu: n = 2, ∇C• Example: a Cayley surface
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Definition of centroaﬃne hypersurfaces
中心アファイン曲面に対する３次形式の平行性と再帰性
藤岡敦ContentsIntroduction
Aﬃne
hypersurfaces
Blaschkehypersurfaces
Centroaﬃne
hypersurfaces
Centroaﬃne
surfaces

Definitionf : M n → Rn+1 : a hypersurface
f : a centroaﬃne hypersurface⇕ def.
The radial vector intersects with the tangent
space transversally at any point.
n+1
∑∂
∂xi
i=1
Weingarten formula:ξ := −xiDX ξ = −f∗ X⇕
S = the identity, τ = 0
The aﬃne fundamental form h is called a centroaﬃne metric.
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Fundamental facts
中心アファイン曲面に対する３次形式の平行性と再帰性Consider nondegenerate centroaﬃne hypersurfaces.
f : M n → Rn+1 : a centroaﬃne hypersurface
C : the cubic form

藤岡敦Contents
Introduction
Aﬃne
hypersurfaces
Blaschkehypersurfaces
Centroaﬃne
hypersurfaces
Centroaﬃne
surfaces

Proposition
C = 0 =⇒ f : a piece of a quadric centered at the origin
RemarkIn general, a nondegenerate aﬃne hypersurface with τ = 0 &C = 0 is a piece of a quadric.
∇: the induced connection
ˆ the Levi-Civita connection for the centroaﬃne metric h∇:ˆ = 0 & flat1991 Li-Wang: ∇C
ˆ = 0 (∇C = 0)2015 Hildebrand: ∇C
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Tchebychev form
中心アファイン曲面に対する３次形式の平行性と再帰性

Introduction

f : M n → Rn+1 : a centroaﬃne hypersurface
h: the centroaﬃne metricC : the cubic form
X ∈ X(M)Define a (1, 1)-tensor AX by

Aﬃne
hypersurfaces

C (X , Y , Z ) = h(AX (Y ), Z )藤岡敦

Contents

Blaschkehypersurfaces
Centroaﬃne
hypersurfaces(∀ Y , Z ∈ X(M)).

Define a 1-form trh C by(trh C )(X ) = trAX .

Centroaﬃnesurfaces

T :=1trh C is called a Tchebychev form.
n14 / 22

Traceless part of the cubic form
中心アファイン曲面に対する３次形式の平行性と再帰性
藤岡敦Contents
Introduction
Aﬃnehypersurfaces
Blaschkehypersurfaces

f : M n → Rn+1 : a centroaﬃne hypersurface
h: the centroaﬃne metricC : the cubic form
T : the Tchebychev form
X , Y , Z ∈ X(M)nC̃ (X , Y , Z ) := C (X , Y , Z ) −(T (X )h(Y , Z )n+2+ T (Y )h(Z , X ) + T (Z )h(X , Y ))(traceless part of C )Centroaﬃne
hypersurfaces
Centroaﬃnesurfaces

RemarkC̃ coincides with the cubic form as a Blaschke hypersurface.
ˆ the Levi-Civita connection for h∇:ˆ C̃ = 01997 Liu-Wang: n = 2, ∇15 / 22

Definiteness and the Euclidean Gaussian curvature
中心アファイン曲面に対する３次形式の平行性と再帰性
藤岡敦

Contents

Propositionf : M 2 → R3 : a centroaﬃne surface
f : definite (resp. indefinite)⇕Introduction
Aﬃne
hypersurfaces
Blaschkehypersurfaces
Centroaﬃne
hypersurfaces
Centroaﬃnesurfaces

the Euclidean Gaussian curvature: positive (resp. negative)
Proof(x1 , x2 ): local coordinates
Gauss formula:fxi xj = Γ1ij fx1 + Γ2ij fx2 − h(∂xi , ∂xj )f(i = 1, 2)

Take the inner product with the unit normal vector field.
16 / 22

Notations
中心アファイン曲面に対する３次形式の平行性と再帰性
藤岡敦

Contents
Introduction
Aﬃne
hypersurfaces
Blaschkehypersurfaces
Centroaﬃne
hypersurfaces
Centroaﬃne
surfaces

Consider the indefinite case.
f : M 2 → R3 : a centroaﬃne surface
K : the Euclidean Gaussian curvature < 0(u, v ): asymptotic line coordinates
φ := h(∂u , ∂v )d: the signed distance from the origin to the tangent plane()1Kρ := − log − 44d/ ffa := φ det  fu  det  fu fuu
fv
/ 
ffb := φ det  fv  det  fv fvv
fu

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Gauss formula in asymptotic line coordinates
中心アファイン曲面に対する３次形式の平行性と再帰性
藤岡敦Contents
Introduction
Aﬃne
hypersurfaces
Blaschkehypersurfaces
Centroaﬃnehypersurfaces
Centroaﬃnesurfaces

Gauss formula)(aφu+ ρu fu + fv
fuu =
φφfuv = −φf + ρv fu + ρu fv
()

φvb

fvv =+ ρv fv + fuφφProposition
The integrability conditions for the above Gauss formula areab
(log φ)uv = −φ − φ2 + ρu ρv
av + ρu φu = ρuu φbu + ρv φv = ρvv φ
18 / 22

Covariant derivative of the cubic form
中心アファイン曲面に対する３次形式の平行性と再帰性
藤岡敦ContentsIntroductionAﬃnehypersurfacesLemma
f : M 2 → R3 : an indefinite centroaﬃne surface∇: the induced connection
C : the cubic form
∇C = 0 ⇐⇒ All the functions below vanish.
−au +3aφu+ 6aρu , −av + 3aρv + 3ρ2u φ,φ−bv +3bφv+ 6bρv , −bu + 3bρu + 3ρ2v φ,
φBlaschkehypersurfaces
Centroaﬃne
hypersurfaces
Centroaﬃne
surfaces

3aρv + ρu φu − ρuu φ + 3ρ2u φ, 3bρu + ρv φv − ρvv φ + 3ρ2v φ,
ab+ 5ρu ρv φ − ρuv φφ19 / 22

Main results
中心アファイン曲面に対する３次形式の平行性と再帰性
藤岡敦ContentsIntroduction

f : M 2 → R3 : a centroaﬃne surface∇: the induced connection
C : the cubic form
Theorem 1 (F.-Hamamoto-Nakai)∇C = 0 =⇒ f : a piece of a quadric centered at the origin(an ellipsoid or a hyperboloid)Aﬃne
hypersurfaces
Blaschkehypersurfaces
Centroaﬃne
hypersurfaces
Centroaﬃne
surfaces

ProofIf f is indefinite, show that C = 0 by use of the integrability
conditions and the above Lemma.
Then f is a piece of a hyperboloid of one sheet centered at theorigin.
If f is definite, similar argument as above shows that f is apiece of an ellipsoid or a hyperboloid of two sheets.
20 / 22

Main results (continued)
中心アファイン曲面に対する３次形式の平行性と再帰性
藤岡敦

Contents
Introduction
Aﬃne
hypersurfaces
Blaschkehypersurfaces
Centroaﬃne
hypersurfaces
Centroaﬃne
surfacesf : M 2 → R3 : a centroaﬃne surface∇: the induced connection
C̃ : the traceless part of the cubic form
Theorem 2 (F.-Hamamoto-Nakai)C̃ : recurrent relative to ∇:C̃ ̸= 0 & ∃ σ ∈ Ω1 (M) s.t. ∇C̃ = σ ⊗ C̃=⇒ f : a piece of a ruled surface given byf (x, y ) = A′ (x) + yA(x)21 / 22

中心アファイン曲面に対する３次形式の平行性と再帰性
藤岡敦

Contents
Introduction
Aﬃne
hypersurfaces

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Blaschkehypersurfaces
Centroaﬃne
hypersurfaces
Centroaﬃne
surfaces

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