Tchebychev
作用素が半単純でない
平坦中心アファイン曲面藤岡敦

Contents

Tchebychev 作用素が半単純でない
平坦中心アファイン曲面

Introduction
Blaschkehypersurfaces

藤岡敦

Centroaﬃne
hypersurfaces
Centroaﬃnesurfaces
Flatcentroaﬃnesurfaces

関西大学システム理工学部数学科

２０１５年７月２５日（土）慶応義塾大学, 非可換幾何学と数理物理学

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Contents
Tchebychev
作用素が半単純でない
平坦中心アファイン曲面

1Introduction

2

Blaschke hypersurfaces

3

Centroaﬃne hypersurfaces

4

Centroaﬃne surfaces

5

Flat centroaﬃne surfaces

藤岡敦

Contents
Introduction
Blaschkehypersurfaces
Centroaﬃne
hypersurfaces
Centroaﬃnesurfaces
Flatcentroaﬃne
surfaces2 / 21

Euclidean diﬀerential geometry and aﬃnediﬀerential geometry
Tchebychev
作用素が半単純でない
平坦中心アファイン曲面
藤岡敦

Contents
Introduction
Blaschkehypersurfaces
Centroaﬃne
hypersurfaces
Centroaﬃne
surfaces
Flat
centroaﬃne
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 position vector field.
· Study properties invariant under the aﬃnetransformation fixing the origin.
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Gauss-Weingarten formula
Euclidean diﬀerential geometry
Tchebychev
作用素が半単純でない
平坦中心アファイン曲面
藤岡敦

Contents
Introduction
Blaschkehypersurfaces
Centroaﬃne
hypersurfaces
Centroaﬃne
surfaces
Flatcentroaﬃne
surfacesf : M n → Rn+1 : a hypersurfaceD: 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

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Gauss-Weingarten formulaAﬃne diﬀerential geometry
Tchebychev
作用素が半単純でない
平坦中心アファイン曲面藤岡敦

Contents
Introduction
Blaschkehypersurfaces
Centroaﬃne
hypersurfaces
Centroaﬃne
surfaces
Flatcentroaﬃnesurfaces

f : 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
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Nondegenerate, definite or indefinite aﬃne
hypersurfaces
Tchebychev
作用素が半単純でない
平坦中心アファイン曲面
藤岡敦

Contents
Introduction
Blaschkehypersurfaces
Centroaﬃne
hypersurfaces
Centroaﬃne
surfaces
Flat
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 = h6 / 21

Definition of Blaschke hypersurfaces
Tchebychev
作用素が半単純でない
平坦中心アファイン曲面
藤岡敦Contents
Introduction
Blaschkehypersurfaces

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

Centroaﬃne
hypersurfaces
Centroaﬃne
surfaces
Flat
centroaﬃne
surfaces

Proposition∃ ξ (unique up to the sign) s.t. τ ≡ 0 & θ = ωh

The 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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Aﬃne hyperspheres
Tchebychev
作用素が半単純でない平坦中心アファイン曲面藤岡敦

Definitionf : M n → Rn+1 : a Blaschke hypersurface
S: the aﬃne shape operator
def.

Contents
Introduction
Blaschkehypersurfaces
Centroaﬃne
hypersurfaces
Centroaﬃne
surfaces
Flatcentroaﬃne
surfacesf : an aﬃne hypersphere ⇐⇒ S: a scalar operator
S = 0: improperS ̸= 0: proper
Propositionf : M n → Rn+1 : an aﬃne hypersphereξ: a Blaschke normal vector field◦ f : improper ⇐⇒ ξ: a constant vector◦ f : proper ⇐⇒ The lines through f with direction ξ meet at
one point (the center).
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Examples
Tchebychev
作用素が半単純でない
平坦中心アファイン曲面
藤岡敦

Example (Quadrics)◦ A proper noncentral quadric is an improper aﬃne
hypersphere.
◦ A proper central quadric is a proper aﬃne hypersphere.

Contents
Introduction
Blaschkehypersurfaces
Centroaﬃne
hypersurfaces
Centroaﬃne
surfaces
Flat
centroaﬃne
surfaces

Theorem (M. A. Magid-P. J. Ryan 1990)
f = (X , Y , Z ) : M 2 → R3 : an aﬃne sphere
If the curvature of the Blaschke metric is 0, then f is one of
the following up to the aﬃne congruence.
◦ Z = X 2 + Y 2 (definite, improper)◦ Z = XY + a(X ), a: an arbitrary function(indefinite, improper)◦ XYZ = 1 (definite, proper)◦ (X 2 + Y 2 )Z = 1 (indefinite, proper)
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Definition of centroaﬃne hypersurfaces
Tchebychev
作用素が半単純でない
平坦中心アファイン曲面
藤岡敦

Contents
Introduction
Blaschkehypersurfaces
Centroaﬃne
hypersurfaces
Centroaﬃne
surfaces
Flatcentroaﬃne
surfaces

Definitionf : M n → Rn+1 : a hypersurface
f : a centroaﬃne hypersurface⇕ def.
The position 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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Tchebychev operator
Tchebychev
作用素が半単純でない
平坦中心アファイン曲面藤岡敦

Contents
Introduction
Blaschkehypersurfaces
Centroaﬃne
hypersurfaces
Centroaﬃne
surfaces
Flat
centroaﬃne
surfaces

Consider nondegenerate centroaﬃne hypersurfaces.
f : M n → Rn+1 : a centroaﬃne hypersurface∇: the induced connection∇h : the Levi-Civita connection for the centroaﬃne metric h
C := ∇ − ∇h : the diﬀerence tensor
1
T := trh C : the Tchebychev vector fieldnϕ(trh C ) = trAϕ (∀ ϕ ∈ Ω1 (M))
A: a (1, 1) tensor s.t.
(ϕ ◦ C )(X , Y ) = h(Aϕ (X ), Y )(∀ X , Y ∈ X(M))

The (1, 1) tensor ∇h T is called a Tchebychev operator.
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Examples
Tchebychev
作用素が半単純でない
平坦中心アファイン曲面
藤岡敦

ContentsIntroductionBlaschke
hypersurfaces
Centroaﬃne
hypersurfaces
Centroaﬃne
surfaces
Flatcentroaﬃne
surfacesf : M n → Rn+1 : a centroaﬃne hypersurface
C : the diﬀerence tensor
T : the Tchebychev vector field∇h T : the Tchebychev operator
Maschke-Pick-Berwald’s theorem
C ≡ 0 ⇐⇒ f : a quadric centered at the origin
PropositionT ≡ 0 ⇐⇒ f : a proper aﬃne hypersphere centeredat the origin
1995 H. L. Liu-C. P. Wang: Classified centroaﬃne surfaces with∇h T ≡ 012 / 21

Definiteness and the Euclidean Gaussian curvature
Tchebychev
作用素が半単純でない
平坦中心アファイン曲面
藤岡敦

Contents
Introduction
Blaschkehypersurfaces

Consider nondegerate centroaﬃne surfaces.
Propositionf : M → R3 : a centroaﬃne surface
f : definite (resp. indefinite)⇕
the Euclidean Gaussian curvature: positive (resp. negative)

Centroaﬃne
hypersurfaces
Centroaﬃne
surfaces
Flat
centroaﬃne
surfacesProof(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.
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Notations
Tchebychev
作用素が半単純でない
平坦中心アファイン曲面
藤岡敦

Contents
Introduction
Blaschkehypersurfaces
Centroaﬃne
hypersurfaces
Centroaﬃne
surfaces
Flat
centroaﬃne
surfaces

Consider the indefinite case.
f : M → R3 : a centroaﬃne surface
K : the Euclidean Gaussian curvature < 0(x, y ): asymptotic line coordinatesψ := h(∂x , ∂y )d: the signed distance from the origin to the tangent plane()
1Kρ := − log − 44d/ ffα := ψ det  fx  det  fx fxx
fy
/ ffβ := ψ det  fy  det  fy fyyfx

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Gauss formula in asymptotic line coordinates
Tchebychev
作用素が半単純でない
平坦中心アファイン曲面

Gauss formula藤岡敦

Contents
Introduction
Blaschkehypersurfaces

()
ψxα


+ ρx fx + fy
fxx =ψψ
fxy = −ψf + ρy fx + ρx fy
()

ψyβ


fyy =+ ρy fy + fx
ψψCentroaﬃne
hypersurfaces
Centroaﬃne
surfaces
Flat
centroaﬃnesurfaces

ProofUseΓxxy = −1 Ky
1 Kx, Γyxy = −
4 K
4K

etc.
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Integrability conditions
Tchebychev
作用素が半単純でない
平坦中心アファイン曲面
藤岡敦Contents
Introduction
Blaschkehypersurfaces
Centroaﬃne
hypersurfaces
Centroaﬃne
surfaces
Flat
centroaﬃne
surfaces

Proposition
The integrability conditions for the above Gauss formula are
αβ(log ψ)xy = −ψ − ψ 2 + ρx ρyαy + ρx ψx = ρxx ψβx + ρy ψy = ρyy ψIf ρ is constant and α, β ̸= 0, changing the coordinates, ifnecessary, we obtain Tzitzéica equation:1(log ψ)xy = −ψ − 2ψProposition
T : the Tchebychev vector field=⇒ T = gradh ρ
16 / 21

Equations of associativity
Tchebychev
作用素が半単純でない
平坦中心アファイン曲面
藤岡敦

Contents
Introduction
Blaschkehypersurfaces
Centroaﬃne
hypersurfaces
Centroaﬃne
surfaces
Flatcentroaﬃne
surfacesf : M → R3 : an indefinite centroaﬃne surface(x, y ): asymptotic line coordinates
κ: the curvature of the centroaﬃne metric h(log ψ)xyκ=−ψConsider the flat case. (κ ≡ 0)
May assume ψ ≡ 1.
=⇒ Equation of associativity in topological field theory:gxxx gyyy − gxxy gxyy + 1 = 0(ρ = gxy , α = gxxx , β = gyyy )2004 E. V. Ferapontov: Showed that the integrability
conditions for flat centroaﬃne
hypersurfaces are equivalent toequations of associativity.
17 / 21

Centroaﬃne surfaces with vanishing Tchebychev
operator
Tchebychev
作用素が半単純でない
平坦中心アファイン曲面
藤岡敦

Contents
Introduction
Blaschkehypersurfaces
Centroaﬃne
hypersurfaces
Centroaﬃnesurfaces
Flatcentroaﬃnesurfaces

f = (X , Y , Z ) : M → R3 : a centroaﬃne surface∇h T : the Tchebychev operator
κ: the curvature of the centroaﬃne metric h∇h T ≡ 0 =⇒ κ ≡ 0 except proper aﬃne spheres centeredat the origin with κ ≡ 1
The case κ ≡ 0: p, q, r ∈ Rp q r◦X{ Y (Z = 1)}
X−1◦ exp −p tan(X 2 + Y 2 )q Z r = 1Y
◦ Z = −X (p log X + q log Y )Y2◦ Z = ±X log X +X
◦ f = (e x , A1 (x)e y , A2 (x)e y ), A1 , A2 : Linearly independent
solutions to the diﬀerential equation:
A′′ − A′ − a(x)A = 0, a: an arbitrary function
18 / 21

Another example
Tchebychev
作用素が半単純でない
平坦中心アファイン曲面
藤岡敦

Contents
Introduction
Blaschkehypersurfaces
Centroaﬃne
hypersurfaces
Centroaﬃne
surfaces
Flat
centroaﬃne
surfaces

Theorem (F. 2014)
f : M → R3 : a flat centroaﬃne surface∇h T : the Tchebychev operator
11r
J := ∥C ∥2 = hip hjq hkr Cijk Cpq: the Pick function
22∇h T : nonsemisimple & J: constant)(∞∞∞∑∑∑=⇒ f =φn,1 (x)y n ,φn,2 (x)y n ,φn,3 (x)y n
n=0

n=0

n=0

(x, y ): asymptotic line coordinates around (x0 , 0) s.t.
x0 ̸= 0
φ0,1 , φ0,2 , φ0,3 : Linearly independent solutions to thediﬀerential equation: xφ′′′ + φ′′ − φ = 0(n + 1)φn+1,i = xφ′′n,i (i = 1, 2, 3)
19 / 21

Characterizations of the surface
Tchebychev
作用素が半単純でない
平坦中心アファイン曲面
藤岡敦Contents
Introduction
BlaschkehypersurfacesCentroaﬃne
hypersurfaces
Centroaﬃnesurfaces
Flatcentroaﬃne
surfaces◦ J ≡ −1
◦ Solutions to the equation of associativity:
11g = xy 3 − x 2 log x + (polynomials with degree ≤ 2)
62
◦ Centroaﬃne minimal: Extremals of the area integral of
the centroaﬃne metric h⇐⇒ tr ∇h T ≡ 0◦ The diﬀerential equation can be solved by using Meijer G
functions:
( 2)
( 2)
x——
2,0 x
1,0
φ = c1 G0,3+ ic2 G0,3 −8 12 , 12 , 0
8 12 , 12 , 0)
( 2x—
1,0
(c1 , c2 , c3 ∈ R)
+ c3 G0,3−
1 10,82, 2
The second and the third terms can be written by using
the generalized hypergeometric function 0 F2 .
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Tchebychev
作用素が半単純でない
平坦中心アファイン曲面藤岡敦

Contents
Introduction
Blaschkehypersurfaces

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Centroaﬃne
hypersurfaces
Centroaﬃne
surfaces
Flat
centroaﬃne
surfaces

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