Centroaﬃne
surfaces withparallel orrecurrent
cubic form
relative to theinducedconnection

Centroaﬃne surfaces with parallel or recurrent
cubic form relative to the induced connection

AtsushiFujioka

Atsushi Fujioka
Contents
Introduction
Aﬃne
hypersurfaces
Blaschkehypersurfaces
Centroaﬃne
hypersurfaces
Centroaﬃne
surfaces

Faculty of Engineering Science
Kansai University

March 15, 2017
Osaka City University
The 2nd OCAMI-KOBE-WASEDA Joint International Workshop
on Diﬀerential Geometry and Integrable Systems
Joint work with K. Hamamoto and Y. Nakai
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Contents
Centroaﬃnesurfaces withparallel orrecurrent
cubic form
relative to the
inducedconnection
AtsushiFujioka1

Introduction

2Aﬃne hypersurfaces

3Blaschke hypersurfaces

4

Centroaﬃne hypersurfaces

5

Centroaﬃne surfaces

Contents
Introduction
Aﬃne
hypersurfaces
Blaschkehypersurfaces
Centroaﬃne
hypersurfaces
Centroaﬃne
surfaces2 / 22

Backgrounds
Centroaﬃne
surfaces withparallel orrecurrent
cubic form
relative to the
inducedconnection
AtsushiFujiokaContents
Introduction
Aﬃne
hypersurfaces
Blaschkehypersurfaces
Centroaﬃnehypersurfaces
Centroaﬃnesurfaces◦ 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

ProblemCentroaﬃne
surfaces withparallel orrecurrent
cubic form
relative to the
inducedconnection
AtsushiFujiokaContents
Introduction
Aﬃne
hypersurfaces
Blaschkehypersurfaces
Centroaﬃne
hypersurfaces
Centroaﬃne
surfaces

Problem
Characterize fundamental centroaﬃne surfaces by the cubicform.
◦ 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
Centroaﬃne
surfaces withparallel orrecurrent
cubic form
relative to the
inducedconnection
AtsushiFujiokaContents
Introduction
Aﬃne
hypersurfaces
Blaschkehypersurfaces
Centroaﬃne
hypersurfaces
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 radial vector field.
· Study properties invariant under the aﬃnetransformation fixing the origin.
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Gauss-Weingarten formula
Euclidean diﬀerential geometry
Centroaﬃne
surfaces withparallel orrecurrent
cubic form
relative to the
inducedconnection
AtsushiFujiokaContents
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

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Gauss-Weingarten formulaAﬃne diﬀerential geometry
Centroaﬃne
surfaces withparallel orrecurrent
cubic form
relative to the
inducedconnection
AtsushiFujiokaContents
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
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Nondegenerate, definite or indefinite aﬃne
hypersurfaces
Centroaﬃne
surfaces withparallel orrecurrent
cubic form
relative to theinducedconnection
AtsushiFujiokaContentsDefinitionf : M n → Rn+1 : an aﬃne hypersurface
h: the aﬃne fundamental formf : nondegenerate (resp. definite, indefinite)⇕ def.
h: nondegenerate (resp. definite, indefinite)

Introduction
Aﬃne
hypersurfaces
Blaschkehypersurfaces

Proposition
The above definition is independent of the choice of thetransversal vector field ξ.

Centroaﬃne
hypersurfaces
Centroaﬃne
surfaces

Proofξ¯ := φξ + f∗ Z (φ : M → R \ {0}, Z ∈ X(M)) =⇒ φh̄ = h
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Cubic form
Centroaﬃne
surfaces withparallel orrecurrent
cubic form
relative to theinduced
connection
AtsushiFujioka
Contents
Introduction
Aﬃne
hypersurfaces
Blaschkehypersurfaces
Centroaﬃne
hypersurfaces
Centroaﬃne
surfacesf : 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
Centroaﬃne
surfaces withparallel orrecurrent
cubic form
relative to the
inducedconnection
AtsushiFujiokaContents
Introduction
Aﬃne
hypersurfaces
Blaschkehypersurfaces
Centroaﬃne
hypersurfaces
Centroaﬃne
surfacesf : 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 fundamentalform h
X1 , . . . , Xn ∈ X(M) s.t. θ(X1 , . . . , Xn ) = 1
1ωh (X1 , . . . , Xn ) := |det(h(Xi , Xj ))| 2
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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Characterizations by the cubic form
Centroaﬃnesurfaces withparallel orrecurrent
cubic form
relative to the
inducedconnection
AtsushiFujioka

f : M n → Rn+1 : a Blaschke hypersurface
C : the cubic form
Maschke-Pick-Berwald’s theorem
C = 0 =⇒ f : a piece of a quadric∇: the induced connection

Contents
Introduction

Theorem (Nomizu-Pinkall 1989)

Aﬃne
hypersurfacesn = 2, ∇C = 0, C ̸= 0 =⇒ f : a piece of a Cayley surface

Blaschkehypersurfaces

1988 Vrancken: n = 3, ∇C = 0, C ̸= 0

Centroaﬃne
hypersurfaces

ˆ the Levi-Civita connection for the aﬃne fundamental form h∇:ˆ = 0, C ̸= 0
1989 Magid-Nomizu: n = 2, ∇C
2011 Hu-Li-Vrancken: n: arbitrary & h: definte
2011 Hu-Li-Li-Vrancken: n: arbitrary & h: LorentzianCentroaﬃne
surfaces

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Definition of centroaﬃne hypersurfaces
Centroaﬃne
surfaces withparallel orrecurrent
cubic form
relative to the
inducedconnection
AtsushiFujioka

Definitionf : M n → Rn+1 : a hypersurface
f : a centroaﬃne hypersurface⇕ def.
The radial vector intersects with the tangentspace transversally at any point.

Contents
Introduction
Aﬃne
hypersurfaces
Blaschkehypersurfaces
Centroaﬃne
hypersurfaces
Centroaﬃne
surfaces

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
Centroaﬃne
surfaces withparallel orrecurrent
cubic form
relative to the
inducedconnection
AtsushiFujiokaContents

Consider nondegenerate centroaﬃne hypersurfaces.
f : M n → Rn+1 : a centroaﬃne hypersurface
C : the cubic form
Proposition
C = 0 =⇒ f : a piece of a quadric centered at the origin

Introduction
Aﬃne
hypersurfaces
Blaschkehypersurfaces
Centroaﬃne
hypersurfaces
Centroaﬃne
surfaces∇: the induced connection
ˆ the Levi-Civita connection for the centroaﬃne metric h∇:ˆ = 0 & flat
1991 Li-Wang: ∇C
ˆ = 0 (∇C = 0)2015 Hildebrand: ∇C
ˆ = 0 & h: definite (geometric)
2017 Cheng-Hu-Moruz: ∇C

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Tchebychev form
Centroaﬃne
surfaces withparallel orrecurrent
cubic form
relative to the
inducedconnection
AtsushiFujiokaf : M n → Rn+1 : a centroaﬃne hypersurface
h: the centroaﬃne metricC : the cubic form
X ∈ X(M)Define a (1, 1)-tensor AX byContents

C (X , Y , Z ) = h(AX (Y ), Z )Introduction
Aﬃne
hypersurfaces(∀ Y , Z ∈ X(M)).

Define a 1-form trh C by

Blaschkehypersurfaces(trh C )(X ) = trAX .

Centroaﬃne
hypersurfaces
Centroaﬃne
surfaces

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

Traceless part of the cubic form
Centroaﬃne
surfaces withparallel orrecurrent
cubic form
relative to the
inducedconnection
AtsushiFujioka
ContentsIntroduction

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 ))Aﬃne
hypersurfaces
BlaschkehypersurfacesCentroaﬃne
hypersurfaces
Centroaﬃnesurfaces(traceless part of C )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
Centroaﬃne
surfaces withparallel orrecurrent
cubic form
relative to theinducedconnection
AtsushiFujiokaContents

Propositionf : M 2 → R3 : a centroaﬃne surface
f : definite (resp. indefinite)⇕the Euclidean Gaussian curvature: positive (resp. negative)

Introduction

Centroaﬃne
hypersurfaces

Proof(x1 , x2 ): local coordinates
Gauss formula:fxi xj = Γ1ij fx1 + Γ2ij fx2 − h(∂xi , ∂xj )f

Centroaﬃne
surfaces

Take the inner product with the unit normal vector field.

Aﬃne
hypersurfaces
Blaschkehypersurfaces(i = 1, 2)

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Notations
Centroaﬃne
surfaces withparallel orrecurrent
cubic form
relative to theinduced
connection
AtsushiFujiokaContents
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
/ 
ff

a := 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
Centroaﬃne
surfaces withparallel orrecurrent
cubic form
relative to theinducedconnection
AtsushiFujiokaContentsGauss formula)(aφu
+ ρu fu + fv
fuu =
φφ
fuv = −φf + ρv fu + ρu fv
()

φvb

fvv =+ ρv fv + fu
φφIntroduction
Aﬃne
hypersurfaces
Blaschkehypersurfaces
Centroaﬃne
hypersurfaces
Centroaﬃne
surfaces

Proposition
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
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Covariant derivative of the cubic form
Centroaﬃne
surfaces withparallel orrecurrent
cubic form
relative to theinducedconnection
AtsushiFujiokaContents
Introduction
Aﬃne
hypersurfaces
Blaschkehypersurfaces
Centroaﬃne
hypersurfaces
Centroaﬃne
surfacesLemma
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,
φ3aρv + ρu φu − ρuu φ + 3ρ2u φ, 3bρu + ρv φv − ρvv φ + 3ρ2v φ,ab+ 5ρu ρv φ − ρuv φφ
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Main results
Centroaﬃne
surfaces withparallel orrecurrent
cubic form
relative to the
inducedconnection
AtsushiFujiokaf : 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)

Contents
Introduction
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)
Centroaﬃne
surfaces withparallel orrecurrentcubic form
relative to the
inducedconnection
AtsushiFujiokaf : M 2 → R3 : a centroaﬃne surface∇: the induced connection
C̃ : the traceless part of the cubic form
Theorem 2 (F.-Hamamoto-Nakai)Contents
Introduction
Aﬃne
hypersurfaces
Blaschkehypersurfaces
Centroaﬃne
hypersurfacesC̃ : 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)Centroaﬃnesurfaces

21 / 22Centroaﬃne
surfaces withparallel orrecurrent
cubic form
relative to the
inducedconnection
AtsushiFujiokaContents

Thank you for your attention.

Introduction
Aﬃne
hypersurfaces
Blaschkehypersurfaces
Centroaﬃne
hypersurfaces
Centroaﬃne
surfaces

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