中心アファイン曲面に対する3次形式の平行性と再中心ア...

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

Centroaffinehypersurfaces

Centroaffinesurfaces


藤岡敦

関西大学システム理工学部数学科

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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藤岡敦

Contents

Introduction

Affinehypersurfaces




Contents

1 Introduction

2 Affine hypersurfaces

3 Blaschke hypersurfaces

4 Centroaffine hypersurfaces

5 Centroaffine surfaces

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藤岡敦

Contents

Introduction

Affinehypersurfaces




Backgrounds

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 is

a piece of a Cayley surface: z = xy − 1

3x3.

· A graph of a cubic polynomial· A ruled surface· Equiaffinely homogeneous· An improper affine sphere

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Contents

Introduction

Affinehypersurfaces




Problem

Problem

Characterize fundamental centroaffine surfaces by the cubicform.

Examples of fundamental centroaffine surfaces• Quadrics• A ruled surface given by

f (x , y) = A′(x) + yA(x)

A is an R3-valued function s.t. det

AA′

A′′

= 0.

· The curvature of the centroaffine metric is 1.· The Pick function vanishes.· Centroaffine minimal· Projective minimal

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藤岡敦

Contents

Introduction

Affinehypersurfaces




Euclidean differential geometry and affinedifferential geometry

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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Contents

Introduction

Affinehypersurfaces




Gauss-Weingarten formulaEuclidean differential geometry

f : Mn → Rn+1: a hypersurfaceD: the standard flat connection on Rn+1

X ,Y ∈ X(M) Euclidean differential geometryn: a unit normal vector field=⇒ Gauss-Weingarten formula:

DX f∗Y = f∗∇XY + h(X ,Y )n (Gauss)

DXn = −f∗SX (Weingarten)

∇: the Levi-Civita connectionh: the second fundamental formS : the shape operator

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藤岡敦

Contents

Introduction

Affinehypersurfaces




Gauss-Weingarten formulaAffine differential geometry

f : Mn → Rn+1: a hypersurfaceD: the standard flat connection on Rn+1

X ,Y ∈ X(M) Affine differential geometryξ: a transversal vector field=⇒ Gauss-Weingarten formula:

DX f∗Y = f∗∇XY + h(X ,Y )ξ (Gauss)

DX ξ = −f∗SX + τ(X )ξ (Weingarten)

f is called an affine hypersurface.∇: the induced connectionh: the affine fundamental form

Considered as a metric.S : the affine shape operatorτ : the transversal connection form

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Contents

Introduction

Affinehypersurfaces




Nondegenerate, definite or indefinite affinehypersurfaces

Definition

f : Mn → Rn+1: an affine hypersurfaceh: the affine fundamental form

f : nondegenerate (resp. definite, indefinite) def.

h: nondegenerate (resp. definite, indefinite)

Proposition

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

Proof

ξ := φξ + f∗Z (φ : M → R \ 0, Z ∈ X(M)) =⇒ φh = h

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Contents

Introduction

Affinehypersurfaces




Cubic form

f : Mn → Rn+1: an affine hypersurface∇: the induced connectionh: the affine fundamental formτ : the transversal connection formX ,Y ,Z ∈ X(M)

Codazzi equation

(∇Xh)(Y ,Z ) + τ(X )h(Y ,Z ) = (∇Y h)(X ,Z ) + τ(Y )h(X ,Z )

The cubic form:

C (X ,Y ,Z ) := (∇Xh)(Y ,Z ) + τ(X )h(Y ,Z )

Defines a symmetric (0, 3)-tensor.

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藤岡敦

Contents

Introduction

Affinehypersurfaces




Definition of Blaschke hypersurfaces

f : Mn → Rn+1: a nondegenerate affine hypersurfaceθ: the volume form induced by the immersion f

θ(X1, . . . ,Xn) := ω(f∗X1, . . . , f∗Xn, ξ) (X1, . . . ,Xn ∈ X(M))

ω: a parallel volume form with respect to the flat connection D

ωh: the volume form with respect to the affine fundamentalform hX1, . . . ,Xn ∈ X(M) s.t. θ(X1, . . . ,Xn) = 1

ωh(X1, . . . ,Xn) := |det(h(Xi ,Xj))|12

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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藤岡敦

Contents

Introduction

Affinehypersurfaces




Characterizations by the cubic form

f : Mn → 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

∇: the Levi-Civita connection for the affine fundamental form h1989 Magid-Nomizu: n = 2, ∇C = 0, C = 0

• Example: a Cayley surface

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Contents

Introduction

Affinehypersurfaces




Definition of centroaffine hypersurfaces

Definition

f : Mn → Rn+1: a hypersurfacef : a centroaffine hypersurface

def.The radial vector intersects with the tangent

space transversally at any point.

ξ := −n+1∑i=1

xi∂

∂xi

Weingarten formula:DX ξ = −f∗X

S = the identity, τ = 0

The affine fundamental form h is called a centroaffine metric.12 / 22


藤岡敦

Contents

Introduction

Affinehypersurfaces




Fundamental facts

Consider nondegenerate centroaffine hypersurfaces.f : Mn → Rn+1: a centroaffine hypersurfaceC : the cubic form

Proposition

C = 0 =⇒ f : a piece of a quadric centered at the origin

Remark

In general, a nondegenerate affine hypersurface with τ = 0 &C = 0 is a piece of a quadric.

∇: the induced connection∇: the Levi-Civita connection for the centroaffine metric h1991 Li-Wang: ∇C = 0 & flat2015 Hildebrand: ∇C = 0 (∇C = 0)

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Contents

Introduction

Affinehypersurfaces




Tchebychev form

f : Mn → Rn+1: a centroaffine hypersurfaceh: the centroaffine metricC : the cubic formX ∈ X(M)Define a (1, 1)-tensor AX by

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

Define a 1-form trhC by

(trhC )(X ) = trAX .

T :=1

ntrhC is called a Tchebychev form.

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Contents

Introduction

Affinehypersurfaces




Traceless part of the cubic form

f : Mn → Rn+1: a centroaffine hypersurfaceh: the centroaffine metricC : the cubic formT : the Tchebychev formX ,Y ,Z ∈ X(M)

C (X ,Y ,Z ) := C (X ,Y ,Z )− n

n + 2(T (X )h(Y ,Z )

+ T (Y )h(Z ,X ) + T (Z )h(X ,Y ))

(traceless part of C )

Remark

C coincides with the cubic form as a Blaschke hypersurface.

∇: the Levi-Civita connection for h1997 Liu-Wang: n = 2, ∇C = 0

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藤岡敦

Contents

Introduction

Affinehypersurfaces




Definiteness and the Euclidean Gaussian curvature

Proposition

f : M2 → R3: a centroaffine surfacef : definite (resp. indefinite)

the Euclidean Gaussian curvature: positive (resp. negative)

Proof

(x1, x2): local coordinatesGauss formula:

fxixj = 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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藤岡敦

Contents

Introduction

Affinehypersurfaces




Notations

Consider the indefinite case.f : M2 → R3: a centroaffine surfaceK : the Euclidean Gaussian curvature < 0(u, v): asymptotic line coordinatesφ := h(∂u, ∂v )d : the signed distance from the origin to the tangent plane

ρ := −1

4log

(− K

d4

)

a := φ det

ffufuu

/det

ffufv

b := φ det

ffvfvv

/det

ffvfu

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藤岡敦

Contents

Introduction

Affinehypersurfaces




Gauss formula in asymptotic line coordinates

Gauss formula fuu =

(φu

φ+ ρu

)fu +

a

φfv

fuv = −φf + ρv fu + ρufv

fvv =

(φv

φ+ ρv

)fv +

b

φfu

Proposition

The integrability conditions for the above Gauss formula are(logφ)uv = −φ− ab

φ2+ ρuρv

av + ρuφu = ρuuφ

bu + ρvφv = ρvvφ

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藤岡敦

Contents

Introduction

Affinehypersurfaces




Covariant derivative of the cubic form

Lemma

f : M2 → R3: an indefinite centroaffine surface∇: the induced connectionC : the cubic form

∇C = 0 ⇐⇒ All the functions below vanish.

−au +3aφu

φ+ 6aρu, −av + 3aρv + 3ρ2uφ,

−bv +3bφv

φ+ 6bρv , −bu + 3bρu + 3ρ2vφ,

3aρv + ρuφu − ρuuφ+ 3ρ2uφ, 3bρu + ρvφv − ρvvφ+ 3ρ2vφ,

ab

φ+ 5ρuρvφ− ρuvφ

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藤岡敦

Contents

Introduction

Affinehypersurfaces




Main results

f : M2 → R3: a centroaffine surface∇: the induced connectionC : 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)

Proof

If f is indefinite, show that C = 0 by use of the integrabilityconditions 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.

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Contents

Introduction

Affinehypersurfaces




Main results (continued)

f : M2 → R3: a centroaffine surface∇: the induced connectionC : the traceless part of the cubic form

Theorem 2 (F.-Hamamoto-Nakai)

C : recurrent relative to ∇:

C = 0 & ∃σ ∈ Ω1(M) s.t. ∇C = σ ⊗ C

=⇒ f : a piece of a ruled surface given by

f (x , y) = A′(x) + yA(x)

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藤岡敦

Contents

Introduction

Affinehypersurfaces




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中心アファイ ン曲面に対す る3次形式の 平行性と再 中心ア...

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中心アファイン曲面に対する3次形式の平行性と再中心ア...