Equivariant
projections
betweenspaces ofclosed equicentroaﬃnecurvesEquivariant projections between spaces ofclosed equicentroaﬃne curves

AtsushiFujioka
Contents
Introduction
EquicentroaﬃnecurvesClosed equicentroaﬃnecurvesAction of thediﬀeomorphism groupof the circle
Equivariant
projections

Atsushi Fujioka
Faculty of Engineering Science
Kansai University

December 25, 2021
Chongqing University of Technology
The 6th China-Japan Geometry Conference
Joint work with T. Kurose and H. Moriyoshi
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Contents
Equivariant
projections
betweenspaces ofclosed equicentroaﬃnecurves1Introduction

2

Equicentroaﬃne curves

3

Closed equicentroaﬃne curves

Closed equicentroaﬃnecurves4

Action of the diﬀeomorphism group of the circle

Action of thediﬀeomorphism groupof the circle

5

Equivariant projections

AtsushiFujiokaContents
Introduction
EquicentroaﬃnecurvesEquivariant
projections
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Backgrounds and main results
Equivariant
projections
betweenspaces ofclosed equicentroaﬃnecurvesAtsushiFujiokaContents
Introduction
EquicentroaﬃnecurvesClosed equicentroaﬃnecurvesAction of thediﬀeomorphism groupof the circle◦ The space of closed equicentroaﬃne plane curves• ∃ Action of the diﬀeomorphism group of the circle• Considered as an action of the Virasoro-Bott group.
• The space is considered as the coadjoint orbit of the dualof the Virasoro algebra.
• Studied from the viewpoint of symplectic geometry.
◦ Today: Consider the space of closed equicentroaﬃne curves ingeneral vector space.
• ∃ Action of the diﬀeomorphism group of the circle• Deﬁne projections into the space of plane or space curves.
• The above projections are equivariant w.r.t. the aboveaction.

Equivariant
projections
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Deﬁnition of the equicentroaﬃne curve
Equivariant
projections
betweenspaces ofclosed equicentroaﬃnecurvesAtsushiFujiokaDeﬁnition
I : an intervaln = 2, 3, 4, . . .
γ : I → Rn \ {0}: an equicentroaﬃne curve

Contents
Introduction
EquicentroaﬃnecurvesClosed equicentroaﬃnecurvesAction of thediﬀeomorphism groupof the circle




det 


m def.
γγ′ 
≡1..
.
γ (n−1)s ∈ I is called an equicentroaﬃne arclength parameter.

Equivariant
projections
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The fundamental theorem of equicentroaﬃnecurvesEquivariant
projections
betweenspaces ofclosed equicentroaﬃnecurvesAtsushiFujiokaContents
Introduction
EquicentroaﬃnecurvesClosed equicentroaﬃnecurvesAction of thediﬀeomorphism groupof the circle
Equivariant
projections

γ : I → Rn \ {0}: an equicentroaﬃne curve=⇒ ∃ κ1 , κ2 , . . . , κn−1 : I → R s.t.
γ (n) + κ1 γ (n−2) + κ2 γ (n−3) + · · · + κn−1 γ = 0For i = 1, 2, . . . , n − 1, we call κi the i-th curvature.
The fundamental theorem of equicentroaﬃne curves
κ1 , κ2 , . . . , κn−1 : I → R=⇒ ∃ γ : I → Rn \ {0}: an equicentroaﬃne curve with the i-thcurvature κi
Unique up to equiaﬃne transformation
ﬁxing the origin.
Equiaﬃne transformation ﬁxing the origin:◦ Multiplication by the element of SL(n, R)◦ Called equicentroaﬃne transformation.
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Example
Equicentroaﬃne plane curves with constant curvature
Equivariant
projectionsbetween
spaces ofclosed equicentroaﬃnecurvesAtsushiFujiokaContents
Introduction
EquicentroaﬃnecurvesClosed equicentroaﬃnecurvesConsider equicentroaﬃne plane curves.
We call the ﬁrst curvature the equicentroaﬃne curvature.
Example 1 (Equicentroaﬃne plane curves with constantcurvature)γ : I → R2 \ {0}: an equicentroaﬃne plane curve
κ: the equicentroaﬃne curvature◦ κ ≡ 0 ⇐⇒ γ is a part of a line:γ(s) = (a + bs, c + ds)(a, b, c, d ∈ R, ad − bc = 1)

Action of thediﬀeomorphism groupof the circle◦ κ: a positive constant ⇐⇒ γ is a part of an ellipse:()(ss )1γ(s) = a cos , b sin(a, b > 0)
κ= 2 2ababa bEquivariant
projections◦ κ: a negative constant ⇐⇒ γ is a part of a hyperbola.
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Notations
Equivariant
projections
betweenspaces ofclosed equicentroaﬃnecurvesAtsushiFujiokaContents
Introduction
EquicentroaﬃnecurvesClosed equicentroaﬃnecurvesAction of thediﬀeomorphism groupof the circleEquivariantprojections◦ Consider an equicentroaﬃne curve with period 2π:
γ : R → Rn \ {0},where
det 

γγ′..
.



 ≡ 1,∀s ∈ R, γ(s + 2π) = γ(s).

γ (n−1)
We call γ a closed equicentroaﬃne curve.
◦ S 1 := R/2πZ (the circle)
We consider γ a map from S 1 .
◦ Mn : the set of all closed equicentroaﬃne curves in Rn \ {0}◦ Mn /SL(n, R): the set of all congruence classes of closed
equicentroaﬃne curves in Rn \ {0}7 / 22

Example
Closed equicentroaﬃne curves with constant curvatures when n is even
Equivariant
projections
betweenspaces ofclosed equicentroaﬃnecurves
AtsushiFujiokaContents
Introduction

Example 2 (Closed equicentroaﬃne curves with constantcurvatures when n is even)γ ∈ M2m (m ∈ N) s.t. κ1 , κ2 , . . . , κ2m−1 : constantγ is given byγ(s) = (cos λ1 s, sin λ1 s, . . . , cos λm s, µ sin λm s) (s ∈ S 1 ),where λ1 , λ2 , . . . , λm are distinct natural numbers,
t 2m + κ1 t 2m−2 + · · · + κ2m−1 = (t 2 + λ21 ) · · · (t 2 + λ2m )EquicentroaﬃnecurvesClosed equicentroaﬃnecurvesAction of thediﬀeomorphism groupof the circle
Equivariant
projections(κ2 = κ4 = · · · = κ2m−2 = 0)
and1 ∏ ∏ 2=λi(λi − λ2j )2 .
µmi=1i<j

A similar example can be given when n is odd.

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Deﬁnition of the action of the diﬀeomorphismgroup of the circle
Equivariant
projections
betweenspaces ofclosed equicentroaﬃnecurvesAtsushiFujiokaContents
Introduction
EquicentroaﬃnecurvesClosed equicentroaﬃnecurvesAction of thediﬀeomorphism groupof the circle
Equivariantprojections

Diﬀ (S 1 ): the group of all orientation preserving
diﬀeomorphisms of S 1
γ ∈ Mn , g ∈ Diﬀ (S 1 )γ̃(s) := (γ · g )(s) := (g ′ (s)) 2 (γ ◦ g )(s) (s ∈ S 1 )1−nProposition
γ · g deﬁnes an action of Diﬀ (S 1 ) on Mn , Mn /SL(n, R).
Proof
For k = 0, 1, 2, . . . , n − 1,γ̃ (k) = (g ′ ) 2 +k (γ (k) ◦ g ) + · · ·1−nandn−1 (∑1−nk=02)+k= 0.
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Notations
Equivariant
projections
betweenspaces ofclosed equicentroaﬃnecurvesAtsushiFujiokaContents
Introduction
EquicentroaﬃnecurvesClosed equicentroaﬃnecurvesAction of thediﬀeomorphism groupof the circle
Equivariant
projections◦ Consider the action of Diﬀ (S 1 ) on Mn :Mn 3 γ 7→ γ̃ = γ · g ∈ Mn

(g ∈ Diﬀ (S 1 ))◦ Derive the transformation rule for the curvatures.
1−n◦ α :=2◦ Write γ (k) ◦ g simply as γ (k) .
◦ h :=g ′′g′◦ Consider h, h′ , h′′ , . . . as independent variables.
◦ Deﬁne the degree of h(k) as (k + 1).
◦ Deﬁne the weighted degree of a polynomial P of h, h′ , h′′ ,....
◦ Denote the weighted degree by degw P.
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Transformation rule for the derivatives of curves
1/3
Equivariant
projections
betweenspaces ofclosed equicentroaﬃnecurvesAtsushiFujiokaContents
Introduction
EquicentroaﬃnecurvesClosed equicentroaﬃnecurves
Action of thediﬀeomorphism groupof the circle
Equivariantprojections

Lemma◦ For k = 1, 2, . . . , n,k−1∑γ̃ (k) =Pk,l γ̃ (l) + (g ′ )α+k γ (k) ,l=0where Pk,l is a homogeneous polynomial of h, h′ , h′′ , . . .
w.r.t. the weighted degree s.t.
degw Pk,l = k − l.
◦ The following three recurrence relations hold:• For k = 1, 2, . . . , n − 1,k−1∑∂Pk,0 (m+1)
h− (α + k)Pk,0 h.
Pk+1,0 =∂h(m)m=011 / 22

Transformation rule for the derivatives of curves
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Equivariant
projections
betweenspaces ofclosed equicentroaﬃnecurvesAtsushiFujiokaContents

Lemma (continued)• For k = 2, 3, . . . , n − 1, l = 1, 2, . . . , k − 1,
k−l−1
∑ ∂Pk,lPk+1,l =h(m+1) + Pk,l−1 − (α + k)Pk,l h.
(m)
∂h
m=0
• For k = 1, 2, . . . , n − 1,Pk+1,k = Pk,k−1 + (α + k)h.

Introduction
EquicentroaﬃnecurvesClosed equicentroaﬃnecurves

ProofFirst, diﬀerentiating the equation

Action of thediﬀeomorphism groupof the circle
Equivariant
projectionsγ̃ = (g ′ )α γ,we haveγ̃ ′ = αhγ̃ + (g ′ )α+1 γ ′ .
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Transformation rule for the derivatives of curves
3/3
Equivariant
projections
betweenspaces ofclosed equicentroaﬃnecurvesAtsushiFujiokaContents
Introduction
EquicentroaﬃnecurvesClosed equicentroaﬃnecurvesAction of thediﬀeomorphism groupof the circle
Equivariant
projections

Proof (continued)
Hence we haveP1,0 = αh.
γ̃ ′ is expressed as above.
P1,0 is a homogeneous polynomial s.t. degw P1,0 = 1.
Next, for k = 1, 2, . . . , n − 1, assume that• γ̃ (k) ’s are expressed as above.
• Pk,l is a homogeneous polynomial s.t. degw Pk,l = k − l.
◦ Compute γ̃ (k+1) using that Pk,l is a polynomial of h, h′ ,. . . , h(k−l−1) .
◦ Recurrence relations are derived.
◦ Compute degw Pk,l .
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Transformation rule for the curvatures
Equivariant
projections
betweenspaces ofclosed equicentroaﬃnecurves
AtsushiFujiokaContents
Introduction
EquicentroaﬃnecurvesClosed equicentroaﬃnecurvesAction of thediﬀeomorphism groupof the circle
Equivariantprojections

Theorem 1γ ∈ Mn
κi : the i-th curvature of γ (i = 1, 2, . . . , n − 1)γ̃ = γ · g (g ∈ Diﬀ (S 1 ))
κ̃i : the i-th curvature of γ̃=⇒ For l = 0, 1, 2, . . . , n − 3,n−2∑(g ′ )n−k κn−k−1 Pk,l .
κ̃n−l−1 = (g ′ )n−l κn−l−1 − Pn,l −k=l+1

κ̃1 = (g ′ )2 κ1 +n(n2 − 1)

S(g ),
12
where S(g ) is the Schwarzian derivative of g :( ′′ )′( )g
1 g ′′ 2S(g ) =−.
g′2 g′
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Transformation rule for the curvatures
Proof
Equivariant
projections
betweenspaces ofclosed equicentroaﬃnecurvesAtsushiFujiokaContents
Introduction
EquicentroaﬃnecurvesClosed equicentroaﬃnecurvesAction of thediﬀeomorphism groupof the circle
Equivariant
projections

Proof
By use of the transformation rule for the derivatives of curves,
we haveγ̃ (n) =n−2∑Pn,l γ̃ (l) − (g ′ )n κn−1 γ̃

l=0n−2∑−k=1((g ′ )n−k κn−k−1γ̃ (k) −k−1∑)
Pk,l γ̃ (l).

l=0Hence we have the ﬁrst equation in Theorem 1 andκ̃1 = (g ′ )2 κ1 − Pn,n−2 .
Use the recurrence relations to compute Pn,n−2 .
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Equivariant projections into the orbit of M2
Equivariant
projections
betweenspaces ofclosed equicentroaﬃnecurves
AtsushiFujiokaContents

n = 3, 4, 5, . . .
γ ∈ Mnκ1 : the ﬁrst curvature of γγ̄ : R → R2 \ {0}: an equicentroaﬃne plane curve s.t.
the equicentroaﬃne curvature =Introduction
EquicentroaﬃnecurvesClosed equicentroaﬃnecurves
Action of thediﬀeomorphism groupof the circle
Equivariantprojections6

κ1n(n2 − 1)Consider the action of Diﬀ (S 1 ).
Theorem 2If γ̄ ∈ M2 , then the correspondence from γ to γ̄ deﬁnes anequivariant map from the orbit of γ into the orbit of γ̄ :γ · g = γ̄ · g(g ∈ Diﬀ (S 1 )).
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Equivariant projections into the orbit of M2
Proof
Equivariant
projections
betweenspaces ofclosed equicentroaﬃnecurvesAtsushiFujiokaContents
Introduction
EquicentroaﬃnecurvesClosed equicentroaﬃnecurvesAction of thediﬀeomorphism groupof the circle
Equivariant
projections

Proof
By Theorem 1, the transformation rule for the ﬁrst curvature isgiven byn(n2 − 1)κ̃1 = (g ′ )2 κ1 +S(g ).
12¯ 1 : the equicentroaﬃne curvature of γ̄, γ · gκ̄1 , κ̃
Thenn(n2 − 1) ¯n(n2 − 1)n(n2 − 1)κ̃1 = (g ′ )2 ·κ̄1 +S(g )6612m¯ 1 = (g ′ )2 κ̄1 + 1 S(g )κ̃2This is the transformation rule for the equicentroaﬃnecurvature when n = 2.

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Example
Equivariant projection from the orbit of M2m
Equivariant
projections
betweenspaces ofclosed equicentroaﬃnecurvesAtsushiFujiokaContents
Introduction
EquicentroaﬃnecurvesClosed equicentroaﬃnecurvesAction of thediﬀeomorphism groupof the circle
Equivariant
projections

Example 3 (Equivariant projection from the orbit of M2m )m = 2, 3, 4, . . .
γ ∈ M2m : the closed equicentroaﬃne curve with constantcurvatures as in Example 2γ(s) = (cos λ1 s, sin λ1 s, . . . , cos λm s, µ sin λm s) (s ∈ S 1 )◦ λ1 , λ2 , . . . , λm : distinct natural numbers
m1 ∏ ∏ 2◦ =λi(λi − λ2j )2µi<j
i=12m
2m−2
◦ t + κ1 t+ · · · + κ2m−1 = (t 2 + λ21 ) · · · (t 2 + λ2m )In particular,
κ1 = λ21 + λ22 + · · · + λ2m .
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ExampleEquivariant projection from the orbit of M2m (continued)
Equivariant
projections
betweenspaces ofclosed equicentroaﬃnecurvesExample 3 (continued)
λ1 := l,λ2 := 3l,6
κ1 = l 22m{(2m)2 − 1}Contents

EquicentroaﬃnecurvesClosed equicentroaﬃnecurvesAction of thediﬀeomorphism groupof the circle
Equivariantprojections(l ∈ N)⇓AtsushiFujioka

Introduction

λm := (2m − 1)l...,Hence we haveγ̄(s) =(cos ls,1sin lsl)(s ∈ S 1 ).

In particular, γ̄ is a closed equicentroaﬃne plane curve withroation number l.
In a similar manner, we have an example of equivariant
projection from the orbit of M2m+1 .
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Equivariant projections into the orbit of M3
Equivariant
projections
betweenspaces ofclosed equicentroaﬃnecurves

n = 4, 5, 6, . . .
γ ∈ Mnκ1 , κ2 : the ﬁrst and the second curvature of γγ̄ : R → R3 \ {0}: an equicentroaﬃne space curve s.t.

AtsushiFujioka

the ﬁrst curvature =Contents24κ1n(n2 − 1)

IntroductionEquicentroaﬃnecurvesClosed equicentroaﬃnecurvesAction of thediﬀeomorphism groupof the circle
Equivariant
projections

the second curvature =24

κ2n(n2 − 1)(n − 2)Consider the action of Diﬀ (S 1 ).
Theorem 3
If γ̄ ∈ M3 , then the correspondence from γ to γ̄ deﬁnes anequivariant map from the orbit of γ into the orbit of γ̄.
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Equivariant projections into the orbit of M3
Proof
Equivariant
projections
betweenspaces ofclosed equicentroaﬃnecurvesAtsushiFujiokaContents
Introduction
Equicentroaﬃnecurves
Closed equicentroaﬃnecurves

Proof
By Theorem 1, the transformation rule for the ﬁrst curvature isgiven byn(n2 − 1)κ̃1 = (g ′ )2 κ1 +S(g ).
12
The transformation rule for the second curvature is given byκ̃2 = (g ′ )3 κ2 − Pn,n−3 − (g ′ )2 κ1 Pn−2,n−3 .
Further computation shows that

Action of thediﬀeomorphism groupof the circle

n(n2 − 1)(n − 2)(S(g ))′ .
24
We can proceed in a similar way as to the proof of Theorem 2.

Equivariantprojections

We can also have a similar example as in Example 3.

Pn−2,n−3 = −(n − 2)h,Pn,n−3 = −21 / 22

Equivariant
projections
betweenspaces ofclosed equicentroaﬃnecurvesAtsushiFujiokaContents

Thank you for your attention.

Introduction
EquicentroaﬃnecurvesClosed equicentroaﬃnecurvesAction of thediﬀeomorphism groupof the circle
Equivariant
projections
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