Equivariant
projections
betweenspaces ofequicentroaﬃnecurvesEquivariant projections between spaces of
equicentroaﬃne curves

AtsushiFujiokaContents
Introduction
EquicentroaﬃnecurvesAction of thediﬀeomorphism
group
Equivariant
projections
Periodic
examplesAtsushi Fujioka
Faculty of Engineering Science
Kansai University

December 3, 2023
Ritsumeikan University
Poisson geometry and related topics 23
Joint work with T. Kurose and H. Moriyoshi
1 / 27

Contents
Equivariant
projections
betweenspaces ofequicentroaﬃnecurvesAtsushiFujioka

1Introduction

2

Equicentroaﬃne curves

3Action of the diﬀeomorphism group4

Equivariant projections

5

Periodic examples

Contents
Introduction
EquicentroaﬃnecurvesAction of thediﬀeomorphism
group
Equivariant
projections
Periodicexamples

2 / 27

Backgrounds and main results
Equivariant
projections
betweenspaces ofequicentroaﬃnecurvesAtsushiFujiokaContents
Introduction
EquicentroaﬃnecurvesAction of thediﬀeomorphism
group
Equivariant
projections
Periodicexamples◦ 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 equicentroaﬃne curves ingeneral vector space.
• ∃ Action of the diﬀeomorphism group of the line
• Deﬁne projections into the space of plane or space curves.
• The above projections are equivariant w.r.t. the above
action.
3 / 27

Deﬁnition of the equicentroaﬃne curve
Equivariant
projections
betweenspaces ofequicentroaﬃnecurves
AtsushiFujioka

Deﬁnition
I : an intervaln = 2, 3, 4, . . .
γ : I → Rn \ {0}: an equicentroaﬃne curve

Contents
Introduction
EquicentroaﬃnecurvesAction of thediﬀeomorphism
group
Equivariant
projections
Periodic
examples




det 


m def.

γγ′ 

=1..
.
γ (n−1)s ∈ I is called an equicentroaﬃne arclength parameter.
4 / 27

The fundamental theorem of equicentroaﬃnecurvesEquivariant
projections
betweenspaces ofequicentroaﬃnecurvesAtsushiFujiokaContents
IntroductionEquicentroaﬃnecurves
Action of thediﬀeomorphism
group
Equivariantprojections
Periodicexamples

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 = 0
For 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.
5 / 27

Example
Equicentroaﬃne plane curves with constant curvature
Equivariant
projections
betweenspaces ofequicentroaﬃnecurves
AtsushiFujiokaContents
Introduction
Equicentroaﬃnecurves
Action of thediﬀeomorphism
group
Equivariant
projections
Periodic
examplesConsider 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)
◦ κ: a positive constant ⇐⇒ γ is a part of an ellipse:
()
(ss )
1
γ(s) = a cos , b sin
a, b > 0, κ = 2 2ababa b◦ κ: a negative constant ⇐⇒ γ is a part of a hyperbola.
6 / 27

Deﬁnition of the action of the diﬀeomorphismgroup of the line
Equivariant
projections
betweenspaces ofequicentroaﬃnecurvesAtsushiFujiokaContents
Introduction
EquicentroaﬃnecurvesAction of thediﬀeomorphism
group
Equivariant
projections
Periodicexamples◦ Mn : the set of all equicentroaﬃne curves from R into
Rn \ {0}◦ Mn /SL(n, R): the set of all congruence classes ofequicentroaﬃne curves from R into Rn \ {0}◦ Diﬀ (R): the group of all orientation preservingdiﬀeomorphisms of R
◦ γ ∈ Mn , g ∈ Diﬀ (R)1−nγ̃(s) := (γ · g )(s) := (g ′ (s)) 2 (γ ◦ g )(s) (s ∈ R)
Proposition
γ · g deﬁnes an action of Diﬀ (R) on Mn , Mn /SL(n, R).
Proof
For k = 0, 1, 2, . . . , n − 1,1−nγ̃ (k) = (g ′ ) 2 +k (γ (k) ◦ g ) + · · · .
7 / 27

Notations
Equivariant
projections
betweenspaces ofequicentroaﬃnecurvesAtsushiFujiokaContents
Introduction
EquicentroaﬃnecurvesAction of thediﬀeomorphism
group
Equivariant
projections
Periodicexamples◦ Consider the action of Diﬀ (R) on Mn :Mn 3 γ 7→ γ̃ = γ · g ∈ Mn

(g ∈ Diﬀ (R))◦ 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.
8 / 27

Transformation rule for the derivatives of curves
1/3
Equivariant
projections
betweenspaces ofequicentroaﬃnecurvesAtsushiFujiokaContents
Introduction
EquicentroaﬃnecurvesAction of thediﬀeomorphism
group
Equivariant
projections
Periodic
examplesLemma◦ For k = 1, 2, . . . , n,k−1∑(k)γ̃ =Pk,l γ̃ (l) + (g ′ )α+k γ (k) ,l=0where Pk,l ’s are homogeneous polynomials of h, h′ , h′′ , . . .
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)
Pk+1,0 =h− (α + k)Pk,0 h.
∂h(m)m=0(R1)

9 / 27

Transformation rule for the derivatives of curves
2/3
Equivariant
projections
betweenspaces ofequicentroaﬃnecurves
AtsushiFujiokaContents

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)∂hm=0(R2)• For k = 1, 2, . . . , n − 1,Pk+1,k = Pk,k−1 + (α + k)h.

Introduction(R3)

EquicentroaﬃnecurvesAction of thediﬀeomorphism
group

Proof
First, diﬀerentiating the equationγ̃ = (g ′ )α γ,

Equivariant
projections
Periodicexamples

we haveγ̃ ′ = αhγ̃ + (g ′ )α+1 γ ′ .
10 / 27

Transformation rule for the derivatives of curves
3/3
Equivariant
projections
betweenspaces ofequicentroaﬃnecurvesAtsushiFujiokaContents
Introduction
EquicentroaﬃnecurvesAction of thediﬀeomorphism
group
Equivariant
projectionsPeriodic
examplesProof (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 ’s are homogeneous polynomials 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+1,l .
11 / 27

Example
Pk+1,kEquivariant
projections
betweenspaces ofequicentroaﬃnecurvesExample 2 (Pk+1,k )k = 1, 2, . . . , n − 1
From P1,0 = αh and (R3):
Pk+1,k = Pk,k−1 + (α + k)h,Atsushi
Fujioka

we have
Contents
Introduction
EquicentroaﬃnecurvesPk+1,k = P1,0 +Periodic
examples(α + l)hl=1)(k= (k + 1) α +h.
2

Action of thediﬀeomorphism
group
Equivariant
projectionsk∑

Since α = 1−n2 , we havePn,n−1 = 0.

(Assume the weighted degree is 1.)
12 / 27

ExampleP2,0 , P3,0
Equivariant
projections
betweenspaces ofequicentroaﬃnecurvesExample 3 (P2,0 , P3,0 )From P1,0 = αh and (R1):

AtsushiFujioka

Pk+1,0 =k−1∑
∂Pk,0∂h(m)m=0h(m+1) − (α + k)Pk,0 h,Contents
Introduction
EquicentroaﬃnecurvesAction of thediﬀeomorphism
group
Equivariant
projections
Periodic
examples

we have∂P1,0 ′h − (α + 1)P1,0 h∂h= αh′ − α(α + 1)h2 .

P2,0 =

Moreover, if n ≥ 3, we haveP3,0 = αh′′ − α(3α + 4)hh′ + α(α + 1)(α + 2)h3 .
13 / 27

Transformation rule for the curvatures
1/3
Equivariant
projections
betweenspaces ofequicentroaﬃnecurvesAtsushiFujiokaContents
Introduction
EquicentroaﬃnecurvesAction of thediﬀeomorphism
group
Equivariant
projections
Periodic
examplesTheorem 1γ ∈ Mnκi : the i-th curvature of γ (i = 1, 2, . . . , n − 1)γ̃ = γ · g (g ∈ Diﬀ (R))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 :( ′′ )′( )g1 g ′′ 2S(g ) =−.
g′2 g′
14 / 27

Transformation rule for the curvatures
2/3
Equivariant
projections
betweenspaces ofequicentroaﬃnecurves

ProofFirst, by use of the transformation rule for the derivatives ofcurves (Lemma), we have

AtsushiFujiokaγ̃ (n) =Contents
IntroductionEquicentroaﬃnecurvesAction of thediﬀeomorphism
group
Equivariant
projections
Periodicexamples

n−2∑Pn,l γ̃ (l) − (g ′ )n κn−1 γ̃

l=0n−2∑−((g ′ )n−k κn−k−1k=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 .
Next, compute Pn,n−2 .
15 / 27

Transformation rule for the curvatures
3/3
Equivariant
projections
betweenspaces ofequicentroaﬃnecurvesAtsushiFujiokaContentsIntroductionEquicentroaﬃnecurvesAction of thediﬀeomorphism
group
Equivariant
projections
Periodicexamples

Proof (continued)
From (R2) and Example 2 (Pk+1,k ), we have
∂Pk,k−1 ′h − (α + k)Pk,k−1 h(∂h)
k −1=k α+h′2()k −1− (α + k)k α +h2 .
2

Pk+1,k−1 − Pk,k−2 =Moreover, from Example 3 (P2,0 ), we haven(n2 − 1)
S(g ).
Pn,n−2 = −1216 / 27

Equivariant projections into M2
1/2
Equivariant
projections
betweenspaces ofequicentroaﬃnecurvesAtsushi
FujiokaContents

n = 3, 4, 5, . . .
γ ∈ Mnκ1 : the ﬁrst curvature of γγ̄ ∈ M2 : an equicentroaﬃne plane curve s.t.
the equicentroaﬃne curvature =Introduction
Equicentroaﬃnecurves
Action of thediﬀeomorphism
group
Equivariant
projections
Periodicexamples

6
κ1n(n2 − 1)Consider the action of Diﬀ (R).
Theorem 2
The correspondence from γ to γ̄ deﬁnes an equivariant map
from Mn into M2 :γ · g = γ̄ · g(g ∈ Diﬀ (R)).

17 / 27

Equivariant projections into M21/2
Equivariant
projections
betweenspaces ofequicentroaﬃnecurvesAtsushiFujiokaContents
Introduction
EquicentroaﬃnecurvesAction of thediﬀeomorphism
group
Equivariant
projections
Periodicexamples

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.
18 / 27

Example
Equicentroaﬃne curves with vanishing higher curvatures
Equivariant
projections
betweenspaces ofequicentroaﬃnecurvesAtsushiFujiokaContents
Introduction
EquicentroaﬃnecurvesAction of thediﬀeomorphism
group
Equivariant
projections
Periodic
examplesExample 4 (Equicentroaﬃne curves with vanishing highercurvatures)n = 3, 4, 5, . . .
ρ : R → R2 \ {0}: a curve s.t. ρ(n−2) is an equicentroaﬃneplane curve with the equicentroaﬃnecurvature κDeﬁne an equicentroaﬃne curve γ : R → Rn \ {0} by()
1 2
1n−3γ(s) = 1, s, s , . . . ,s,ρ(s ∈ R).
2!
(n − 3)!
=⇒ κ1 = κ, κ2 = κ3 = · · · = κn−1 = 0=⇒ The projection into M2 is given by)(√6(n−2)sγ̄(s) = ρn(n2 − 1)(s ∈ R).
19 / 27

Equivariant projections into M3
1/3
Equivariantprojections
betweenspaces ofequicentroaﬃnecurves
AtsushiFujiokaContents

n = 4, 5, 6, . . .
γ ∈ Mnκ1 , κ2 : the ﬁrst and the second curvature of γγ̄ ∈ M3 : an equicentroaﬃne space curve s.t.
24
the ﬁrst curvature =κ12n(n − 1)

Introduction
EquicentroaﬃnecurvesAction of thediﬀeomorphism
group
Equivariant
projections
Periodic
examplesthe second curvature =24

κ2n(n2 − 1)(n − 2)Consider the action of Diﬀ (R).
Theorem 3
The correspondence from γ to γ̄ deﬁnes an equivariant map
from Mn to M3 .
20 / 27

Equivariant projections into M3
2/3
Equivariant
projections
betweenspaces ofequicentroaﬃnecurvesAtsushiFujiokaContents

Proof
By Theorem 1, the transformation rule for the ﬁrst curvature isgiven byn(n2 − 1)
S(g ).
κ̃1 = (g ′ )2 κ1 +12Introduction
EquicentroaﬃnecurvesAction of thediﬀeomorphism
group
Equivariantprojections

The transformation rule for the second curvature is given byκ̃2 = (g ′ )3 κ2 − Pn,n−3 − (g ′ )2 κ1 Pn−2,n−3 .
From Example 2 (Pk+1,k ), we havePn−2,n−3 = −(n − 2)h.

Periodic
examples21 / 27

Equivariant projections into M3
2/3
Equivariant
projections
betweenspaces ofequicentroaﬃnecurvesAtsushiFujiokaContents

Proof (continued)
From (R2), we have
Pk+1,k−2 =∂Pk,k−2 ′ ∂Pk,k−2 ′′h +h + Pk,k−3 − (α + k)Pk,k−2 h.
∂h∂h′Hence we have

Introduction
EquicentroaﬃnecurvesAction of thediﬀeomorphism
group
Equivariant
projections
Periodic
examplesPn,n−3 = P3,0+n−1 (∑∂Pk,k−2k=3∂hh′ +)∂Pk,k−2 ′′h−(α+k)P
h.
k,k−2
∂h′

Further computation shows thatn(n2 − 1)(n − 2)(S(g ))′ .
Pn,n−3 = −24
22 / 27

Example
Closed equicentroaﬃne curves with constant curvatures when n is even
Equivariant
projections
betweenspaces ofequicentroaﬃnecurvesAtsushiFujioka

Example 5 (Closed equicentroaﬃne curves with constantcurvatures when n is even)
λ1 , λ2 , . . . , λm ∈ N, λi 6= λj (i 6= j)Deﬁne an equicentroaﬃne curve γ : R → R2m \ {0} byγ(s) = (cos λ1 s, sin λ1 s, . . . , cos λm s, µ sin λm s) (s ∈ R),

Contents
Introduction

whereEquicentroaﬃnecurves
Action of thediﬀeomorphism
group
Equivariant
projections
Periodicexamples

1 ∏ ∏ 2=λi(λi − λ2j )2 .
µm

i=1i<jThent 2m + κ1 t 2m−2 + · · · + κ2m−1 = (t 2 + λ21 ) · · · (t 2 + λ2m )(κ2 = κ4 = · · · = κ2m−2 = 0).
23 / 27

Example
Closed equicentroaﬃne curves with constant curvatures when n is even(continued)
Equivariant
projections
betweenspaces ofequicentroaﬃnecurvesAtsushiFujiokaContents
Introduction
Equicentroaﬃnecurves
Action of thediﬀeomorphism
group
Equivariant
projections
Periodic
examplesExample 5 (continued)l ∈N
λ1 := l,λ2 := 3l,...,

λm := (2m − 1)l⇓6
κ1 = l 22m{(2m)2 − 1}
Hence the projection into M2 is given by()1γ̄(s) = cos ls, sin ls(s ∈ R).
lMoreover, the projection into M3 is given by()1γ̄(s) = cos 2ls, sin 2ls, 3(s ∈ R).
8l
24 / 27

Example
Closed equicentroaﬃne curves with constant curvatures when n is odd
Equivariant
projections
betweenspaces ofequicentroaﬃnecurves
AtsushiFujioka

Example 6 (Closed equicentroaﬃne curves with constantcurvatures when n is odd)
λ1 , λ2 , . . . , λm ∈ N, λi 6= λj (i 6= j)Deﬁne an equicentroaﬃne curve γ : R → R2m+1 \ {0} byγ(s) = (cos λ1 s, sin λ1 s, . . . , cos λm s, sin λm s, µ)(s ∈ R),

Contents
Introductionwhere

EquicentroaﬃnecurvesAction of thediﬀeomorphism
group
Equivariant
projections
Periodic
examples1 ∏ 3∏ 2=λi(λi − λ2j )2 .
µm

i=1i<jThent 2m+1 + κ1 t 2m−1 + · · · + κ2m = t(t 2 + λ21 ) · · · (t 2 + λ2m )(κ2 = κ4 = · · · = κ2m = 0).
25 / 27

Example
Closed equicentroaﬃne curves with constant curvatures when n is odd(continued)
Equivariant
projections
betweenspaces ofequicentroaﬃnecurves
AtsushiFujiokaContents
IntroductionEquicentroaﬃnecurvesAction of thediﬀeomorphism
group
Equivariant
projections
Periodic
examplesExample 6 (continued)l ∈N
λ1 := 2l,λ2 := 4l,...,

λm := 2ml⇓6
κ1 = l 2(2m + 1){(2m + 1)2 − 1}
Hence the projection into M2 is given by()1γ̄(s) = cos ls, sin ls(s ∈ R).
lMoreover, the projection into M3 is given by()1γ̄(s) = cos ls, sin ls, 3(s ∈ R).
l26 / 27

Equivariant
projections
betweenspaces ofequicentroaﬃnecurvesAtsushiFujiokaContents

Thank you for your attention.

Introduction
EquicentroaﬃnecurvesAction of thediﬀeomorphism
group
Equivariant
projections
Periodic
examples27 / 27