An H2 Riemannian metric on the space of planar curves modulo similitudes

Advances in Applied Mathematics 51 (2013) 483–506

Contents lists available at SciVerse ScienceDirect

Advances in Applied Mathematics www.elsevier.com/locate/yaama

An H 2 Riemannian metric on the space of planar curves modulo similitudes Jayant Shah Mathematics Department, Northeastern University, Boston, MA, United States

a r t i c l e

i n f o

Article history: Received 24 June 2011 Accepted 3 June 2013 Available online 28 June 2013 MSC: 58B20 58D10 58E40 Keywords: Infinite dimensional manifolds Planar loops Geodesics Sectional curvature

a b s t r a c t Analyzing shape manifolds as Riemannian manifolds has been shown to be an effective technique for understanding their geometry. Riemannian metrics of the types H 0 and H 1 on the space of planar curves have already been investigated in detail. Since in many applications, the basic shape of an object is understood to be independent of its scale, orientation or placement, we consider here an H 2 -metric on the space of planar curves modulo similitudes. The metric depends purely on the bending and stretching of the curve. Equations of the geodesic for parametrized curves as well as un-parametrized curves and bounds for the sectional curvature are derived. Equations of gradient descent are given for constructing the geodesics between two given curves numerically. © 2013 Elsevier Inc. All rights reserved.

1. Introduction One of the approaches to shape analysis is to study the Riemannian geometry of the shape manifold. Examples of this approach may be seen in [2–6,8–16]. In particular, there has been a great deal of progress in understanding the geometry of planar shapes exemplified by silhouettes and MRI. Mathematically, this space is the space of smooth planar curves. Michor and Mumford [3,4] have analyzed this space in considerable detail. In particular, they derive the geodesic equation for a general Sobolev metric. A more detailed analysis including the sectional curvature of specific metrics is carried out in [9,16]. Conformal versions of the H 0 -metric of Michor and Mumford are studied in [9] while an H 1 -metric is studied in [16]. It is useful to separate out the pose and scale of an object from its inherent geometry. This leads us to study the space of planar curves modulo translation, rotation and scale. An example of such a formulation is an H 1 -metric analyzed in [16] by Younes et al. To obtain a metric which depends

E-mail address: [email protected]. 0196-8858/$ – see front matter © 2013 Elsevier Inc. All rights reserved. http://dx.doi.org/10.1016/j.aam.2013.06.003

484

J. Shah / Advances in Applied Mathematics 51 (2013) 483–506

purely on the bending and stretching of the curve, what is needed is an H 2 -metric. Klassen et al. studied such a metric in [2]. This was further investigated by the author in [8]. The current paper is a more thorough analysis of a closely related H 2 -metric in the framework of Michor and Mumford. We derive the geodesic equation and give the equation of gradient descent for deforming a given curve into a geodesic. Using the techniques developed in [3,4,16], we compute the sectional curvature and derive an absolute bound which guarantees the existence of minimal geodesics. This paper is organized as follows. In Section 2, we define the space of parametrized closed curves and describe the action of the group of similitudes. We define Sobolev bilinear forms on the tangent bundle which induces Riemannian metrics on the quotient space. In the rest of the paper, we restrict the analysis to the case of H 2 -metric. In Section 3, we derive the geodesic equation for the space of parametrized curves modulo similitudes. In Section 4, the action of the reparametrization group is considered and the equation for the geodesics on the quotient by the group is derived. Computation of the sectional curvature is rather involved and is carried out in several steps. In  which includes open Section 5.1, local charts are constructed by means of a much bigger space Ω . The space Ω is the quotient of Ω  by these curves. Rotations of the curves induce translations in Ω translations. The space of closed curves modulo similitudes appears as a submanifold Ω0 of Ω . In  and Ω . We identify a subbundle of the tangent Section 5.2, we define the tangent bundles on Ω  with the tangent bundle of Ω , allowing us to carry out all our computations on this space of Ω  which then defines a Sobolev metric on Ω . subbundle. We carry over the Sobolev bilinear form to Ω  and Ω . The metric In Section 5.3, we describe the action of the group of reparametrization on Ω on Ω induces a metric on its quotient by the group. Christoffel symbols are computed in Section 5.4. The sectional curvature of Ω is computed in Section 5.5 and an application of Gauss Lemma in Section 5.6 gives a formula for the sectional curvature of the submanifold Ω0 . In Section 5.7, we use O’Neill’s formula to compute the sectional curvature of Ω0 modulo the group of reparametrizations. In Section 5.7, we derive an absolute bound on the sectional curvature. 2. Sobolev metrics Let Imm( S 1 , R2 ) denote the space of all C ∞ immersions c : S 1 → R2 . Let the unit circle S 1 be parametrized by θ . Let s denote the arc-length along the image curve of c in R2 . The infinitesimal arc-length ds = |c θ | dθ . (Subscripts θ , s, t denote the derivative.) A tangent vector h at a point c ∈ Imm( S 1 , R2 ) is just a vector field in R2 along c in R2 . If a, b are vectors in R2 , a · b will denote their dot product. Let v and n denote the vector fields of unit tangent and unit normal vectors along c. As vector fields along c viewed as a curve in the complex plane C, v = c s and n = ic s . We will often represent h s as a complex function:





h s = (h s · v ) v + (h s · n)n = (h s · v ) + i (h s · n) c s The vector (h s · v , h s · n) as a complex function takes the form

h s · v + ih s · n = h s /c s We will use the following notations interchangeably:

(ms · v , ms · n) · (h s · v , h s · n) = (ms · v )(h s · v ) + (ms · n)(h s · n) =

ms cs

·

hs cs

Let Σ denote the group of similitudes. Its action on c as a curve in C is given by

α , β are (complex) constants. The vertical vectors at c of the quotient map 







ρ : Imm S 1 , R2 → Imm S 1 , R2 /Σ

α c + β where

J. Shah / Advances in Applied Mathematics 51 (2013) 483–506

485

have the form α c + β . If h is a vertical vector, then h s /c s is constant, that is, h s · v and h s · n are constant. We define Σ -horizontal vectors at c as

{h | h = h s · v = h s · n = 0}

(2.1)

where the bar over a variable denotes its average value:

f =

1





 f ds,

=

c

ds c

Let

f0 = f − f The horizontal and the vertical vectors define a decomposition of the tangent bundle of Imm( S 1 , R2 ) into two subbundles such that the tangent map









T c Imm S 1 , R2 → T ρ (c ) Imm S 1 , R2 /Σ is an isomorphism when restricted to the horizontal vectors. We identify the tangent vectors at ρ (c ) with the horizontal tangent vectors at c. The projection from the tangent space at c onto the space of horizontal tangent vectors is given by the formula

h → hΣ = h0 − (h s /c s )c 0 Note that c 0 is the curve c translated so that its center of gravity is at the origin. Given a path c (t ) in Imm( S 1 , R2 ), there exists a horizontal path c Σ (t ) = α (t )c (t ) + β(t ) where and β(t ) are given by equations

(2.2)

α (t )

αt + α cts /c s = 0 βt + (α c )t = 0 such that c (t ) and c Σ (t ) project to the same path in Imm( S 1 , R2 )/Σ . We now define a bilinear form on the tangent space at c:

m, h = 2p −3





p −1 

Ds



p −1 

mΣ s · v , Ds

mΣ s ·n

  p −1  Σ  p −1  Σ  · Ds hs · v , D s h s · n ds

c 0 Σ 0 where D s = d/ds,  is the length of c and p  1. Note that hΣ s · v = (h s · v ) and h s · n = (h s · n) . Moreover h, h = 0 if and only if h is a vertical vector. m, h is non-degenerate on the horizontal subbundle and defines a Riemannian metric on it and hence on Imm( S 1 , R2 )/Σ . The case p = 1 has been treated in [16] using a representation of Imm( S 1 , R2 )/Σ by grassmanp −1

p −1

Σ (h s · v , h s · n). We may assume  = 1 since the metric is nians. If p  2, D s (hΣ s · v , h s · n) = D s scale-invariant. Setting p = 2 from now on, we consider the metric induced by the bilinear form

 m, h =

(ms · v , ms · n)s · (h s · v , h s · n)s ds c

(2.3)

486

J. Shah / Advances in Applied Mathematics 51 (2013) 483–506

Given a path c (t ) : [0, 1] → Imm( S 1 , R2 ), we define its length by setting





L c (t ) =

1 

ct , ct  dt

0

The length of a path is invariant under the action of Σ .

R2 )/Σ 3. Geodesic equation for the H 2 -metric on Imm( S 1 ,R Let c (t ) : [0, 1] → Imm( S 1 , R2 ) be a horizontal lift of a path in Imm( S 1 , R2 )/Σ . We may assume that for each t, the curve c (t ) has unit length. Deformation of a curve by a horizontal tangent vector preserves its length. We derive the geodesic equation by calculating the first variation of the energy

E (c ) =

1

1 ct , ct  dt

2 0

=

1

1 



2 0

(cts · v )s

2

2  + (cts · n)s ds dt

c

where cts = (ct )s = D s D t c. Let m(t ) be a field of horizontal tangent vectors along c (t ), vanishing at its end-points. We have the following formulas for derivatives in the direction of m:

D m |c θ | = (ms · v )|c θ |, D m v = (ms · n)n,

D m D s = −(ms · v ) D s + D s D m D m n = −(ms · n) v

D m (cts · v ) = D m D t log |c θ | = D t D m log |c θ | = (ms · v )t ,

D m E (c ) =

1

1

 Dm

2 0

1  = 0 S1

−

1

S1

1  0 S1

1  =−

−

1

(cts · v )2θ + (cts · n)2θ

 dθ dt |c θ |

  dθ (ms · v )θ t (cts · v )θ + (ms · n)θ t (cts · n)θ dt |c θ |

2

0



D m (cts · n) = (ms · n)t



 dθ  (ms · v ) (cts · v )2θ + (cts · n)2θ dt |c θ |

 (ms · v )s (cts · v )st + (ms · n)s (cts · n)st ds dt

c

1 

2 0

c

  (ms · v ) (cts · v )2s + (cts · n)2s ds dt

J. Shah / Advances in Applied Mathematics 51 (2013) 483–506

487

where the last step follows from integration by parts. Since m is horizontal, ms /c s has zero mean. Hence,

1  D m E (c ) = − 0

−

1



 0  0  (ms · v )s (cts · v )st + (ms · n)s (cts · n)st ds dt

c

1 

2 0

0  (ms · v ) (cts · v )2s + (cts · n)2s ds dt

c

1 1 The operator D − is uniquely defined on zero-mean functions by requiring that D − s f = 0. Applying s integration by parts,

1  D m E (c ) = − 0

+

1



 0  0  (ms · v )s (cts · v )st + (ms · n)s (cts · n)st ds dt

c

1 

2 0

1 =−

 0 1 (cts · v )2s + (cts · n)2s ds dt (ms · v )s D − s

c





m, γ (c ) dt

0

where the symbol γ denotes the geodesic curvature of the path in Imm( S 1 , R2 )/Σ corresponding to the path c (t ). An explicit expression for γ may be derived as follows. Using the identity f st = −(cts · v ) f s + f ts , we get

(cts · v )st = (cts · v )ts − (cts · v )(cts · v )s   = ctts · v − (cts · v )2 + (cts · n)2 s − (cts · v )(cts · v )s

0  3 = ctts · v − (cts · v )2 + (cts · n)2 = (cts · v )st 2

s

(cts · n)st = (cts · n)ts − (cts · v )(cts · n)s   = ctts · n − 2(cts · v )(cts · n) s − (cts · v )(cts · n)s  0   0  1 (cts · n)st = ctts · n − 2(cts · v )(cts · n) − D − (cts · v )(cts · n)s s s Let Λ be the vector (Λ1 , Λ2 ) where

Λ1 =

3 2

 0 1 2 (cts · v )2s + (cts · n)2s (cts · v )2 − (cts · n)2 + D − s 2

 0  1 (cts · v )(cts · n)s Λ2 = 2(cts · v )(cts · n) + D − s 

488

J. Shah / Advances in Applied Mathematics 51 (2013) 483–506

Then,

1  D m E (c ) = −

Since

 c

·

cs

c

0



ms

ctts cs

s

−Λ

ds dt s

ms ds = 0, we have

 0=

(ms /c s )∗ c ∗s ds =



c

  1 ∗ (ms /c s )∗s D − c s ds s

c

where the superscript ∗ denotes the complex conjugate. Therefore,



ms



cs

c

  1 ∗ · D− c s ds = 0 s s

and

1  D m E (c ) = − 0

=−

for all

ms

·

ctts cs

s

·

cs

c





cs

c

1  0

ms

γs cs

s

  2 ∗ − Λ − α D− c ds dt s s s

ds dt s

2 ∗ α ∈ C. Let γs /c s = ctts /c s − Λ − α D − s (c s ). Since we must have



α=  c

c

 c

γs ds = 0, set

Λc s ds

−1

| D s (c s )|2 ds

Finally,



  

1 2 ∗ Λ + α D− γ = ctt − D − cs cs s s

(3.1)

A path c (t ) is a geodesic if and only if

    1 2 ∗ Λ + α D− ctt = D − cs cs s s

(3.2)

A path c (t ) may be deformed into a geodesic by gradient descent: ∂∂τc ∝ γ . An alternative form of the geodesic equation and the gradient descent without using the explicit form of γ may be derived as follows:

Let μ = (cts · v )st − Then,

0 1 −1  (cts · v )2s + (cts · n)2s , D 2 s

ν = (cts · n)st

(3.3)

J. Shah / Advances in Applied Mathematics 51 (2013) 483–506



ctts cs



489

  = μ, ν 0

−Λ s



m, γ  =

ms

·

cs

c





= c





ctts cs

s

−Λ

ds s

 (ms · v )s μ + (ms · n)s ν 0 ds



  (ms · v )μs + (ms · n)νs ds

=− c



  (ms · v ) v + (ms · n)n · (μs v + νs n) ds

=− c

 =−

ms · (μs v + νs n) ds c

 =

m · (μs v + νs n)s ds

(3.4)

c

for all vectors m ∈ T c Imm( S 1 , R2 ).



0 1 (μs v + νs n)s = (cts · v )sts v + (cts · n)sts n − (cts · v )2s + (cts · n)2s v 2

s

  1  (cts /c s )s 2 0 c s = (cts /c s )sts c s s − s 



(3.5)

2

Since

γ (c ) = 0







m, γ (c ) = 0 for all m ∈ T c Imm S 1 , R2

⇐⇒



a path c (t ) is a geodesic if and only if



0 1 (cts · v )sts v + (cts · n)sts n − (cts · v )2s + (cts · n)2s v = 0 2

(3.6)

s

Since



(μs v + νs n)s

Σ

   , γ = (μs v + νs n)s , γ  0 and 1

D mΣ E (c ) = −



1



mΣ , γ dt = −

0

m, γ  dt 0

the gradient descent equation takes form

∂c ∝ ∂τ

(cts · v )sts v + (cts · n)sts n −

1 2

0 (cts · v )2s + (cts · n)2s v

Σ (3.7) s

490

J. Shah / Advances in Applied Mathematics 51 (2013) 483–506

The geodesic equation may also be written in terms of the “momentum” vector u as follows:



u = (cts · v )ss v + (cts · n)ss n



ut = (cts /c s )ss c s



 s

  = (cts /c s )ss c s s

st

  = −(cts · v )u + (cts /c s )ss c s ts     = −(cts · v )u + (cts /c s )sst c s s + i (cts · n)(cts /c s )ss c s s

where we have used the formula c st = D t (c s ) = i (cts · n)c s . Therefore,



ut = −(cts · v )u + −(cts · v )(cts /c s )ss c s + (cts /c s )sts c s

  + i (cts · n)u + (cts · n)s (cts /c s )ss c s



(cts /c s )sts c s

 s

  = (−2cts · v + icts · n)u + (−cts · v + icts · n)s (cts /c s )ss c s + (cts /c s )sts c s s

 s

= ut + (2cts · v − icts · n)u + (cts · v − icts · n)s (cts /c s )ss c s

(3.8)

Substituting (3.8) into (3.5) we get

(μs v + νs n)s = ut + (2cts · v − icts · n)u + (cts · v − icts · n)s (cts /c s )ss c s −

  (cts /c s )s 2 0 c s s

1  2

Noting D s c s = i κ c s ,

  1  (cts /c s )s 2 0 c s (cts · v − icts · n)s (cts /c s )ss c s − s 2

 0 κ = −i (cts · v )ss (cts · n)s − (cts · v )s (cts · n)ss + (cts · v )2s + (cts · n)2s cs 2

where

κ is the curvature of c defined by the equation v s = κ n.

(μs v + νs n)s = ut + (2cts · v − icts · n)u

 0 κ − i (cts · v )ss (cts · n)s − (cts · v )s (cts · n)ss + (cts · v )2s + (cts · n)2s cs 2

(3.9)

The geodesic equation takes the form

ut + (2cts · v − icts · n)u

 0 κ = i (cts · v )ss (cts · n)s − (cts · v )s (cts · n)ss + (cts · v )2s + (cts · n)2s cs 2

(3.10)

R2 )/(Σ × Diff( S 1 )) 4. Geodesic equation for the H 2 -metric on Imm( S 1 ,R The group of diffeomorphisms Diff( S 1 ) acts on Imm( S 1 , R2 ) by composition from the right. The action commutes with the action of Σ and hence Diff( S 1 ) acts on Imm( S 1 , R2 )/Σ . The quotient Imm( S 1 , R2 )/(Σ × Diff( S 1 )) is the space of un-parametrized curves modulo similitudes which inherits a Riemannian metric making the quotient map a submersion.

J. Shah / Advances in Applied Mathematics 51 (2013) 483–506

491

The tangent bundle of Imm( S 1 , R2 )/(Σ × Diff( S 1 )) splits into vertical and horizontal subbundles which are orthogonal with respect to the metric on Imm( S 1 , R2 )/Σ . The geodesics on Imm( S 1 , R2 )/(Σ × Diff( S 1 )) are just the horizontal geodesics on Imm( S 1 , R2 )/Σ . The action of an infinitesimal diffeomorphism corresponds to a vector field bv along c where b ∈ R. A tangent vector h ∈ T ρ (c ) Imm( S 1 , R2 )/Σ is Diff( S 1 )-horizontal

 ⇐⇒ ⇐⇒

bv , h = 







bv · (h s · v )ss v + (h s · n)ss n s ds = c

v · (h s · v )ss v + (h s · n)ss n





s

= v · (h s /c s )ss c s



bv · (h s /c s )ss c s

 s

ds = 0

c

 s

= (h s · v )sss − κ (h s · n)ss = 0

Suppose c (t ) is horizontal with respect to Σ × Diff( S 1 ). Then



v · (cts /c s )ss c s

 s

=v ·u=0

Therefore, the momentum vector u = an, a(t ) ∈ R. Substituting this in (3.9), we get

(μs v + νs n)s = at + 2(cts · v )a − (cts · v )ss (cts · n)s + (cts · v )s (cts · n)ss

κ

−

2

(cts · v )2s + (cts · n)2s

0

 n

The geodesic equation takes the form

at + 2(cts · v )a = (cts · v )ss (cts · n)s − (cts · v )s (cts · n)ss +

κ 2

(cts · v )2s + (cts · n)2s

0

(4.1)



Since bv , γ  = c bv · (μs v + νs n)s ds = 0, the geodesic curvature γ (c ) is horizontal with respect to Σ × Diff( S 1 ). A horizontal path c (t ) may be constructed as follows. Lift C (t ) to a path c (t ) on Imm( S 1 , R2 ). We now construct a path ϕ (t , θ) ∈ Diff( S 1 ) such that c ◦ ϕ is horizontal. Write ϕt (t , θ) = ξ(t , θ) ◦ ϕ (t , θ) and let η = |c θ |ξ . Note that (c ◦ ϕ )t = (ct + η v ) ◦ ϕ and D c ◦ϕ ( f ◦ ϕ ) = ( D c f ) ◦ ϕ (see [3]). The path c ◦ ϕ is horizontal with respect to Diff( S 1 )

⇐⇒ ⇐⇒



(ct + η v )s · v

 sss

  − κ (ct + η v )s · n ss = 0

L η + (cts · v )sss − κ (cts · n)ss = 0

where

L η = ηssss − κ (κη)ss

(4.2)

The operator L is self-adjoint and it is positive definite provided that c is not a circle. Assuming that the path does not pass through a circle, η may be calculated by minimizing



1 2







2 ηss + (κη)2s + (cts · v )sss − κ (cts · n)ss η ds

c

by gradient descent. Then, integrate the equation zontal path (c ◦ ϕ )Σ .

ϕt (t , θ) = ξ(t , θ) ◦ ϕ (t , θ) to obtain ϕ and the hori-

492

J. Shah / Advances in Applied Mathematics 51 (2013) 483–506

5. Sectional curvature 5.1. Local charts Let c : S 1 → C be an immersion. c θ is translation invariant. Σ acts on c θ by multiplication and on log c θ by translation. Hence, (log c θ )θ is Σ -invariant.

 (log c θ )θ dθ = log c θ |2π − log c θ |0 = 2π J i S1

where J is the rotation index of c. The space of (log c θ )θ provides local charts of Imm( S 1 , R2 )/Σ as follows. Fix an integer J , the rotation index. Let

       Ω = z ∈ C ∞ S 1 , R2  z1 dθ = 0, z 2 d θ = 2π J S1

S1

  2π   ∞ 2  z1  Ω = z ∈ C R, R  z1 , z2 − J x 2π -periodic, e dx = 1 0

where z1 , z2 are the components of z. The map

→Ω χ :Ω is defined by setting

 defines a curve c z : R → C: χ (z) = zx . A point z ∈ Ω

x c z (x) =

e z1 +iz2 dx

0

c z is a closed curve of length 1 if c z (2π ) = 0. The arc-length s of c z is given by s(x) =

x

unit tangent vector field v and the unit normal vector field n along c z are given by e respectively. The curvature κ = z2s . Let

e z1 dx. The

0 iz2

and ie iz2

0 = { z: c z is closed} Ω 0 ) Ω0 = χ (Ω Define a map

  Ω0 → Imm S 1 , R2 /Σ as follows. Let ζ ∈ Ω0 . χ −1 (ζ ) consists of pairs ( z1 , z2 ) such that z1 is uniquely determined and z2 is unique up to a constant. Let c χ −1 (ζ ) = {c z : z ∈ χ −1 (ζ )}. The curves c χ −1 (ζ ) lie in the same Σ -orbit and hence map to a unique point in Imm( S 1 , R2 )/Σ .

J. Shah / Advances in Applied Mathematics 51 (2013) 483–506

493

5.2. Tangent bundles

. If a function f (x) is 2π -periodic, we will also regard it as a function on S 1 as well as a Let z ∈ Ω  and T χ (z) Ω are given by function on the circle  = {e 2π is : 0  s  1}. The tangent spaces T z Ω

    T z Ω = h = (h1 , h2 ): h1 , h2 2π -periodic, h1 ds = 0 

 T χ (z) Ω = {hθ : h ∈ T z Ω} The vertical vectors at z are of the form (0, a), a ∈ R. Let







 = h: T z0 Ω

h ds = 0 

We have the decomposition

 T z0 Ω ⊕R T zΩ such that

 T χ (z) Ω T z0 Ω . We will carry out all the calculaEvery tangent vector field on Ω lifts uniquely to a section of T 0 Ω . tions below in terms of the sections of T 0 Ω Let



m ∈ T c Imm S 1 , R2



Then,



D m c θ = D m |c θ | v



= (ms · v )|c θ | v + (ms · n)n|c θ |   = (ms · v ) + i (ms · n) c θ , m maps onto (ms · v , and D m log c θ = (ms · v ) + i (ms · n). Therefore, if zc is the image of c in Ω . The sub-Riemannian metric on Imm( S 1 , R2 ) induces a sub-Riemannian metric on Ω . m s · n) ∈ T zc Ω , If m, h ∈ T z Ω

 m, h =

ms · h s ds 

 and hence, induces a Riemannian metric on Ω . The metric is non-degenerate on T 0 Ω 5.3. Action of the group of reparametrization Let Diff + (R) be the group of increasing C ∞ diffeomorphisms ϕ : R → R such that ϕ (x + 2π ) = ϕ (x) + 2π for all x. Let Diff 0 (R) be the central group subgroup of translations: ϕ (x) = x + 2π n. The

494

J. Shah / Advances in Applied Mathematics 51 (2013) 483–506

 as quotient Diff + (R)/Diff 0 (R) is the group of diffeomorphisms of S 1 , Diff + ( S 1 ). Diff + (R) acts on Ω follows: ϕ

z1 = z1 ◦ ϕ + log ϕx ϕ

z2 = z2 ◦ ϕ The action commutes with the translations z2 → z2 + a and hence Diff + (R) acts on Ω . The action of Diff 0 (R) on Ω is trivial so that Diff + ( S 1 ) acts on Ω . The metric on Ω is defined independently of parametrization. Therefore the quotient map

  Π : Ω → Ω/Diff + S 1 is a Riemannian submersion. An infinitesimal diffeomorphism, a, of S 1 induces a tangent vector field  where L 1 = ( D s , κ ). The bv along c z where b = e z1 a and maps to a vertical vector ( L 1 b)0 ∈ T z0 Ω  to be Π -horizontal takes the form L ∗ · h ss = 0 where the adjoint L ∗ = (− D s , κ ). condition for h ∈ T z0 Ω 1 1 , let h V and h H denote its horizontal and vertical components. h V has the form ( L 1 b)0 . If h ∈ T z0 Ω Since L ∗1 · (h − L 1 b)ss = 0,

L ∗1 · h ss = L ∗1 · ( L 1 b) = −b ssss + κ (κ b)ss = − Lb where L is the self-adjoint operator defined earlier (Eq. (4.2)). We have





h V = − L 1 · L −1 L ∗1 · h ss

0

provided that c z is not a circle. 5.4. Christoffel symbols

 such that hθ , kθ , mθ are constant vector fields on Ω . We calculate Let h, k, m be sections of T 0 Ω  using the two gradients, K (m, h) and H (h, k) of D m h, k: the Christoffel symbol Γ (h, k) ∈ T 0 Ω











D m h, k = K (m, h), k = m, H (h, k)



K s (m, h) · k s ds = D m

h s · k s ds







= Dm

h θ · k θ e − z1 d θ

S1

 =−

h θ · k θ m1 e − z1 d θ

S1



=−

(m1 h s )0 · ks ds

h s · k s m1 ds − 

Therefore, K s (m, h) = −(m1 h s )0 . Similarly,

 

J. Shah / Advances in Applied Mathematics 51 (2013) 483–506

495



 H s (h, k) · ms ds = − 

h s · k s m1 ds 



(h s · ks )0m1 ds

=− 



=

1 0 m1s D − s (h s · k s ) ds

 1 0 Hence, H s (h, k) = ( D − s (h s · k s ) , 0). We have the identity

Γ (h, k) =

1 2



K (h, k) + K (k, h) − H (h, k)

Therefore,

Γs (h, k) = −

1 2

  1 0 (h1ks )0 + (k1 h s )0 + D − s (h s · k s ) , 0

(5.1)

5.5. Sectional curvature of Ω

 such that mθ , hθ are constant vector fields on Ω . We have the identity Let m, h be sections of T 0 Ω for the sectional curvature at the two-dimensional subspace of the tangent space at ζ ∈ Ω spanned by m and h: 1

1

2

2

2 2 2 κζ,Ω (m, h) = D m ,h m, h  − D m,m h , h  − D h,h m, m

    + Γ (m, h), Γ (m, h) − Γ (m, m), Γ (h, h) In evaluating the right-hand side, we will use the formula

D m h = −m1 h obtained as follows: Let  h=

θ 0

h = h−

hθ dθ . Then,





 h ds,

Dmh = −

 hm1 ds = −



 hm1 ds 

 which are sections of T 0 Ω  and which are constant Let m, h, k, p be tangent vector fields on Ω on Ω .

 2 Dm , p h , k

= −D p  =

h θ · k θ m1 e − z1 d θ

S1

h θ · k θ m1 p 1 e

− z1

 dθ + m1 p 1

S1

S1



=

m1 p 1 h s · k s ds + m1 p 1 h, k 

h θ · k θ e − z1 d θ

496

J. Shah / Advances in Applied Mathematics 51 (2013) 483–506

We may assume that m, h are orthonormal at ζ . Then,

 2 Dm ,h m, h 

=

m1 h1 ms · h s ds 

1

1

2

2

1

1

2

2

2 − Dm ,m h , h  = −

 m21 h s · h s ds −

1 2

m21



− D h2,h m, m = −



h21 ms · ms ds −

1 2

h21



The expression Γ (m, h), Γ (m, h) − Γ (m, m), Γ (h, h) simplifies to

1

 |h1ms − m1 h s |2 ds +

2 

−

1



  −1   D (ms · h s )0 2 − D −1 (ms · ms )0 D −1 (h s · h s )0 ds s

4

s

s



1 4

 |h1ms + m1 h s |2 + m21 + h21 + m1ms · h1 h s

Summing all the terms, we get

κζ,Ω (m, h) =

1



  −1   D (ms · h s )0 2 − D −1 (ms · ms )0 D −1 (h s · h s )0 ds s

4

s

s



−

1 4

  |h1ms + m1 h s |2 + 3 m21 + h21 + m1ms · h1 h s

(5.2)

where m, h are orthonormal. 5.6. Sectional curvature of Ω0 Now let ζ ∈ Ω0 . Since Ω0 is a submanifold of Ω , its sectional curvature may be calculated using Gauss Lemma. 5.6.1. The normal bundle . Let z ∈ Ω

2π

e z1 +iz2 dx

Q = Q1 + iQ2 = 0

0 if and only if Q = 0. Now assume that z ∈ Ω 0 such that z∈Ω

2π

. χ (z) = ζ . Let h ∈ T z0 Ω

e z1 +iz2 (h1 + ih2 ) dx

Dh Q = 0

 he iz2 ds

= 

J. Shah / Advances in Applied Mathematics 51 (2013) 483–506

497

Let

u 1 = (cos z2 , −sin z2 ),

u 2 = (sin z2 , cos z2 )

Then, in vector notation,

 Dh Q =

(h · u 1 , h · u 2 ) ds 



=−









2 −2 hs · D − s u1 s , hs · D s u2

 s

ds



Therefore, the gradient 2 0 ∇ Q i = −D− s ui ∈ T z Ω

As a complex valued function



1 −1 iz2 D− s u1 = D s e

∗

 ∗ = c 0z

where the superscript ∗ indicates complex conjugation. c 0z is the curve c z translated so that its center 1 0 ∗ 0 ∗ of gravity is at the origin. Similarly, D − s u 2 = i (c z ) which is just a rotation of (c z ) by π /2. Clearly, ∇ Q 1 , ∇ Q 2 are orthogonal. The normal vector space defined by the gradients ∇ Q 1 and ∇ Q 2 is invariant under change in z2 by an additive constant and hence defines the normal vector space at ζ ∈ Ω0 . A choice of z corresponds to a choice of a basis of the normal vector space at ζ . For i = 1, 2,



2

∇ Q i  = ∇ Q i , ∇ Q i  =

 0 2 c  ds z



∇ Q i 2 is the polar moment of the curve c 0z , invariant under rotation of c 0z and hence, invariant under change in z2 by an additive constant. 5.6.2. The second fundamental form The second fundamental form S (m, h) has the decomposition:

 S (m, h), ∇ Q 1 ∇ Q 1  S (m, h), ∇ Q 2 ∇ Q 2 + ∇ Q 1 2 ∇ Q 2 2   S (m, h), ∇ Q i = − D m ∇ Q i , h = −∇ 2 Q i (m, h)

S (m, h) =

The Hessian ∇ 2 Q i (m, h) = D m D h Q i − D Γ (m,h) Q i

2π he z dx

Dm Dh Q = Dm 0

2π = m1 h  =

mhe z dx

e dx + 0

mhe 

2π z

0 iz2

ds

0 at z ∈ Ω

498

0 , since, at z ∈ Ω

J. Shah / Advances in Applied Mathematics 51 (2013) 483–506

 2π 0

e z dx = 0.

 Γ (m, h)e iz2 ds

D Γ (m,h) Q = 

Therefore,

 



 







mh − Γ (m, h) e iz2 ds

S (m, h), ∇ Q = − 

S (m, h), ∇ Q i = Γ (m, h) − (mh), ∇ Q i



Therefore,

S (m, h) = Γ (m, h) − (mh)

   1 1 1 0 Γ (m, h) = − D − (m1 h s )0 + (h1ms )0 + D − s s (m s · h s ) , 0 2

(mh) = (m1 h1 − m2 h2 , m1 h2 + m2 h1 ) 5.6.3. Gauss Lemma









κζ,Ω0 (m, h) = κζ,Ω (m, h) + S (m, m), S (h, h) − S (m, h), S (m, h)      = κζ,Ω (m, h) + Γ (m, m) − m2 , Γ (h, h) − h2   − Γ (m, h) − (mh), Γ (m, h) − (mh)

This may be simplified to take the form

1

κζ,Ω0 (m, h) = −3h1ms − m1 h s 20 − h2ms − m2 h s 20 2

− 9(h1m2 − m1 h2 )(h1s m2s − m1s h2s ) −

 1 2 m1 + h21 2

(5.3)

where m, h are orthonormal. 5.7. Sectional curvature of Ω0 /Diff + ( S 1 )

0 map to ζ . Let a, b be a pair of orthonormal tangent vectors at Π(ζ ) ∈ Let ζ ∈ Ω0 . Let z ∈ Ω 0 . Let m# , h# be sections of Ω0 /Diff + ( S 1 ). Let m, h be orthonormal horizontal lifts of a, b in T z0 Ω 0 T Ω0 in a neighborhood of z which are horizontal extensions of m, h. Then, by O’Neill’s formula [7], we have



3 κΠ (ζ ),Ω0 /Diff + ( S 1 ) (a, b) = κζ,Ω0 (m, h) + m#θ , h#θ 4

V ζ,Ω0

2

(5.4)

# where [m# θ , h θ ] is the Lie bracket and as before, the superscript V denotes its Π -vertical component. # V We now derive an explicit expression for [m# θ , h θ ]ζ,Ω0 .  since [m# , h# ]ζ,Ω0 = [m# , h# ]ζ,Ω . We may regard the vector fields m# , h# as vector fields on Ω θ θ θ θ We now construct the vector fields m# , h# . An important fact implicit in O’Neill’s formula is that V [m#θ , h#θ ]ζ,Ω is independent of the choice of the horizontal extensions m# , h# . Extend the vectors m, h

J. Shah / Advances in Applied Mathematics 51 (2013) 483–506

499

 such that mθ , hθ are constant vector fields on Ω and denote them again as m, h as sections of T 0 Ω respectively. Set m# = m H and h# = h H . Since D mθ hθ = D hθ mθ = 0,



V

# H H V V m# θ , h θ ζ,Ω0 = D mθ h θ − D hθ mθ = D hθ mθ − D mθ h θ

Let D ζ,h f denote the derivative of f at the point ζ in the direction h.







D ζ,hθ mθV = − D ζ,h L 1 L −1 L ∗1 · mss

 θ

   = − L 1 L −1 D ζ,h L ∗1 · mss θ

since L ∗1 · mss = 0 at ζ . Therefore,



V







# −1 m# θ , h θ ζ,Ω0 = L 1 L ψ θ





where ψ = D ζ,m L ∗1 · h ss − D ζ,h L ∗1 · mss



We use the following formulas:

D ζ,h ms = D ζ,h e −z1 mθ = −h1 ms D ζ,h mss = −2h1mss − h1s ms D ζ,h κ = D ζ,h z2s

= −h1 z2s + h2s = −h1 κ + h2s D ζ,h L ∗1 = −h1 L ∗1 + (0, h2s ) We recall that

κ without subscripts denotes the curvature of c z and equals z2s . Since L ∗1 · mss = 0 at ζ , 







D ζ,h L ∗1 · mss = D ζ,h L ∗1 · mss + L ∗1 · D ζ,h mss

= h2s m2ss − 2L ∗1 · (h1mss ) − L ∗1 · (h1s ms ) = h2s m2ss + 2h1s m1ss + (h1s m1s )s − κ (h1s m2s ) taking into account again that L ∗1 · mss = 0 at z. We get

ψ = (m2s h2ss − h2s m2ss ) + 2(m1s h1ss − h1s m1ss ) + κ (h1s m2s − m1s h2s )  # #  V m ,h θ

θ ζ,Ω0

2 =





L 1 L −1 ψ

 s

  · L 1 L −1 ψ s ds





=



L −1 ψ



L −1 ψ





= 

ψ L −1 ψ ds

 ssss

   − κ κ L −1 ψ ss ds

500

J. Shah / Advances in Applied Mathematics 51 (2013) 483–506

κΠ (ζ ),Ω0 /Diff + ( S 1 ) (a, b) = κζ,Ω0 (m, h) +

3



4

ψ L −1 ψ ds

(5.5)



where a, b are orthonormal. 5.8. Absolute bounds We derive the following bounds for the sectional curvature. Let ζ ∈ Ω .

|κζ,Ω | 

9 2

19

+

(5.6)

8π 2

If ζ ∈ Ω0 , then

|κζ,Ω0 |  72 +

1

(5.7)

2π 2

 map to ζ ∈ Ω0 . Let a, b be a pair of orthonormal tangent vectors at Π(ζ ) ∈ Let z ∈ Ω 0 . Fix h. Then, for all a, Ω0 /Diff + ( S 1 ). Let m, h be orthonormal horizontal lifts of a, b in T z0 Ω

 κ

 1 3  + Π (ζ ),Ω0 /Diff + ( S 1 ) (a, b )  72 + 2 2π

4



3 8π

 +

2

3 8π

2 +C



6h2 + 2κ  L ∞

2

(5.8)

where √

C=

2π J 2π J 1 + 3 π 3 κ  L ∞ κ 2 (1 + κ  ∞ + √ ) 4 3κ 2 L

2π 2 (1 − κ A −1 κ )

κ is the curvature of the curve c z corresponding to the point z and J is its rotation index. 5.8.1. Sobolev spaces Let C ∞,0 (, R) = { f ∈ C ∞ (, R): f = 0}. Define norms  f n , n ∈ Z, by setting

  f s =

 n 2  D f  ds

 12 (5.9)

s



Let H n (, R) denote the completion of C ∞,0 (, R) in the norm  · n . If ( f 1 , f 2 ) ∈ C ∞,0 (, R2 ), its norm

( f 1 , f 2 ) = n



 n 2  n 2   D f 1  +  D f 2  ds s

 12

s



defines the corresponding space H n (, R2 ). If f ∈ H n (, R) and g ∈ H −n (, R), f g ∈ L 1 (, R):

 | f g | ds   f n  g −n 

(5.10)

J. Shah / Advances in Applied Mathematics 51 (2013) 483–506

501

The following estimate may be proved by means of the Fourier series of f ∈ H n (, R):

 f m 

 f n , (2π )n−m

m
(5.11)

For n > 0, H n (, R) is an algebra in which the multiplication is defined as ( f g )0 .

√ ( f g )0  2 f 1  g 1 1 √ ( f g )0  4 3 f n  g n

(5.12) if n > 1

π n −2

n

(5.13)

In what follows, we will refer to the function spaces defined above simply as H n , L 1 , etc. and omit the specification (, R) or (, R2 ). 5.8.2. An upper bound for |κζ,Ω |

κζ,Ω (m, h) =

1



  −1   D (ms · h s )0 2 − D −1 (ms · ms )0 D −1 (h s · h s )0 ds s

4

s

s



−

1 4

  |h1ms + m1 h s |2 + 3 m21 + h21 + m1ms · h1 h s

where m, h are orthonormal. We estimate below typical terms in the expression. If f ∈ L 1 ,

 s   s      1 1     0  f ds   f ds + |s f | for −  s      2 2 0

0

3 3 1 0 0 It follows that | D − s ( f )|  2  f  L 1 and hence,  f −1  2  f  L 1 . Since m and h are unit vectors,

 −1   D (ms · h s )0 2  9 s

 −1   D (ms · ms )0 D −1 (h s · h s )0   9 s

s

m21

=

m1 20



m1 1

2

2π



1 4π 2

  1 |h1m1s |  h1 0 m1 1 + m1 1 

π

Substituting these inequalities in the expression for κζ,Ω , we get

|κζ,Ω | 

9 2

+

19 8π 2

(5.14)

502

J. Shah / Advances in Applied Mathematics 51 (2013) 483–506

5.8.3. An upper bound for |κζ,Ω0 |

1

κζ,Ω0 (m, h) = −3h1ms − m1 h s 20 − h2ms − m2 h s 20 2

− 9(h1m2 − m1 h2 )(h1s m2s − m1s h2s ) −

 1 2 m1 + h21 2

where m, h are orthonormal.

3

h1ms 0  h1 L ∞ ms 0  h1ms − m1 h s 0  3 |h1m2 − m1 h2 | 

2 9 2

    (h1m2 − m1 h2 )(h1s m2s − m1s h2s )  9 (h1s m2s − m1s h2s )  9 |h s ||ms | 

2 9 2

2



h s 0 ms 0 

|κζ,Ω0 |  3 · 9 +  72 +

9 2 1

+9·

9 2

+

9 2 1 2π 2 (5.15)

2π 2

5.8.4. An upper bound for |κΠ(ζ ),Ω0 /Diff + ( S 1 ) |

κΠ (ζ ),Ω0 /Diff + ( S 1 ) (a, b) = κζ,Ω0 (m, h) +

3



4

ψ L −1 ψ ds



where

ψ = (m2s h2ss − h2s m2ss ) + 2(m1s h1ss − h1s m1ss ) + κ (h1s m2s − m1s h2s ) and m, h are orthonormal.

 ψ L −1 ψ =



 



L −1 ψ L L −1 ψ ds





=



L −1 ψ

2 ss

 2  + κ L −1 ψ s ds



For ψ L −1 ψ to be finite, L −1 ψ ∈ H 2 or ψ ∈ H −2 . Orthonormality of m, h implies that h s , ms ∈ H 0 and h ss , mss ∈ H −1 . This is not sufficient to place the terms like m2s h2ss in H −2 . We have to assume additional regularity at least for one of the tangent vectors m, h. We assume that h ∈ H 2 . We fix h and derive an upper bound with respect to m. There is an additional complication because ψ need not be equal to zero. We will see that under this assumption, ψ ∈ L 1 . Therefore,

|ψ|  ψL 1 0 ψ  1 ψ 0  1 · 3 ψ 1 L −2 −1 2π

2π

2

J. Shah / Advances in Applied Mathematics 51 (2013) 483–506

503

Let u = L −1 ψ so that ψ L −1 ψ = u ψ = uLu. Since u ψ = u ψ + u 0 ψ 0 ,

ψ L −1 ψ  |u ||ψ| + u 0 2 ψ 0 −2 Since uLu = u 2ss + (κ u )2s  (u 0ss )2 , u 0 2 

 ψ L −1 ψ . We get



ψ L −1 ψ ψ 0 −2  3  |u |ψL 1 + ψ L −1 ψψL 1 4π 

2

2 |u | 3 3 − 1 ψL ψ  + + ψ2L 1 8π 8π ψL 1 ψ L −1 ψ

 |u ||ψ| +

(5.16)

We now proceed to estimate u, and ψ L 1 . Let L 0 = D 4s − 4π 2 J 2 D 2s where J is the rotation index. We have κ = κ 0 + 2π J . Let M = L 0 − L so 1 0 −1 ψ . Note that 4π 2 J 2 v = that Mb = 2π J ((κ 0 b)ss + κ 0 b ss ) + κ 0 (κ 0 b)ss . Let v = L − 0 ψ and let w = L 2 0 −1 ψ . We have D 2s v − D − ∗ s ψ so that v = 0. Let ψ∗ = M v. Let u ∗ = L

u = w + v + u∗ u = w + u∗ Let λ be the smallest eigenvalue of L.

λ w 20  w L w  | w ||ψ|  |ψ| w 0 Therefore,

 w 0 

|ψ| λ

Note that the norms  ·  extend to C ∞ (, R) if n  0. Since | w |   w 0 ,

|w| 

|ψ| λ

Since ψ ∈ H −2 , ψ 0 ∈ H −2 , u , v ∈ H 2 , ψ∗ ∈ H 0 and u ∗ ∈ H 4 .

λu ∗ 20  u ∗ Lu ∗  u ∗ ψ∗  u ∗ 0 ψ∗ 0 |u ∗ |  u ∗ 0 

ψ∗ 0 λ

It follows that

|u |  Let



 1 ψ∗ 0 |ψ| + ψ∗ 0  1+ ψL 1 λ λ ψL 1 1

(5.17)

504

J. Shah / Advances in Applied Mathematics 51 (2013) 483–506

2 0 D− s ψ =



ak e 2π iks

k=0

v=



bk e 2π iks

k=0 2 0 Substituting these in the equation v ss − 4π 2 J 2 v = D − s ψ , we get

bk = − v ss =

ak 4π 2 (k2 + J 2 )

 k=0

k2 k2 + J 2

 v 2 =  v ss 0 =

ak e 2π iks

 k=0



 k=0

1/2 ak2

k2 k2 + J 2

2 1/2 ak

 ψ 0 −2

ψ∗ = 2π J ((κ 0 v )ss + κ 0 v ss ) + κ 0 (κ 0 v )ss .

√  0  κ v = κ 0 v  4 3 κ 0  v 2 ss 0 2 2 √ 0  4 3κ 2 ψ −2 0 κ v ss  κ L ∞ ψ 0 0 −2 0 0    κ κ v  κ L ∞ κ 0 v ss 0 ss 0 √  4 3κ L ∞ κ 2 ψ 0 −2 We also have ψ 0 −2  43π ψ L 1 . We get

√

ψ∗ 0 



2π J 2π J ψL 1 κ L ∞ κ 2 1 + + √ π κ L ∞ 4 3κ 2

3 3

(5.18)

Estimates for typical terms in the expression for ψ are as follows.

m2s h2ss L 1  m2s 0 h2ss 0  h2 m2ss h2s L 1  m2ss −1 h2s 1  h2 κ m1s h2s L 1  κ L ∞ m1s 0 h2s 0  κ L ∞ Therefore,

ψL 1  6h2 + 2κ L ∞

(5.19)

Substituting the estimates (5.17), (5.18), (5.19) and a lower bound (5.20) for λ derived below in the inequality (5.16), we obtain an estimate for O’Neill’s bracket.

J. Shah / Advances in Applied Mathematics 51 (2013) 483–506

505

5.8.5. A lower bound for λ In this section, H n denotes the completion of C ∞ (, R) or C ∞ (, R2 ) in the norm  · n . Recall that L 1 = ( D s , κ ). For b, b1 , b2 ∈ H 2 ,

 b1 , b2  L =

 b1 Lb2 ds =



( L 1 b1 )s · ( L 1 b2 )s ds 



b2L = b, b L =

( L 1 b)s · ( L 1 b)s ds =  L 1 b21 

λ=

b2L min b=0 b 2 0

= min b=0

 L 1 b21 b20

Let A be the positive definite operator − D 2s + κ 2 . Let

 b1 , b2  A =

 b1 Ab2 ds =





b2A = b, b A =

( L 1 b1 ) · ( L 1 b2 ) ds 

( L 1 b) · ( L 1 b) ds =  L 1 b20 

We have the inequality 2 ( L 1 b)0 2   L 1 b1 0 2

4π

and the following estimate from [1]:

( L 1 b) 2 = b2  1 b2 A 0 0 2

Therefore,

λ

2π 2 ( L 1 b)0 20

( L 1 b)20

We need to relate ( L 1 b)0 0 and  L 1 b0 . . Then, L 1 b = ( L 1 b)0 + κ bU . L 1 b = ( L 1 b)0 if and only if Let U = (0, 1) ∈ T z Ω

 0=



κ b ds = 







b A A −1 κ ds = b, A −1 κ

 A



Let β = A −1 κ . κ β = β2A = (βs2 + κ 2 β 2 )  κ β20 . We also have κ β  κ β0 by Schwartz inequality. Therefore, κ β  (κ β)1/2 and hence, β A  1. Equality requires β to be constant which, in turn, implies that κ is constant, that is, c z is a circle. Write

b = b 0 + aβ

where a =

b, β A β2A

506

J. Shah / Advances in Applied Mathematics 51 (2013) 483–506

Then,

( L 1 b) 2 = b2 = b0 2 + a2 β2 A A A 0 and since L 1 β = ( L 1 β)0 + κ β U = ( L 1 β)0 + β2A U ,

( L 1 b)0 = ( L 1 b0 ) + a( L 1 β)0 = ( L 1 b0 ) + aL 1 β − aβ2A U   ( L 1 b)0 2 = b0 2 + a2 β2 1 − β2 0

A

A

A

Therefore,

( L 1 b)0 20 ( L 1 b)20

=

b0 2A + a2 β2A (1 − β2A ) b0 2A + a2 β2A

 1 − β2A  1 − κ A −1 κ

and

  λ  2π 2 1 − κ A −1 κ

(5.20)

References [1] R.D. Benguriaand, M. Loss, Connection between the Lieb–Thirring conjecture for Schrödinger operators and an isoperimetric problem for ovals in the plane, Contemp. Math. 362 (2004) 53–61. [2] E. Klassen, A. Srivastava, W. Mio, S.H. Joshi, Analysis of planar shapes using geodesic paths on shape spaces, IEEE Trans. PAMI 26 (2004) 372–383. [3] P. Michor, D. Mumford, Riemannian geometries on spaces of plane curves, J. Eur. Math. Soc. (JEMS) 8 (2006) 1–48. [4] P. Michor, D. Mumford, An overview of the Riemannian metrics on spaces of curves using the Hamiltonian approach, Appl. Comput. Harmon. Anal. 23 (2007) 74–113. [5] W. Mio, A. Srivastava, Elastic-string models for representation and analysis of planar shapes, in: Computer Vision and Pattern Recognition, CVPR 2004, vol. 2, 2004, pp. 10–15. [6] W. Mio, A. Srivastava, S.H. Joshi, On shape of plane elastic curves, Int. J. Comput. Vis. 73 (2007) 307–324. [7] B. O’Neill, The fundamental equations of a submersion, Michigan Math. J. 13 (1966) 459–469. [8] J. Shah, An H 2 type Riemannian metric on the space of planar curves, in: Workshop on the Mathematical Foundation of Computational Anatomy, MICCAI2006, in: Lecture Notes in Comput. Sci., vol. 4190, Springer, 2006, pp. 40–46. [9] J. Shah, H 0 type Riemannian metrics on the space of planar curves, Quart. Appl. Math. 66 (2008) 123–137. [10] A. Trouvé, L. Younes, Diffeomorphic matching problems in one dimension: designing and minimizing matching functionals, in: D. Vernon (Ed.), ECCV 2000, in: Lecture Notes in Comput. Sci., vol. 1982, Springer, 2000, pp. 573–587. [11] A. Trouvé, L. Younes, On a class of optimal matching problems in 1 dimension, SIAM J. Control Optim. 39 (2001) 1112–1135. [12] A. Yezzi, A. Mennucci, Metrics in the space of curves, arXiv:math.DG/0412454v2, 2004. [13] A. Yezzi, A. Mennucci, Conformal metrics and true “gradient flows” for curves, in: Proceedings of the 10th IEEE International Conference on Computer Vision, ICCV 2005, 2005, pp. 913–919. [14] L. Younes, Computable elastic distances between shapes, SIAM J. Appl. Math. 58 (1998) 565–586. [15] L. Younes, Optimal matching between shapes via elastic deformations, Image Vis. Comput. 17 (1999) 381–389. [16] L. Younes, P. Michor, J. Shah, D. Mumford, A metric on shape spaces with explicit geodesics, Rend. Lincei Mat. Appl. 19 (2008) 25–57.