K\"ahler structures on spaces of framed curves

1.6 MB·

by  Tom Needham

KA¨HLER STRUCTURES ON SPACES OF FRAMED CURVES
TOMNEEDHAM
7 Abstract. We consider the space M of Euclidean similarity classes of framed loops
1 in R3. Framed loop space is shown to be an infinite-dimensional Ka¨hler manifold by
0 identifying it with a complex Grassmannian. We show that the space of isometrically
2 immersedloopsstudiedbyMillsonandZombroisrealizedasthesymplecticreductionof
Mbytheactionofthebasedloopgroupofthecircle,givingasmoothversionofaresult
n
of Hausmann and Knutson on polygon space. The identification with a Grassmannian
a
allowsustodescribethegeodesicsofMexplicitly. Usingthisdescription,weshowthat
J
Manditsquotientbythereparameterizationgrouparenonnegativelycurved. Wealso
1
showthattheplanarloopspacestudiedbyYounes, Michor, ShahandMumfordinthe
1
contextofcomputervisionembedsinMasatotallygeodesic,Lagrangiansubmanifold.

TheactionofthereparameterizationgrouponMisshowntobeHamiltonianandthis
]
isusedtocharacterizethecriticalpointsoftheweightedtotaltwistfunctional.
G
D
.
h
t
a
m 1. Introduction
[ Let Polpn,(cid:126)rq denote the space of n-edge polygons in R3 with fixed edgelengths given by

1 (cid:126)r “pr1,...,rnq, where two polygons are identified if they differ by a rigid motion. In [15],
v KapovichandMillsonrealizePolpn,(cid:126)rqasthesymplecticreductionofaproductof2-spheres
3 by the diagonal action of SOp3q. Remarkably, essentially the same construction extends
8 to the infinite-dimensional space of smooth curves. Millson and Zombro show in [21] that
1 the space IsoImmpS1,R3q of smooth arclength-parameterized loops, identified up to rigid
3
motions, is realized as the symplectic reduction of the loop space of the 2-sphere by the
0
. rotation action of SOp3q. Illustrating a symplectic Gelfand-Macpherson correspondence,
1
Hausmann and Knutson show in [13] that Polpn,(cid:126)rq can alternatively be realized as the
0
7 symplectic reduction of the Grassmannian of 2-planes Gr2pCnq by the natural action of
1 Up1qn{Up1q. This idea was taken further by Howard, Manon and Millson in [14] to identify
: Gr pCnq with a moduli space of framed polygons called the space of spin-framed n-gons.
v 2
i Theaimofthispaperistogivesmoothversionsoftheconstructionsof[13,14]andtoshow
X
a relationship between these ideas and recent work in the field of computer vision.
r We begin by giving a construction of a K¨ahler structure on the space of smooth, param-
a
eterized, framed paths in R3—a framing of a smooth curve is a choice of smooth normal
unitvectorfieldalongthecurve. Thebasicideaoftheconstructionistorepresentaframed
path in R3 as a path in C2 (or the quaternions) via the Hopf map. This trick is well-known
to the computer graphics community [12] and has applications to contact geometry [1], but
we take the novel viewpoint that this representation is a local diffeomorphism of infinite-
dimensional manifolds. This locally embeds the space of framed paths into the path space
of C2—a complex Fr´echet vector space denoted PC2. We endow PC2 with a Hermitian
L2 metric, and the space of framed paths is thereby shown to have a rather transparent
K¨ahler structure. Moreover, various natural moduli spaces of framed curves are realized as
symplectic reductions of PC2.
1
2 TOMNEEDHAM
We are particularly interested in the moduli space of framed loops,
M:“trelatively framed loops in R3u{Sim,
where Sim denotes the group of Euclidean similarities. A relative framing of a loop is an
equivalence class of framings which is determined up to a choice of initial conditions. Our
first main result says that each connected component of M is identified in our coordinate
system with an infinite-dimensional complex Grassmannian (Theorem 3.15). This follows
fromthefactthattheclosureconditionforaframedpathinthecomplexcoordinatesystemis
simply L2-orthonormality—this is in stark contrast to the traditional curvature and torsion
functional coordinates on the space of Frenet-framed paths, in which the known closure
characterizations are impractical to check [9, 18].
The based loop group of the circle C8pS1,S1q{S1 acts on M by frame twisting; that is,
it acts transitively on the set of relative framings of a fixed base curve. We show in Theo-
rem 4.7 that symplectic reduction by this group action produces the Millson-Zombro space
IsoImmpS1,R3q. The proof involves showing that M has the stucture of a principal bundle
over the space of unframed loops (Theorem 4.1). An immediate corollary formalizes a phe-
nomenon observed in computer graphics literature [5, 11, 28]: any attempt to continuously
assign a framing which works for all immersed loops will necessarily fail.
Since M consists of parameterized framed curves, we can also consider the action of the
reparameterization group Diff`pS1q. On the symplectic side, we show that the action is
Hamiltonian with a momentum map that records the twisting of the vector field. This is
usedtocharacterizethecriticalpointsofanaturalgeneralizationoftheclassicaltotaltwist
functional (Theorem 5.2). The interplay of the Riemannian part of the K¨ahler structure of
M and the action of Diff`pS1q shows connections with recent work on shape recognition
applications. The Riemannian metric of M is Diff`pS1q-invariant, meaning it induces a
well-defined metric on the quotient space of unparameterized framed loops M{Diff`pS1q.
The methods of the recently developed field of elastic shape analysis [2, 20, 27, 30] can
be employed to approximate geodesic distance in M{Diff`pS1q (see Section 3.4), thereby
giving a shape recognition algorithm for framed loops.
TheshaperecognitionalgorithmrequiresefficientcomputationofgeodesicsinM. Fortu-
nately,thefactthatMisidentifiedwithaGrassmannianallowsustodescribeitsgeodesics
explictly. Using this description, we show that the planar loop space studied by Younes,
Michor, Shah and Mumford in [30] in the context of object recognition embeds as a totally
geodesicLagrangiansubmanifoldofM(Proposition6.5). ThespacesMandM{Diff`pS1q
are then shown to be nonnegatively curved (Theorem 6.12), echoing prior results in the
shape recognition literature for spaces of planar curves [2, 30].
Thepaperis organizedas follows. Section2introducesthe basicspaces offramedcurves
of interest and Section 3 describes the complex coordinate system for framed curve space.
Sections 4 and 5 treat the actions of C8pS1,S1q{S1 and Diff`pS1q, respectively. Section 6
is devoted to the Riemannian geometry of M.
1.1. Notation. For a finite-dimensional manifold M, we will use the notation PM :“
C8pr0,2s,Mq for the path space of M—the interval r0,2s is chosen as the domain of any
path as a convenient normalization. Let LM :“ C8pS1,Mq denote the loop space of M.
We identify S1 with r0,2s{p0 „ 2q so that LM includes naturally into PM. The spaces
PM andLM aretameFr´echetmanifolds(see[10]forageneralreferenceonFr´echetspaces)
and the inclusion LM ãÑ PM embeds LM into PM as a smooth submanifold of infinite
codimension. A smooth map f : M Ñ N induces a smooth map Pf : PM Ñ PN by the
formula pPfpγqqptq“fpγptqq. A similar statement holds for the loop spaces.
KA¨HLER STRUCTURES ON SPACES OF FRAMED CURVES 3
To distinguish from S1 “ r0,2s{p0 „ 2q, we use Sn to denote the standard radius-1 n-
sphere embedded in Rn`1 with its induced Riemannian metric. We reserve (cid:104)¨,¨(cid:105) and }¨}
for the Euclidean inner product and norm in R3, respectively. Other inner products will be
given specialized notation. We will use R` to denote the positive real numbers, considered
as a Lie group under multiplication.
2. Framed Curve Spaces
2.1. Framed Path Space. We begin with a formal definition of framed path space.
Definition 2.1. A framed path is a pair pγ,Vq of smooth maps r0,2s Ñ R3 such that the
base curve γ is an immersion and the framing V is a unit normal vector field along γ. The
moduli space of framed paths is the quotient space
O :“tframed pathsu{R3,
where R3 acts on a framed path by translation.
It will frequently be convenient to represent elements of O as framed paths pγ,Vq with
γp0q “ (cid:126)0. This representation is equivalent to choosing a global section of the R3-bundle
tframed pathsuÑO. This convention will be assumed unless otherwise noted.
A simple but useful observation is that it is possible to representa framed path pγ,Vq as
an element of PpSOp3qˆR`q via the frame map F:O ÑPpSOp3qˆR`q defined by
` ˘ γ1
(1) F:pγ,VqÞÑ pT,V,T ˆVq,}γ1} , T “ .
}γ1}
SinceO containsframedpathsdefineduptotranslation,Fisinvertible. Ifwetakeelements
of O to be based at the origin, then the inverse is given explicitly by
ˆ ż ˙
t ` ˘ ` ˘
(2) F´1 :ppU,V,Wq,rqÞÑ tÞÑ r t˜ U t˜ dt˜,V .
0
Here we are denoting an element of PSOp3q as a triple pUptq,Vptq,Wptqq of paths in R3
which are pairwise orthonormal for all t. The frame map gives an identification of O with
PpSOp3q ˆ R`q and we conclude that O naturally has the structure of a tame Fr´echet
manifold.
To simplify notation, we denote the group of Euclidean similarities by Sim :“ R3 ˆ
SOp3qˆR` and the subgroup of similarities that fix the origin by Sim :“SOp3qˆR`.
0
2.2. Framed Loop Space. ThemanifoldO ofopenframedpathscontainsthemoreinter-
estingsubmanifoldofclosed framedloops. Itwillbeconvenienttointroduceanintermediate
space.
Definition 2.2. A frame-periodic framed path is a framed path pγ,Vq such that γ1 and V
are closed smooth curves. If a frame-periodic framed path also satisfies γp0q“γp2q, then it
is called a framed loop. The collections of frame-periodic framed paths and framed loops,
considered up to translation, will respectively be denoted O and C.
fp
As in the case of O, we will typically represent elements of O and C as framed
fp
paths/loops which are based at the origin.
If pγ,Vq is a frame-periodic framed path, then its image under the frame map (1) is a
loop in Sim . The identification O « PSim restricts to an identification O « LSim
0 0 fp 0
and it follows that O Ă O is a submanifold of infinite codimension. It is apparent that
fp
the frame map restricts to give an embedding of C into LSim which is not surjective. In
0
4 TOMNEEDHAM
fşact, the inverse frame map takes an element ppU,V,Wq,rq of LSim0 into C if and only if
rUdt “(cid:126)0—this is simply the closure condition for γ. A straightforward application of
S1
Hamilton’s Implicit Function Theorem [10, Section III, Theorem 2.3.1] proves the following
proposition. We omit the proof, but its structure is similar to the proof of Proposition 4.5,
which is given below.
Proposition 2.3. Framed loop space C is a codimension-3 submanifold of frame-periodic
framed path space O .
fp
TheidentificationO «LSim showsthatO hastwopathcomponents. Thesubman-
fp 0 fp
ifold C therefore has at least two path components and an argument similar to the classical
proof of the Whitney-Graustein theorem [29] can be used to show that it has exactly two.
The path component of a framed loop pγ,Vq with embedded base curve γ is determined
by the linking number of γ with a small pushoff γ`(cid:15)V, modulo 2. Accordingly, the path
componentsofC aredenotedC andC foroddandevenself-linkingnumber,respectively.
od ev
3. Complex Coordinates for Framed Curve Spaces
3.1. Complex Coordinates for Framed Paths. The goal of this section is to provide
a dictionary between the framed curve spaces O and C and various submanifolds of the
path space PC2. We begin by introducing some notation. Elements of the complex vector
space PC2 will be denoted Φ “ pφ,ψq, where φ,ψ P PC. We endow C2 with its standard
Hermitian inner product (cid:104)¨,¨(cid:105) and PC2 with the L2 Hermitian (weak) inner product
C2
ż
2
(cid:104)Φ ,Φ (cid:105) :“ (cid:104)Φ ptq,Φ ptq(cid:105) dt.
1 2 L2 1 2 C2
0
The associated norm will be denoted } ¨ } . This Hermitian structure trivially gives a
L2
K¨ahlerstructureonPC2 withRiemannianmetricgL2 :“Re(cid:104)¨,¨(cid:105) ,symplecticformωL2 :“
L2
´Im(cid:104)¨,¨(cid:105) and complex structure given by pointwise multiplication by the imaginary unit
L2
i. Since ωL2 doesn’t depend on its basepoint, it is obviously closed in the sense of [4,
Section 1.4]. This K¨ahler structure restricts to the linear subspace LC2 and to the open
submanifolds
P˝C2 :“PpC2zt(cid:126)0uq and L˝C2 :“LpC2zt(cid:126)0uq.
We will abuse notation and continue to use (cid:104)¨,¨(cid:105) , gL2 and ωL2 to denote the restrictions
L2
of these objects to the subspaces. Abusing notation even further, we will denote the L2
Hermitian inner product on PC (and its restriction to any subspaces) by
ż
2
(cid:104)φ,ψ(cid:105) :“ φψdt,
L2
0
and the induced norm by }¨} .
L2
We require the definition of a new subspace of PC2. A path ΦPPC2 is called smoothly
antiperiodic if
ˇ ˇ
dk ˇˇ dk ˇˇ
ˇ Φ“´ ˇ Φ for all k “0,1,2,....
dtk dtk
t“2 t“0
The space of antiperiodic paths in C2, or the antiloop space of C2, is denoted AC2. The
antiloop space is a complex tame Fr´echet vector space. We define AC and ApC2zt(cid:126)0uq “:
A˝C2 similarly. There is a biholomorphism from LC to AC given by φptqÞÑeit{2φptq.
KA¨HLER STRUCTURES ON SPACES OF FRAMED CURVES 5
Proposition3.1. ThereisasmoothdoublecoveringP˝C2 ÑO whichrestrictstoasmooth
double covering L˝C2 \A˝C2 Ñ O . By transfer of structure, O and O are K¨ahler
fp fp
manifolds.
Theproofofthepropositionreliesonthewell-knowntrickofrepresentingaframedpathas
apathinthequaternionsviatheframe-Hopfmap(see,e.g.,[1,12]). LetH“spanRt1,i,j,ku
denote the quaternions. The frame-Hopf map is the map Hopf :HÑR3ˆ3 defined by
(3) Hopfpqq“pqiq,qjq,qkqq.
In the above, q denotes the quaternionic conjugate of q. Each entry on the right side
of (3) lies in spanRti,j,ku, which we identify with R3. We can identify C2 with H via
pz,wq Ø q “ z`wj and i Ø i, and under this identification the frame-Hopf map is given
by the formula
¨ ˛
|z|2´|w|2 2Impzwq ´2Repzwq
(4) Hopfpz,wq:“˝ 2Impzwq Repz2`w2q Impz2`w2q ‚.
2Repzwq Imp´z2`w2q Repz2´w2q
It easy to check that each column of Hopf restricts to give a Hopf fibration S1 ãÑS3 ÑS2.
Theframe-Hopfmaphasseveralusefulproperties. Welistafewoftheminthefollowing
lemma. Eachassertionfollowsbyanelementarycomputation. Inthelemmaandthroughout
the rest of the paper we use Hopf , j Pt1,2,3u to denote the j-th column of Hopf.
j
Lemma 3.2. The frame-Hopf map has the following properties:
(i) Hopf restricts to a map C2zt0uÑR3ˆ3 with Hopfpz,wq“Hopfpz1,w1q if and only
if pz,wq“˘pz1,w1q.
(ii) Hopf has the scaling property
Hopfpr¨pz,wqq“r2Hopfpz,wq, r PR`.
In particular, each entry Hopf , j “1,2,3, squares norms:
j
}Hopf pz,wq}“}pz,wq}2 .
j C2
(iii) The entries Hopf are mutually orthogonal and have the same norm.
j
Proof of Proposition 3.1. It follows from Lemma 3.2 that Hopf induces a smooth double-
cover Hzopf :C2zt0uÑSim defined by
0
ˆ ˙
Hzopfpz,wq:“ 1 Hopfpz,wq,}pz,wq}2 ,
}pz,wq}2 C2
C2
z z
whereHopfpz,wq“Hopfpz1,w1qifandonlyifpz,wq“˘pz1,w1q. Applyingthepathfunctor
producesasmoothmapPHzopf :P˝C2 ÑPSim satisfyingPHzopfpΦ q“PHzopfpΦ qifand
0 1 2
only if Φ “ ˘Φ for Φ P P˝C2. In light of the identification PSim « O via the inverse
1 2 j 0
z
frame map F´1, this completes the proof of the first claim. The map PHopf restricts to
a double covering L˝C2 \A˝C2 Ñ LSim « O , where each factor in the disjoint union
0 fp
covers one of the two path components of LSim . (cid:3)
0
Since the maps in the proof will be used frequently throughout the rest of the paper, we
will use the simplified notation
H:“PHzopf and Hp :“F´1˝H.
6 TOMNEEDHAM
p
3.2. Projective Spaces. ThefactthatHisadoublecoversuggeststhatitwouldbeuseful
to projectivize.
?
Definition 3.3. For V “ PC, LC or AC, let SpV2q denote the radius- 2 L2-sphere in
V2. Let ProjRpV2q denote the projective space of real lines in V2, obtained from SpV2q by
identifying antipodal points. Let Proj˝RpV2q denote the open submanifold obtained as the
corresponding quotient of the open submanifold S˝pV2q:“tΦPSpV2q|Φptq‰(cid:126)0@tu.
Remark 3.4. Adapting the usual finite-dimensional charts, one is able to show that the
spheres and projective spaces defined above are Fr´echet manifolds. We will define infinite-
dimensional complex projective spaces, Stiefel manifolds and Grassmannians below. These
spacesarealsoFr´echetmanifolds,andwewillrefertothemassuchwithoutfurthercomment.
Moreover, one can show that the complex projective spaces and complex Grassmannians are
complex Fr´echet manifolds by adapting the classical holomorphic charts.
Phrasing the definition differently, ProjRpV2q is obtained as the quotient of V2zt(cid:126)0u by
the action of Rzt0u by pointwise multiplication. The path component R` ĂRzt0u acts on
framed path space by scaling base curves: that is, r P R` acts on pγ,Vq P O according to
the formula pr¨pγ,Vqqptq “ prγptq,Vptqq. Part (b) of Lemma 3.2 immediately implies the
following.
p
Lemma 3.5. The map H is equivariant with respect to the pointwise multiplication action
of Rzt0u on P˝C2 and the scaling action of R` on O in the sense that Hppr¨Φq“r2¨HppΦq.
For the sake of concreteness, we will identify the quotient O{R` with the global cross-
section consisting of framed curves whose base curve has fixed length 2 (this normalization
willbeconvenientlateron). ThenHp restrictstogiveadoublecoveringofO{R` byS˝pPC2q
with fibers of the form t˘Φu. Indeed, let Φ P S˝pPC2q and let pγ,Vq “ HppΦq. Then
?
}Φ} “ 2 implies
L2
ż ż ż
2 2 2
(5) lengthpγq“ }γ1ptq}dt“ }Hopf pΦptqq}dt“ }Φptq}2 dt“2.
1 C2
0 0 0
The same argument holds for the loop space and antiloop spaces, and we conclude:
p
Corollary 3.6. The map H induces diffeomorphisms
Proj˝RpPC2q«O{R` and Proj˝RpLC2q\Proj˝RpAC2q«Ofp{R`.
Taking the real projectivization ignores the complex structure, and this suggests that we
should further quotient by the S1 factor of Czt0u « R`ˆS1. More precisely, S1 Ă C acts
on PC2 by pointwise multiplication in each factor:
peiθ¨pφ,ψqqptq:“peiθφptq,eiθψptqq, eiθ PS1 ĂC.
This action restricts to the various spheres SpV2q.
Definition 3.7. For V “LC or PC, let ProjCpV2q denote the projective space of complex
linesinV. The projective space is given by the quotient of SpV2q by the pointwise S1-action.
Let Proj˝CpV2q:“S˝pV2q{S1.
Once again, this group action has a natural interpretation for framed curves. The action
is by constant frame twisting: eiθ PS1 acts on pγ,Vq according to the formula
γ1
eiθ¨pγ,Vq“pγ,cosp2θqV `sinp2θqWq, W “ ˆV.
}γ1}
The following lemma can be verified by a simple calculation.
KA¨HLER STRUCTURES ON SPACES OF FRAMED CURVES 7
Lemma 3.8. The first coordinate of the frame-Hopf map Hopf is invariant under the
1
diagonal S1-action on C2 by multiplication; that is, for all eiθ PS1,
Hopf peiθz,eiθwq“Hopf pz,wq.
1 1
The other coordinates of Hopf satisfy
Hopf peiθpz,wqq“cosp2θqHopf pz,wq`sinp2θqHopf pz,wq,
2 2 3
Hopf peiθpz,wqq“´sinp2θqHopf pz,wq`cosp2θqHopf pz,wq.
3 2 3
It follows that the map Hp is equivariant with respect to the pointwise S1-action on PC2 and
the constant frame-twisting action of S1 on O; that is, HppeiθΦq“eiθ¨HppΦq
An equivalence class of this S1-action on O will be called a relatively framed path. A
relatively framed path can be viewed as a path endowed with a framing which is well-
defined up to a choice of initial conditions. A well-known example of a relative framing is
the Bishop framing [3], obtained for a given base curve γ by evolving an initial vector Vp0q
along γ with no intrinsic twisting. Other examples of relative framings include the writhe
framing of [6] and the constant twist minimizing framing introduced in Section 4.4.
The lemma implies that the complex projective spaces correspond to relatively framed
path spaces. As in the finite-dimensional case, the complex projective spaces are K¨ahler
manifolds with Fubini-Study metrics inherited from their respective affine spaces. Summa-
rizing, we have shown:
p
Corollary 3.9. The map H induces diffeomorphisms
Proj˝CpPC2q«O{pR`ˆS1q and Proj˝CpLC2q\Proj˝CpAC2q«Ofp{pR`ˆS1q.
It follows that the relatively framed path spaces are K¨ahler manifolds.
3.3. Complex Coordinates for Framed Loops. Amajorbenefitofthecomplexcoordi-
natesthatwehavedevelopedforframedpathsisthatthenecessaryandsufficientconditions
for the periodicity of the resulting framed path are surprisingly nice. We have the following
key lemma, which can be seen as an infinite-dimensional analogue of the discussion in [13,
Section 3.5] regarding finite-dimensional polygon spaces.
Lemma 3.10. A path Φ“pφ,ψqPP˝C2 corresponds to a framed loop under Hp if and only
if
(i) The path Φ is either smoothly closed or smoothly antiperiodic and
(ii) the maps φ and ψ have the same L2 norm and are L2-orthogonal:
ż ż ż
2 2 2
|φ|2 dt“ |ψ|2 dt and φψ dt“0.
0 0 0
p
Proof. It follows from Proposition 3.1 that a path Φ satisfies HpΦq P O if and only if
fp
Φ P L˝C2 \ A˝C2. In this case, let ppT,V,Wq,rq :“ HpΦq P LSim . Then the frame
0
p
periodic path HpΦq “ F´1ppT,V,Wq,rq PşOfp lies in the submanifold C Ă Ofp of framed
loops if and only if the closure condition rT dt “(cid:126)0 holds. We see from the formula for
S1
the frame-Hopf map given in (4) that the closure condition is written in terms of φ and ψ
as ż ż ż
2 2 2
|φ|2´|ψ|2 dt“ 2Impφψqdt“ 2Repφψqdt“0.
0 0 0
This is equivalent to the conditions given in (ii). (cid:3)
8 TOMNEEDHAM
This lemma has a symplectic interpretation. In the following, let V “ LC or AC. The
group Up2q acts by isometries on the Hermitian vector space V2 by pointwise right multi-
plication. This action is also Hamiltonian with momentum map
µ :V2 Ñup2qp«up2q˚q
Up2q ˆ ˙
(cid:104)φ,φ(cid:105) ´(cid:104)φ,ψ(cid:105)
pφ,ψqÞÑi L2 L2 .
´(cid:104)ψ,φ(cid:105) (cid:104)ψ,ψ(cid:105)
L2 L2
Theconditionsinpart(ii)ofLemma3.10canthenbephrasedintermsoftheentriesofthe
entries of µ pΦq.
Up2q
Definition 3.11. For V “ LC or AC, the Stiefel manifold of L2-orthonormal 2-frames in
V is the Fr´echet manifold
ˆ ˙
(cid:32) (
i 0
St pVq:“ pφ,ψqPV2 |(cid:104)φ,φ(cid:105) “(cid:104)ψ,ψ(cid:105) “1 and (cid:104)φ,ψ(cid:105) “0 “µ´1 .
2 L2 L2 L2 Up2q 0 i
Let St˝pVq denote the open submanifold
2
St˝pVq:“tpφ,ψqPSt pVq|pφptq,ψptqq‰p0,0q@tu.
2 2
ThescalingactionofR`onOrestrictstothespaceofclosedframedloopsC. Wesimilarly
identify the quotient C{R` with the space of closed loops of fixed length 2.
p
Proposition 3.12. The restriction of H gives double coverings
St˝pLCq݈ÝÑ2 C {R` and St˝pACq݈ÝÑ2 C {R`
2 od 2 ev
Proof. A path Φ P P˝C2 lies in one of the Stiefel manifolds if and only if it satisfies the
?
conditionsofLemma3.10and}Φ} “ 2. Thecalculation(5)showsthatthisisequivalent
L2
to HppΦq P C{R`. To see that Hp maps the path components as claimed it suffices to check
an example. For
1
Φptq“ ? pcospπtq´isinpπtq,1qPSt˝pLCq
2 2
p
we have HpΦq“pγ,Vq, where
1
γptq“ p0,cospπtq´1,sinpπtqq, Vptq“p´sinpπtq,cos2pπtq,cospπtqsinpπtqq.
π
Thus the image of Φ is a framed circle with linking number 1. (cid:3)
Definition 3.13. For V “LC or AC, the Grassmannian is the Fr´echet manifold Gr pVq of
2
2-dimensionalcomplexsubspacesofV. ItcanberepresentedasthequotientspaceGr pVq:“
2
St pVq{Up2q. Let Gr˝pVq denote the open submanifold Gr˝pVq:“St˝pVq{Up2q.
2 2 2 2
The Grassmannian Gr pVq is a complex manifold (see Remark 3.4). Moreover, Gr pVq
2 2
is obtained as the symplectic reduction of V2 by the isometric action of Up2q and there-
fore inherits a natural K¨ahler structure. We will use double slash notation for symplectic
reductions for the rest of the paper; e.g., Gr pVq“V2 Up2q.
2
Thenextlemmashowsthattheactionofthesubgrou(cid:12)pSUp2qĂUp2qonPC2 corresponds
to the rotation action of SOp3q on framed path space; that is, an element A of the group
SOp3q acts on a framed path or loop pγ,Vq pointwise by the formula A¨pγ,Vqptq :“ pA¨
γptq,A¨Vptqq. ForU PSUp2q,wedefineHopfpUqPR3ˆ3 byidentifyingSUp2qwithS3 ĂC2
via ˆ ˙
z w
Øpz,wq.
´w z
KA¨HLER STRUCTURES ON SPACES OF FRAMED CURVES 9
The first statement of the lemma is well-known (see, e.g., [8, Section I.1.4]).
Lemma 3.14. The restriction of Hopf to SUp2q Ă C2 gives an anti-homomorphic double
coverofSOp3q. ItfollowsthatthemapHp hasthepropertythatforΦPP˝C2 andU PSUp2q,
p p
HpΦ¨Uq“HopfpUq¨HpΦq.
This brings us to our first main result and to the space which the rest of the paper will
be primarily concerned with, the moduli space of framed loops
M:“C{pSim ˆS1q“trelatively framed loops in R3u{Sim.
0
The moduli space M has two path components corresponding to the components of C.
We correspondingly denote these components M and M . For most of the paper we
ev od
will denote elements of M by rγ,Vs, where pγ,Vq is a framed loop with γp0q “ (cid:126)0 and
lengthpγq “ 2. The brackets denote the equivalence class of pγ,Vq under the action of
SOp3qˆS1 by rotations and constant frame twisting.
p
Theorem 3.15. The map H induces diffeomorphisms
Gr˝pLCq«M and Gr˝pACq«M .
2 od 2 ev
By transfer of structure, M is a K¨ahler manifold.
Proof. Lemma 3.14 implies that passing from St pVq to the quotient St pVq{SUp2q has the
2 2
effect of modding out by the rotation action of SOp3q on C{R`. As we have already seen,
the Up1q « S1 factor of Up2q « SUp2qˆUp1q corresponds to the constant frame twisting
action on C. Further quotienting by S1, we obtain Gr pVq “ St pVq{Up2q. Modding out
2 2
by S1 in particular identifies antipodal paths, thus the double cover Hp : St˝pLCq ݈ÝÑ2 C
2 od
induces a diffeomorphism Gr˝pLCq « M . Similarly, we have an induced diffeomorphism
2 od
Gr˝pACq«M . (cid:3)
2 ev
We will denote elements of Gr pVq by rΦs for Φ “ pφ,ψq P St pVq. The tangent spaces
2 2
to St pVq are
2
! )
T St pVq“ pδφ,δψqPV2 |gL2pφ,δφq“gL2pψ,δψq“(cid:104)φ,δψ(cid:105) ´(cid:104)δφ,ψ(cid:105) “0 .
Φ 2 L2 L2
It will frequently be convenient to represent the tangent space to rΦs P Gr pVq as the
2
horizontal tangent space to Φ P St pVq—this is the subspace of tangent vectors which are
2
gL2-orthogonal to the Up2q-orbit of Φ given by
(cid:32) (
ThorSt pVq“ pδφ,δψqPV2 |(cid:104)φ,δφ(cid:105) “(cid:104)ψ,δψ(cid:105) “(cid:104)φ,δψ(cid:105) “(cid:104)ψ,δφ(cid:105) “0 .
Φ 2 L2 L2 L2 L2
The identifications of this section are summarized below. The spaces in the first column
have K¨ahler structures.
P˝C2 ݈ÝÑ2 O
L˝C2\A˝C2 ݈ÝÑ2 O Proj˝RpPC2q«O{R`
fp
Proj˝CpPC2q«O{pR`ˆS1q Proj˝RpLC2q\Proj˝RpAC2q«Ofp{R`
Proj˝CpLC2q\Proj˝CpAC2q«Ofp{pR`ˆS1q St˝2pLCq\St˝2pACq݈ÝÑ2 C{R`
Gr˝pLCq\Gr˝pACq«M
2 2
10 TOMNEEDHAM
3.4. Elastic Shape Analysis. We now take a brief detour to discuss the field of Elastic
Shape Analysis, a novel approach to shape recognition that has been developed over the
last decade [2, 20, 27, 30]. In this discussion we restrict our attention to shape recognition
for loops in Rd, although the methods of Elastic Shape Analysis have also been applied
to curves in manifolds [19], surfaces in R3 [16], et cetera. The main idea is to endow
the space of immersions ImmpS1,Rdq with a Riemannian metric which is invariant under
the reparameterization action of Diff`pS1q. Then the metric descends to a well-defined
metric on the quotient ImmpS1,Rdq{Diff`pS1q. Geodesic distance with respect to this
Riemannian metric is then interpreted as a measure of dissimilarity between shapes. In
practice, the geodesic distance between the shapes represented by parameterized loops γ
1
andγ istypicallycomputedbyfinding(orapproximating)theinfimumofgeodesicdistance
2
between the Diff`pS1q-orbits of γ and γ in the total space; that is, by computing
1 2
(6) inf distpγ ˝ρ ,γ ˝ρ q“ inf distpγ ,γ ˝ρq,
1 1 2 2 1 2
ρ1,ρ2PDiff`pS1q ρPDiff`pS1q
wheredistisgeodesicdistanceinImmpS1,Rdqandtheequalityfollowsfromtheassumption
thatDiff`pS1qactsbyisometries. Inthisappliedfield,efficientcomputabilityofthedistance
(6)isparamountandonethereforeseeksaRiemannianmetriconimmersionspaceforwhich
geodesics are easily computable.
Theelasticmetrics ofMio,SrivastavaandJoshi[22]forma2-parameterfamilyofmetrics
ga,b on ImmpS1,R2q with theoretically convenient properties. These are given explicitly for
γ PImmpS1,R2q, δγ ,δγ PT ImmpS1,R2q«C8pS1,R2q and a,bą0 by the formula
1 2 γ
ż
(cid:28) (cid:29)(cid:28) (cid:29) (cid:28) (cid:29)(cid:28) (cid:29)
d d d d
(7) ga,bpδγ ,δγ q“ a δγ ,T δγ ,T `b δγ ,N δγ ,N ds.
γ 1 2 ds 1 ds 2 ds 1 ds 2
S1
In the above, d{ds denotes derivative with respect to arclength, ds denotes arclength mea-
sure, T “ γ1{}γ1} is the unit tangent to γ and N is the oriented unit normal. In [30],
Younes, Michor, Shah and Mumford show that, with parameter choice a “ b, the space
ImmpS1,R2q{Sim of Euclidean similarity classes of immersions is locally isometric to an
infinite-dimensional Grassmannian with a natural L2 metric. This is remarkable because it
allows for computation of explicit geodesics in ImmpS1,R2q{Sim, whence the infimum pro-
cedure in (6) can be approximated very efficiently via a dynamic programming algorithm.
The relationship between our space M and the work of Younes et. al. will be expounded
upon in Section 6.5.
3.5. Induced Geometric Structures. Proposition 3.1 states that O inherits a K¨ahler
structure from P˝C2 by transfer of structure under the local diffeomorphism :Hp :P˝C2 Ñ
O. We will give explicit formulas for the various parts of the K¨ahler structure here.
For pγ,Vq P O and pδγ ,δV q P T O, j “ 1,2, we define a metric gO on O by the
j j pγ,Vq
formula
ż
1 2(cid:28) d d (cid:29)
gO ppδγ ,δV q,pδγ ,δV qq“ δγ , δγ `(cid:104)δV ,W(cid:105)(cid:104)δV ,W(cid:105) ds,
pγ,Vq 1 1 2 2 4 ds 1 ds 2 1 2
0
where d “ 1 d denotes arclength derivative and ds“}γ1}dt denotes arclength measure.
ds }γ1}dt
Thefactorof 1 isincludedonlyasaconvenientnormalizationwhoseutilityismadeapparent
4
by the following proposition. The proof of the proposition is a tedious but essentially
straightforward calculation, so we omit it.
Proposition 3.16. The pullback of gO to P˝C2 is gL2.