Analysis and geometry of shape spaces.pdf

8/11/2019 Analysis and geometry of shape spaces.pdf

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Analysis and geometry of shape spacesI - V

Peter W. Michor

Winterschool Geometry and Physics, Srni, Jan. 1724, 2009


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Content:Introduction: What are shape spaces and what

are they good for. A little bit of history.Shape spaces of plane curves:The topology of shape space.Hamiltonian background and conserved momenta.A bunch of metrics, their geodesics and curvatures:The L2 -metric and its vanishing of geodesic dis-tance.Almost local metrics.Immersion Sobolev metrics.The scale invariant Sobolov H 1 -metric and its rela-tion to the Grassmannian of 2-planes in an innite

dimensional space, and Neretin geodesics.A covariant formula for curvature and its rela-tion to ONeills curvature formulas.


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Shape spaces as quotients of diffeomorphism

groups:

Right invariant metrics on diffeomorphism groups,their geodesics and curvatures.Landmark space, geodesics and curvatures.Spaces of submanifolds (plane curves and higherdimensional ones)High dimensional shape space Imm( M, N ) / Diff( M )Vanishing geodesic distance on diffeomorphism

groups. Burgers equation corresponds to this phe-nomenon. The Camassa-Holm equation has topositive geodesic distance. For the Korteweg-deVries equation we do not know.The universal Teichmueller space with the Weil-Peterssen metric as shape space. (not done)


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Based on:P.M. and D. Mumford. Riemannian geometries on spaces of plane curves.J. Eur. Math. Soc. (JEMS) 8 (2006), 1-48, arXiv:math.DG/0312384.

H. Kodama, P.M. The homotopy type of the space of degree 0 im-mersed curves. Revista Matematica Complutense 19 (2006), 227-234.arXiv:math/0509694.P.M. and D. Mumford. An overview of the Riemannian metrics on spacesof curves using the Hamiltonian approach. Appl. Numerical Harmonic Analysis 23 (2007), 74-113.arXiv:math.DG/0605009P.M., David Mumford, Jayant Shah, Laurent Younes: A Metric on ShapeSpace with Explicit Geodesics. Rend. Lincei Mat. Appl. 9 (2008) 25-57.arXiv:0706.4299Mario Micheli, P.M., David Mumford: Landmarks. In preparation.David Mumford: Lectures at the Chennai Mathematical Institute.

P.M., David Mumford: Vanishing geodesic distance on spaces of subman-ifolds and diffeomorphisms, Documenta Math. 10 (2005), 217245.arXiv:math.DG/0409303V.Cervera, F.Mascaro, P.M.: The action of the diffeomorphism group onthe space of immersions. Diff. Geom. Appl. 1 (1991), 391401For background material: Peter W. Michor: Some Geometric EvolutionEquations Arising as Geodesic Equations on Groups of Diffeomorphism,Including the Hamiltonian Approach. IN: Phase space analysis of Par-tial Differential Equations. Birkhauser Verlag 2006. Pages 133-215.arXiv:math/0609077


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Introduction:

What are shapes, why are they interesting, andhow are they arranged in shape spaces.


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Shape spaces of plane curves:

Some spaces:Diff( S 1 ) a regular Lie group, = Diff + ( S 1 )Diff ( S

1 ).

Emb = Emb( S 1 , R 2 ), the manifold of all smooth

embeddings S 1 R 2 .T Emb( S 1 , R 2 ) = Emb( S 1 , R 2 ) C ( S 1 , R 2 ).Imm = Imm( S 1 , R 2 ), the manifold of all smooth

immersions S 1

R 2

.T Imm( S 1 , R 2 ) = Imm( S 1 , R 2 ) C ( S 1 , R 2 ).Imm free = Imm free ( S 1 , R 2 ), the manifold of all free

smooth immersions S 1

R 2 , i.e., those with triv-

ial isotropy group for the right action of Diff( S 1 )on Imm( S 1 , R 2 ).


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B e = Be( S 1 , R 2 ) = Emb( S 1 , R 2 ) / Diff( S 1 ), the man-ifold of 1-dimensional connectedsubmanifolds of R 2 ,

B i = B i( S 1 , R 2 ) = Imm( S 1 , R 2 ) / Diff( S 1 ), an in-nite dimensional orbifold

B i, free = Imm free ( S 1 , R 2 ) / Diff( S 1 ), a manifold, the

base of a principal ber bundle,


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Notation. We work mostly with arclength ds , ar-clength derivative Ds and the unit tangent vector

v to the curve:ds = |c|d

D s = / |c|v = c

/

|c

|Attention: Given a family of curves c( , t ), then and t commute but Ds and t dont. Rota-tion through 90 degrees (complex multiplication by

1) will be denoted by:J = 0 11 0 .

The unit normal vector to the image curve is thusn = Jv.


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The degree of immersions. The degree or rota-tion degree of an immersion c : S 1 R 2 is the wind-ing number around 0 of the tangent c : S 1 R 2 .Imm( S

1,

R 2) decomposes into the disjointunion of the open submanifolds Imm k( S 1 , R 2 ) for

k Z according to the degree k. These are con-nected according to a theorem of Whitney and

Graustein (1931-32)


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Theorem. The manifold Imm k( S 1 , R 2 ) of immersed

curves of degree k contains S 1 as a strong smoothdeformation retract.For k = 0 the manifold B ki ( S

1 , R 2 ) := Imm k( S 1 , R 2 ) / Diff + ( S 1 )is contractible.For k = 0 we have (surprise, Kodama-M.)

1 ( B 0 ( S 1 , R 2 )) = Z ,2 ( B 0 ( S 1 , R 2 )) = Z ,k( B

0 ( S 1 , R 2 )) = 0 for k > 2 .


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The tangent bundle isT Imm( S 1 , R 2 ) = Imm( S 1 , R 2 ) C ( S 1 , R 2 ), thecotangent bundle isT Imm( S 1 , R 2 ) = Imm( S 1 , R 2 ) D( S 1 ) 2where the second factor consists of periodic distri-butions.


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We consider smooth Riemannian metricson Imm( S 1 , R 2 ), i.e., smooth mappings

G : Imm( S 1 , R 2 ) C ( S 1 , R 2 ) C ( S 1 , R 2 ) R( c,h,k ) Gc( h, k ) , bilinear in h, kGc( h, h ) > 0 for h

= 0 .

Each such metric is weak in the sense that Gc,viewed as bounded linear mapping

Gc : T c Imm( S 1

,R 2

) = C ( S 1

,R 2

) T c Imm( S 1 , R 2 ) = D( S 1 ) 2G : T Imm( S 1 , R 2 ) T Imm( S 1 , R 2 )G ( c, h ) = ( c, G c( h, ))

is injective, but can never be surjective.


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The fundamental symplectic form onT Imm( S 1 , R 2 ) pulled back from the canonical sym-plectic form on the contangent bundle via the map-ping G : T Imm( S 1 , R 2 ) T Imm( S 1 , R 2 ) is then:

( c,h ) (( k1 , 1 ) , ( k2 , 2 )) == dG c( k1 )( h, k 2 ) Gc( 1 , k2 )

+ dG c( k2 )( h, k 1 ) + Gc( 2 , k1 )

= Gc( k2 , H c( h, k 1 ) K c( k1 , h ))+ Gc( 2 , k1 ) Gc( 1 , k2 )


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The geodesic equation. The Hamiltonian vectoreld of the Riemann energy function

E ( c, h ) = 12

Gc( h, h ) , E : T Imm( S 1 , R 2 ) Ris the geodesic vector eld:

grad 1 ( E )( c, h ) = hgrad 2 ( E )( c, h ) =

12 H c( h, h ) K c( h, h )

and the geodesic equation becomes:

ct = hh t = 12 H c( h, h ) K c( h, h )

ctt = 12 H c( ct , ct ) K c( ct , ct )


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The momentum mapping for a G-isometric group

action. Consider a (possibly innite dimensionalregular) Lie group with Lie algebra g with a rightaction g r g by isometries on Imm( S 1 , R 2 ). Fun-damental vector eld mapping : g

X(Imm( S 1 , R 2 ))

a bounded Lie algebra homomorphism, given by

X ( c) = t |0 r exp( tX ) ( c) .momentum map j : g

C

G (T Imm( S 1 , R 2 ) , R ):

jX ( c, h ) = Gc( X ( c) , h ) .

J : T Imm( S 1 , R 2 )

g,

J ( c, h ) , X = jX ( c, h ) .


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It ts into the following commmutative diagram

and is a homomorphism of Lie algebras:

0 H 0 i C G grad

X H 1 0

g j

T Imm

J is equivariant for the group action. Along anygeodesic t c( t, ) this momentum mapping isconstant, thus for any X

g

J ( c, c t ) , X = jX ( c, c t ) = Gc( X ( c) , ct )is constant in t.


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We can apply this construction to the followinggroup actions on Imm( S 1 , R 2 ).

The smooth right action of the group Diff( S 1 ) onImm( S 1 , R 2 ), given by composition from the right:c c for Diff( S 1 ). For X X( S 1 ) thefundamental vector eld is then given by

Diff X (c) = X ( c) = t |0 ( c Fl X t ) = c.X.The reparametrization momentum , for any vector

eld X on S 1

is thus: jX ( c, h ) = Gc( c.X,h ) .

Assuming the metric is reparametrization invariant,

it follows that on any geodesic c( , t ), the expres-sion Gc( c.X,c t ) is constant for all X .


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The left action of the Euclidean motion groupM (2) = R 2 SO (2) on Imm( S 1 , R 2 ) given by c eaJ c+ B for ( B, e aJ ) R 2SO (2). The fundamentalvector eld mapping is

( B,a ) ( c) = aJc + BThe linear momentum is thus Gc( B, h ) , B R 2 andif the metric is translation invariant, Gc( B, c t ) will

be constant along geodesics. The angular momen-tum is similarly Gc( Jc,h ) and if the metric is rota-tion invariant, then Gc( Jc,c t ) will be constant alonggeodesics.


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The action of the scaling group of R given byc er c, with fundamental vector eld a ( c) = a.c .If the metric is scale invariant, then the scaling momentum Gc( c, c t ) will also be invariant alonggeodesics.


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If the Riemannian metric G on Imm is invariantunder the action of Diff( S 1 ) it induces a metric on

the quotient B i as follows. For any C 0 , C 1 B i ,consider all liftings c0 , c1 Imm such that ( c0 ) =C 0 , ( c1 ) = C 1 and all smooth curves t ( c( t, )) in Imm( S 1 , R 2 ) with c(0 , ) = c0 and c(1 , ) =c1 . Since the metric G is invariant under the actionof Diff( S 1 ) the arc-length of the curve t ( c( t, ))in B i( S 1 , R 2 ) is given by

LhorG (c) := LG ( ( c( t,

)))

= 10 G ( c) ( T c.c t , T c.c t ) dt=

1

0

Gc( ct , ct ) dt

dist B i ( S 1 ,R 2 )

G (C 1 , C 2 ) = inf c LhorG (c) .


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The simplest ( L2 -) metric.

G0c ( h, k ) = S 1 h, k ds = S 1 h, k |c|dWe compute the G0 -gradients of c G0c ( h, k ):dG

0

( c)( m )( h, k ) = G0c ( K

0c ( m, h ) , k ) = G

0c m, H

0c ( h, k ) ,

K 0c ( m, h ) = D s ( m ) , v h, D s =

|c|, v =

c

|c|.

H 0c ( h, k ) =

D s h, k v

Geodesic equation

ctt =

1

2 |c| |ct |

2 c

|c|

1

|c|2 ct , c ct .


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Horizontal Geodesics for G0

ct , c = 0 and ct = an = aJ c

|c

| for a

C ( S 1 , R ).

We use functions a, s = |c|, and , only holonomicderivatives:

s t =

as, a t = 12 a

2 ,

t = a 2 + 1s

a s

= a 2 + as2

ass3

.

We may assume s|t=0 1. Let v( ) = a(0 , ), theinitial value for a. Thens ts = a = 2 a ta , so log( sa 2 ) t = 0, thuss ( t, ) a ( t, ) 2 = s(0 , ) a (0 , ) 2 = v( ) 2 ,a conserved quantity along the geodesic. We sub-

stitute s = v2

a 2 and = 2 a ta 2 to get


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a tt 4 a2t

a a6

a 2 v4

+ a6

a vv5 a

5a

2

v4 = 0 ,

a (0 , ) = v( ) ,

a nonlinear hyperbolic second order equation. Notethat wherever v = 0 then also a = 0 for all t. Sosubstitute a = vb. The outcome is

( b3

) tt = v2

2 (b3

) 2 vv( b3

) 3vv

2 b3

,b(0 , ) = 1 .

This is the codimension 1 version where

Burgers equation is the codimension 0 version.


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The general almost local metric G .

Gc (h, k ) := S 1 ( c, c( )) h ( ) , k ( ) ds.The metric G is invariant under the reparame-tization group Diff( S 1 ) and under the Euclideanmotion group.


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The horizontal geodesic equation.

Let c( , t ) be a horizontal geodesic for the metricG . Then ct ( , t ) = a( , t ) .n ( , t ). Denote the in-tegral of a function over the curve with respect toarclength by a bar. Then the geodesic equation forhorizontal geodesics is:

a t = 12 +

2 2 a 2

D 2s 2 a 2 + 2 2 aD 2s ( a )2 1 ( a ) a + ( 1 a 2 )


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Curvature on B i for G .

Let W ( 1 , 2 ) = h( 1 ) m ( 2 )

h ( 2 ) m ( 1 )

so that its second derivative 2 W ( 1 , 1 ) = W 2 ( 1 , 1 ) = h( 1 ) m ( 1 )h ( 1 ) m ( 1 )is the Wronskian of h and m .


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R 0 ( m,h,m,h ) = G0 ( R 0 ( m, h ) m, h ) =

= . 2 2 + 2 . 2 2( 2 )

2

( 2 )2

2( 1 ) W 2 ( 1 , 1 ) 2 d1

+ 22 ( 1 )2 W 22 ( 1 , 1 ) 2 d1+

1 2

1 2 1 2 ( 1 ) W 2 ( 1 , 1 ) W (

1 , 2 ) ( 2 ) d2 d1

+ 1 2 12 ( 1 ) W 22 ( 1 , 1 ) W ( 1 , 2 ) ( 2 ) d2 d1+

1 ( 1 )

21

2 .

(2 ) W 1 ( 1 , 2 ) 2 d2 d1

+ 2 . 3 2 .4 24 + 2 .2 + 28 . 1 ( 1 ) 1 ( 2 ) W ( 1 , 2 ) 2 d2 d1

+

112

21

4 ( 1 ) 1 ( 1 )

12 ( 2 )

( 2 ) ( 3 ) W ( 1 , 2 ) W ( 1 , 3 ) d2 d1 d3


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Special case: the metric GA .If we choose ( c, c) = 1 + A 2c then we obtainthe metric we have investigated before:

GAc (h, k ) = S 1 (1 + A c( ) 2 ) h ( ) , k ( ) ds.The horizontal geodesic equation for the GA-metricreduces to

a t = 11 + A 2c 12 ca

2

+ A a 2 (D 2s ( c) + 12 3c )

4 D s ( c) aD s ( a ) 2 cD s ( a ) 2


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Area swept out bound.If c is any path from C 0 to C 1 , then

area of the regionswept out by the

variation c max

t ( c( t, )) Lhor GA ( c) .Maximum distance bound.Consider < min { A/ 4 , 3 / 4 / 8} and let = 4( 3 / 4 A

1 / 4 + 1 / 4 ) . Then for any path c

starting at C 0 whose length Lhor GA is , the nal curve lies in the tubular neighborhood of C 0 of width .More precisely, if we choose the path c( t, ) to be

horizontal, thenmax |c(0 , ) c(1 , ) | < .


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Corollary.For any A > 0, the map from B i( S 1 , R 2 ) in the GA

metric to the space B conti (S 1 , R 2 ) in the Frechet

metric is continuous, and, in fact, uniformly contin-uous on every subset where the length is bounded.In particular, GA is a separating metric on B i( S 1 , R 2 ) .Moreover, the completion B i( S 1 , R 2 ) of B i( S 1 , R 2 )

in this metric can be identied with a subset of B lipi (S 1 , R 2 ) .


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Explicit equicontinuity bounds, under appropriateparameterization.

Corollary.If a path c( , t ) , 0 t 1 satises:

|c( , t ) | ( t ) / 2 for all , t , ct , c (0 , t ) 0 in a base point 0 for all t C t (1 + A 2C t ) | ct , ic |2 d/ |c| L2 for all t,then|c( 1 , t 1 ) c( 2 , t 2 ) |

max2 |1 2 |+

+ 7( 3 / 4max /A 1 / 4 + 1 / 4max ) L ( t1 t2 ) (1)

whenever |t1 t2 | min(2 Amin , 3 / 2min ) / (8 L ) .


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A numerical simulation of the geodesic connect-ing two circles. Minimize E horG1 (c) for variations cwith initial and end curves unit circles at distance3 produced the following image for the geodesic:

The geodesic joining 2 random shapes of sizeabout 1 at distance 5 apart with A = .25 (using20 time samples and a 48-gon approximation forall curves).


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The forward integration of the geodesic equationwhen A = 0, starting from a straight line in thedirection given by a smooth bump-like vector eld.Note that two corner like singularities with curva-ture going to are about to form.


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Top Row: Geodesics in 3 metrics joining the same two ellipses. Ellipseshave eccentricity 3, same center and are rotated at 60 degree.

A = 1; A = 0 .1; A = 0 .01.

Bottom Row: Geodesic triangles in Be formed by joining three ellipses atangles 0, 60 and 120 degrees, for the same three values of A. Here the

intermediate shapes are just rotated versions of the geodesic in the toprow but are laid out on a plane triangle for visualization purposes.


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The sectional curvature on B i

R 0 ( a,b,a,b ) = GA0 ( R 0 ( a, b ) a, b ) =

= S 1 12 ( A 2 1)( aba b) 2 + A( aba b) 2 d+ S 1

A 2

A2 4 + 2 A2

4 A2

2

1 + A 2 (aba b) 2 d= S 1 ( A 2 1) 2 + 4 A2 8 A2 22(1 + A 2 ) W ( a, b ) 2 d+ S 1 A W ( a, b ) 2 dwhere W ( a, b ) = aba b is the Wronskian of a and

b.


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Special case: the conformal metrics( ( c) , ( c)) = ( ( c)), metric proposed by Menucciand Yezzi and, for linear, independently by Shah:

Gc (h, k ) = ( c)

S 1

h, k ds = ( c) G0c ( h, k ) .

All these metrics are conformally equivalent to thebasic L2 -metric G0 .As they show, the inmum of path lengths in thismetric is positive so long as saties an inequality( ) C. for some C > 0.


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More precisely (Shah), if Area( c) is area swept overby the path c,

distG

( C 0 , C 1 ) = inf c

Area( c)

Ae. inf c Area( c) dist GeA( C 0 , C 1 ) Ae.e Amax inf c Area( c)


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The horizontal geodesic equation reduces to:

a t = 2

a 2 + 1

12 a2 .ds .a.ds a

If we change variables and writeb( s, t ) = ( ( t )) .a ( s, t ), then this equation simpli-es to:

bt = 2 b

2

1

b2

ds


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Curvature on B i for the conformal metrics.Sectional curvature has been computed by J. Shah.

Let g, h be orthonomal, then

Curv. in plane g.h=

2 ( g.D s ( h )

h.D s ( g)) 2 +

1

4 g2 . 2 + h2 . 2

+ 3 1 2 2 . 21

4 2 ( g. )2 + ( h. )

2

1

2 D s ( g) 2 + D s ( h ) 2 +

1

2 2 2

Note that the rst two lines are positive while thelast line is negative. The rst term is the curvatureterm for the H 0 -metric. The key point about thisformula is how many positive terms it has.


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Special case: the smooth scale invariant met-ric GSI

( , ) = 3 + A 2 gives the metric:

GSI c (h, k ) = S 1 13c + A 2c

ch, k ds.

The beauty of this metric is that (a) it is scaleinvariant and (b) log( ) is Lipschitz, hence the in-mum of path lengths is always positive.


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Horizontal geodesics in this metric as special case

of the equation for G :

a t = 1

1 + A( ) 2 1 + A( )2 a 2

2

2 A2

D s ( a )2

4 A2

D s ( ) aD s ( a )+ 3 + A( ) 2 ( a ) a

32

( a 2 )

A2

2 (a ) 2

where the overline stands now for the average of a function over the curve, i.e.

ds/ .


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Immersion-Sobolev metrics on Imm( S 1 , R 2 ) and

on B iNote that Ds = |c| is anti self-adjoint for the metric

G0 , i.e., for all h, k C ( S 1 .R 2 ) we have

S 1 D s ( h ) , k ds = S 1 h, D s ( k) dsThe metric:G imm ,nc (h, k ) =

S 1

( h, k + A. D ns h, Dns k ) .ds

= S 1 Ln ( h ) , k ds whereLn ( h ) or Ln,c ( h ) = I + ( 1) n A.D 2 ns (h )


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Geodesics in the H imm ,n -metric

( Ln ( ct )) t = Ln ( ct ) , D s ( ct ) v

|ct

|2 ( c)2 n D s ( ct ) , v Ln ct

+ A2

.2 n1 j =1

(1) n + j D 2 n js ct , D js ct ( c) n


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ct = u =: X 1 ( c, u )

u t = L1n,c Ln,c ( u ) , D s ( u ) D s ( c)

|ct

|2 ( c)2 JD s ( c) D s ( u ) , D sc u

+ A2

.2 n1 j =1

(1) n + j D 2 n js u, D js u ( c) JD s ( c)

+ ( 1) n A.2 n1 j =1

D js D s ( u ) , D s ( c) D2 n js (u )

=: X 2 ( c, u )


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The conserved momenta of Gimm ,n along anygeodesic t c( t, ):

c, L n,c ( ct )

|c( )

| X( S 1 ) repar. moment.

S 1 Ln,c ( ct ) ds R 2 linear moment. S 1 Jc,L n,c ( ct ) ds R angular moment.


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Horizontality for Gimm ,n h T c Imm( S 1 , R 2 ) isG imm ,nc -orthogonal to the Diff( S 1 )-orbit through cif and only if

0 = Gimm ,nc (h, X ( c)) = S 1 X. Ln,c ( h ) , c dsfor all X X( S 1 ). So the Gimm ,n -normal bundle isgiven by

N nc = {h C ( S, R 2 ) : Ln,c ( h ) , v = 0 }.The Gn -orthonormal projection T c Imm N nc , de-noted by h h = h,G

nand the complemen-

tary projection h h T c( c Diff( S 1 )) are 1-dimensional pseudo-differential operators.

They are determined as follows:


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They are determined as follows:

h = X ( h ) .v where Ln,c ( h ) , v = Ln,c ( X ( h ) .v) , vThus we are led to consider the linear differentialoperators associated to Ln.c

Lc , Lc : C ( S 1 ) C ( S 1 ) ,Lc ( f ) = Ln,c ( f.v ) , v = Ln,c ( f.n ) , n ,Lc ( f ) = Ln,c ( f.v ) , n = Ln,c ( f.n ) , v .

The operator Lc is of order 2 n and also unbounded,self-adjoint and positive on L2 ( S 1 ,

|c

|d). In par-

ticular, Lc is injective. Lc , on the other hand is of order 2 n 1 and is skew-adjoint. For example, if n = 1, then one nds that:Lc = A.D

2s + (1 + A.

2

) .I Lc = 2 A..D s A.D s ( ) .I


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The operator Lc : C ( S 1 ) C ( S 1 ) is invertible.This is by deformation invariance of the index.

We want to go back and forth between the naturalhorizontal space of vector elds a.n and the Gimm ,n -horizontal vector elds {h | Lh,v = 0 }: We useC c : C ( S 1 , R 2 ) C ( S 1 ) given by

C c( h ) := ( Lc )1

Lc ,

a pseudo-differential operator of order -1 so that

a.n + C ( a ) .v is H imm ,n -horizontal


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The restriction of the metric Gimm ,n to horizontalvector elds hi = ai .n + bi .v can be computed likethis:

G imm ,nc

(h1 , h 2 ) =

S 1

Lh 1 , h 2 .ds

= S 1 L + LC a 1 .a 2 .ds.Thus the metric restricted to horizontal vector elds

is given by the pseudo differential operator Lred =L + L ( L)1 L.


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The metric on the cotangent space to B i , is simple.On the smooth cotangent spaceC ( S 1 , R 2 ) = G0c ( T c Imm( S 1 , R 2 )) D( S 1 ) 2the dual metric is given by convolution with theelementary kernel K n .

Gnc ( a 1 , a 2 ) = S 1S 1 K n ( s1 s2 ) .. n c( s1 ) , n c( s2 ) .a 1 ( s1 ) .a 2 ( s2 ) .ds 1 ds 2 .


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Horizontal geodesicsFor any smooth path c in Imm( S 1 , R 2 ) there exists a smooth path in Diff( S 1 ) with ( t, ) = Id S 1depending smoothly on c such that the path e given

by e( t, ) = c( t, ( t, )) is horizontal: Ln,c ( et ) , e =0 .

We may specialize the general geodesic equation to

horizontal paths and then take the v and n partsof the geodesic equation. For a horizontal path wemay write Ln,c ( ct ) = an for a ( t, ) = Ln,c ( ct ) , n .The v part of the equation turns out to vanish


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identically and then n part gives us

a t = |ct |2 ( c)

2 D sct , v a++

( c)

2

2 n1 j =1

(

1) n + j D 2 n js ct , D js ct

A Lipschitz bound for arclength in Gimm ,n

| ( C 1 ) ( C 0 ) | C ( A, n )2 dist B iGn ( C 1 , C 0 )


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The scale invariant Sobolov H 1 -metric and itsrelation to the Grassmannian of 2-planes in an in-nite dimensional space, and Neretin geodesics.

Gc( h, k ) = limA

1

AG imm ,scal ,1

c (h, k )

= 1( c) S 1 D sh, D sk ds

= 1

( c) S 1

h,

D 2

sk ds

on Imm / translations or {c Imm : c(1) = 0 }.

Geodesics in this metric


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ctt = 12 D 2s cn c ct 2Gc 12 D 1s |D sct |2 vc 1c c ct , n c ds ct D 1s D sct , vc D sct

The conserved momenta of Gimm ,n along anygeodesic t c( t, ):1( c)

c, D2s ( ct ) |c( ) | X( S 1 ) repar. moment.

1( c) S 1 D 2s ( ct ) ds = 0 R 2 linear moment.1( c) S 1 ic,D 2s ( ct ) ds R angular moment.1( c) S 1 c, D 2s ( ct ) ds = t log( ( t )) scaling moment.


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Thm. For each k 3/ 2 this geodesic equationhas unique local solutions in the Sobolev space of H k-immersions. The solutions depend C on t and on the initial conditions c(0 , . ) and ct (0 , . ) . The domain of existence (in t)is uniform in k and thus this also holds in Imm

:=

{c

Imm( S 1 , R 2 ) : c(1) = 0

}.

Sphere Stiefel and Grassmannian


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Sphere, Stiefel, and GrassmannianV := {f C ( R , R ) : f ( x + 2 ) = f ( x)}below only : odd case. +: even case.f 2 = 2 0 f 2 dx weak inner product on V .Gr (2 , V ) Grassmannian of oriented 2-planes.T W Gr = L( W, W ) with metric

v 2 = tr( vv) = v( e) 2 + v( f ) 2 ,e, f orthonormal basis of W .

For W Gr (2 , V ) letZ ( V ) = {x : f ( x ) = 0 f W }.Gr 0 (2 , V ) =

{W

Gr (2 , V ) : Z ( W ) =

} open in

Gr (2 , V ).


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The Stiefel manifoldSt

(2 , V ) of orthonormal pairsin V .St 0 (2 , V ) = {( e, f ) St : Z ( e, f ) = } open in St .T ( e,f ) St = {( e,f ) V 2 : 0 = e,de = f ,f =e,f + f ,e }Metric ( e,f ) 2 = e 2 + f 2 .

St (2 , V )

S ( V 2

open) sphere of radius 2.

V open = C ([0 , 2 ], R ).


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The basic bijection

( e, f ) = c( ) = 12 0 ( e + if ) 2 dx

: S 0 2fold Imm opentransl.,scalings

: St 0 2fold

Imm oddtransl.,scalings

: Gr0

Imm oddtransl.,rot.,scalings

: Gr 0 /U ( V ) B i,oddtransl.,rot.,scalings


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Reparameterizations


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Let U ( V ) be the group of all unitary operators on V of the form

f (f

) for all smooth

: R

R

with ( x) > 0 and ( x + 2 ) = ( x) + 2 , i.e. liftsof Diff + ( S 1 ).The innitesimal action on V of a periodic vector

eld X on R is f 12 X .f + X.f .Prop. ( e, f ) = ( e ) , ( f ) .A tangent vector (

e,f )

T ( e,f )St is

perpendicular to the rotation orbits iff e, df V = f,de V = 0 .

It is perpendicular to the reparameterization orbit

iff W ( e,e ) + W ( f ,f ) = 0where W ( a, b ) = a.b a .b is the Wronskian.


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Neretin geodesics on Gr (2 , V )Y.A.Neretin: On Jordan angles and the triangle inequality in Grassmannmanifolds, Geom. Dedicata 86 (2001)

If W 0 , W 1 Gr (2 , V ), use the singular value decom-position of the orthonormal projection

p :

W 0 W 1 . This gives ONB ( e0 , f 0 ) of W 0 and ( e1 , f 1 ) of W 1 such that p( e0 ) = cos( ) e1 , p( f 0 ) = cos( ) f 1 ,e0f

1 and f 0e1 for

0 , / 2 the Jordan angles .The metric is then given bydist( W 0 , W 1 ) =

2 + 2


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We apply this to compute the distance betweencurves in Imm od / (sim) and B i, od / (sim). We write

c0 = r0 ( ) ei0 ( ) and c1 = r1 ( ) ei

1 ( ) . We put

e0 = 2 r 0 cos 02 f 0 = 2 r 0 sin 02 ,e

1= 2 r 1 cos

12 f

1= 2 r 1 sin

12 ,

lifting the curves to 2-planes in the Grassmannian.The 2 2 matrix M ( c0 , c1 ) of the orthogonal projection from the space

{e0 , f 0

} to

{e1 , f 1

} in these

bases is:

S 1 2 r 0 .r 1 . cos 02 cos 12 d S 1 2 r 0 .r 1 . cos 02 sin 12 d

S 1 2

r 0 .r 1 . sin 02 cos 12 d

S 1 2

r 0 .r 1 . sin 02 sin 12 d

Notations:


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C :=

S 1

r 0 .r 1 cos

0 12 d= 12 M ( c0 , c1 ) 11 )M ( c

0 , c1 ) 22

S := S 1 r 0 .r 1 sin 0 12 d=

12 M ( c

0, c

1) 21 M ( c

0, c

1) 12

We have to diagonalize this matrix by rotating thecurve c0 by a constant angle 0 , i.e., the basis

{e0 , f 0} by the angle 0 / 2; and similarly c1 by aconstant angle 1 . So replace 0 by 0 0 and 1 by 1 1 such that (for both signs)

0 =

S 1

r 0 .r 1 sin

( 0 0 ) ( 1 1 )2

d

= S . cos 0 12 C . sin

0 12


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Thus

0 1 = 2 arctan ( S /C ) .In the newly aligned bases, the diagonal elements of the matrix will be the cosines of the Jordan angles.

The following lemma gives you a formula for them:

If M = a bc d , C = 12 ( ad) , S =

12 ( c b) , then

the singular values of M are:

C 2 + S 2 C 2+ + S 2+ .


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This gives the formula

D od ,rot ( c0 , c1 ) 2 =

= arccos 2

S 2+ + C

2+ +

S 2

+ C 2

+ arccos 2 S 2 + C 2 S 2+ + C 2+ .This is the distance in the space

Imm od ( S 1 , C )/(transl, rot., scalings).


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Horizontal Neretin distances.If we want the distance in the quotient spaceB i, od / (transl, rot., scalings) by the groupDiff( S 1 ) we have to take the inmum of this dis-

tance over all reparametrizations.To simplify, we assume that the initial curves c0 , c1

are parametrized by arc length so that r 0 r 1 1 / 2 .


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To describe the inf, we can use that geodesics in


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B i are horizontal geodesics in Imm.Consider the Neretin geodesic t

{e( t ) , f ( t )

} in

Gr (2 , V ) described above

W ( t ) =

e( t ) = sin((1 t ))

sin( ) .e0 +

sin( t )sin( )

.e1

f ( t ) = sin((1 t ) )sin( ) .f 0 + sin( t )

sin( ) .f 1

for

e0 =

cos ( 0) 02 e1 = 1 cos 1 12 ,

f 0 = sin ( 0) 02 f 1 = 1 sin 1 12 ,where the rotations 0 and 1 must be computedfrom c0 and c1 .


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This is an ordinary differential equation for which

is coupled to the (integral) equations for calculat-ing the s as functions of . If it is non-singular(i.e., the coefficient function of does not van-ish for any ) then there is a solution , at leastlocally. But the non-existence of the inf describedfor open curves above will also affect closed curvesand global solutions may actually not exist. How-

ever, for closed curves that do not double back onthemselves too much geodesics do seem to usuallyexist.


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The generic way in which a family of open immersions crosses

the hypersurface where Z = . The parametrized straight linein the middle of the family has velocity with a double zero at

the black dot, hence is not an immersion.


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100fold blowup in middle

This is a geodesic of open curves running from the curve with the kinkat the top left to the straight line on the bottom right. A blow up of thenext to last curve is shown to reveal that the kink never goes away itmerely shrinks. Thus this geodesic is not continuous in the C 1 -topologyon Bopen . The straight line is parametrized so that it stops for a wholeinterval of time when it hits the middle point and thus it is C 1 -continuous

in Imm open .


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A great circle geodesic on Bod . The geodesic begins at the circle at thetop left, runs from left to right, then to the second row and nally thethird. It leaves Bod twice: at the top right and bottom left, in both of which the singularity of the rst gure occurs in 2 places. The index of the curve changes from +1 to 3 in the middle row.


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Curvature. Let W Gr (2 , V ) be a xed 2-plane.Let : V V be the isomorphism which equals1 on W and 1 on W satisfying = 1 . ThenGr is the symmetric space O( V ) / ( O ( W ) O ( W ))with involutive automorphism : O( V )

O( V )

given by ( U ) = .U. For the Lie algebra in theV = W W -decomposition we have

1 00 1

x yT y U

1 00 1

= x yT

y U Here x L( W, W ) , y L( W, W ). The xed pointgroup is O( V ) = O( W ) O ( W ).

The reductive decomposition g = k + p is given by


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x yT y U

= x 0

0 U , x

so(2)

+ 0 yT y 0 , y L( W, W )For the sectional curvature we have (where we as-sume that Y 1 , Y 2 is orthonormal):

kspan( Y 1 ,Y 2 ) = B ( Y 2 , [[Y 1 , Y 2 ], Y 1 ])= tr W ( y

T 2 y2 y

T 1 y1 + y

T 2 y1 y

T 1 y2 2 y

T 2 y1 y

T 2 y1 )= 12 y

T 2 y1 yT 1 y2 2L2 ( W,W )

+ 12 y2 yT 1 y1 yT 2 2L2 ( W ,W ) 0 .

where L2 stands for the space of Hilbert-Schmidtoperators. Note that there are many orthonormal


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pairs Y 1 , Y 2 on which sectional curvature vanishesand that its maximum value 2 is attained when yi

are isometries and y2 = Jy 1 where J is rotationthrough angle / 2 in the image plane of y1 .


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We have explicit formulas for the ONeill term and

thus for the sectional curvature kB i/(sim)span( h1 ,h 2 ) at a

curve C B i/(sim) and tangent vector hi . We alsohave an explicit upper bound for this as a functionof h1 . This shows that geodesics have at least asmall interval before they meet another geodesic.The size of this interval can be controlled by anupper bound that involves the supremum norm of

the rst two derivatives of h1 .

See the paper for this.


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Curve evolution with and without the closednessconstraint. Lower and upper bounds for the geodesic

distance: 0.433 and 0.439


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Curve evolution with and without the closednessconstraint. Lower and upper bounds for the geodesic

distance: 0.498 and 0.532


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Curve evolution with and without the closednessconstraint. Lower and upper bounds for the geodesicdistance: 0.513 and 0.528


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The geodesic equation on Diff( R 2 ) is V.Arnoldsequation EPDiff:

t ( t, ) Diff( R 2 )v( t ) = ( t) 1 X( R 2 ) , u ( t ) = L( v( t )) ,u it

+ j

v j .u ix j

+ u j .v j

x i+ div v.u i = 0 .

The quotient Emb( S 1 , R 2 ) .ff( 2 ) b( 1 2 )


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Diff( R 2 ) Emb( S 1 , R 2 ) i , where i : S

1

R

2.

If c = i , the ber through is.{ : i = i} = { : c = c}..The tangent space to the ber is (right translatedby )

{X X ( R 2 ) : X c = 0 }.The horizontal subspace is the translate by of

{Y : R 2 LY,X dx = 0 , if X c = 0 }.If Y is C then Y = 0. So we need

LY = c( p( ) .ds ) for p C ( S 1 , R 2 ), a distribution

carried by c. Thus

Y ( x) = S 1 F ( x c( )) p( ) ds

Y ( x) = S 1 F ( x c( )) p( ) ds


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Mapped to T c Emb we get( Y c)( ) = S 1 F ( c( ) c( 1 )) .p( 1 ) .|c( 1 ) |d1=: ( F c p)( ) whereF c( 1 , 2 ) := F ( c( 1 )

c( 2 ))

is an elliptic pseudo differential operator kernel of order 2 n + 1 which is real and positive, so theoperator p

F c

p is self-adjoint and positive, so

injective, and by index deformation it is bijectivebetween the Sobolev spaces on S 1 . The inverseoperator ( F c )

1 has kernel Lc( , 1 ) which is a

pseudo differential operator kernel of order 2 n 1.

Write h = Y c T c Emb and express the horizontal


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lift Y = Y h in terms of h:

h = Y h c = F ( c( p.ds )) = F c p so p = Lch

Y = Y h = F ( c(( Lch ) .ds ))Y h ( x) = S 1 F ( x c( )) .. S 1 Lc( , 1 ) h ( 1 ) |c( 1 ) |d1 |c( ) |d

Finally the metric:Gdiff ,nc (h, k ) = R 2 LY h , Y k dx=

S 1

S 1

Lc( , 1 ) h ( 1 ) , k ( ) ds 1 ds.

We can now compute K and H and the geodesicequation It becomes simpler if written for the 1


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equation. It becomes simpler if written for the 1-

current Lc

ct = p.|c| =: q:qt ( 0 ) = S 1 F c( 0 , 1 ) q( 0 ) , q( 1 ) d1

where F c( 1 , 2 ) = grad F ( c( 1 ) c( 2 )).Existence of geodesics. Theorem.Let n

1. For each k > 2n

12 the geodesic equa-

tion has unique local solutions in the Sobolev space of H k-embeddings. The solutions are C in t and in the initial conditions c(0 , . ) and ct (0 , . ) . The domain of existence (in t) is uniform in k and thus this also holds in Emb( S 1 , R 2 ) .


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Geometry of landmark spaceand of spaces of currents

The diffeomorphism groupDiff = Diff S( R

n ): the regular Lie group of all dif-


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S ( ) g g pfeomorphisms which are rapidly falling towards theidentity.Its Lie algebra is the space XS ( R

n ) of all smoothvector elds which decrease rapidly, with the neg-ative of the usual bracket as Lie bracket.

We consider XS ( R n ) as pre Hilbert space H L withinner productX, Y H L = R n LX,Y dx

where L : XS ( R n ) XS ( R n ) is an invertible lin-ear (elliptic) scalar differential operator or pseudo-differential operator which is self-adjoint with re-spect to the weak inner product

G0 ( X, Y ) = R n X, Y dx


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on XS ( Rn ) and which is applied to each component

of a vector eld separately.

For example:For the Laplacian = 2i and constant A, let


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p

i A,

L = (1 A) l =| |l

(A) | |l! ! ( l )! 2

= 1 + + nl

(A) 1 + + n l! 1 ! . . . n ! ( l 1 )! . . . ( l n )!

2 1x1 . . . 2 nxn

The Fourier transform is Lu = (1 + A||2 ) lu ( ).Thus the fundamental solution K of LK = in thespace S ( R n ) of tempered distributions is

K ( x) = 1

(2 ) n R n ei x, 1(1 + ||2 ) l dn which can be expressed in terms of the classicalmodied Bessel functions K l1 ( |x|/ A). It satises


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( L1 u )( x) = R n K ( xy) u ( y) dn y for each tempereddistribution u.


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Or:We consider a kernel function K : R n R n R withgood properties (for example smooth and rapidlydecreasing off the diagonal) and its associated op-erator K ( f )( x) = R n K ( x, y ) f ( y) dy which we as-sume to be invertible on C S (R n ) on the space of of smooth functions with compact support, andthen we put L = K

1.


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T 0 ev q0 .( Y 0 ) = Y ( qk) k = i( K ( qk qi) .P i) k

C id P ( P ) T L dN


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Consider a tangent vector P = ( P k) T q Land N .Its horizontal lift with footpoint 0 is P hor 0where the vector eld P hor on R n is given as follows:Let K 1 ( q) ki be the inverse of the ( N N )-matrixK ( q) ij = K ( qi

q j ). Then

P hor ( x) =i,j

K ( x qi) K 1 ( q) ij P jL ( P hor ( x)) =

i,j( x

qi) K 1 ( q) ij P j

Note that P hor is a vector eld of class H 2 l1 .

The Riemannian metric on Land N induced by thegL metric on Diff ( R n ) is


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g -metric on Diff S ( R ) isgLq ( P, Q ) = G

L0

( P hor , Q hor ) = R n L ( P hor ) , Q hor dx

= R n i,j ( x qi) K 1 ( q) ij P j , k,l K ( x qk) K 1 ( q) kl Q l dx=

i,j,k,lK 1 ( q) ij K ( qi qk) K 1 ( q) kl P j , Q l

So the metric is given by:

gLq ( P, Q ) = k,l K

1( q) kl P k , Q l .

Recall: K 1 ( q) ki is the inverse of the ( N N )-matrix K ( q)

ij = K ( q

i q j

).

Lemma Let X, Y XS ( R n ) be a vector elds withsupport in a compact box B

R n . Let q1 , q2 , q3 , . . .


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pp p q1 , q2 , q3 ,be an equidistributed sequence in B : For eachBorel subset U B we require

limN

# {i N : qi U }N

= Vol( U )Vol( B )

.

For each N consider the initial part qN = ( q1 , . . . , q N )as a point in the landmark space Land N of N points in R n . Then we have

limN

Vol( B ) 2

N 2N

i,j =1K 1 ( q) i,j X ( qi) , Y ( q j ) =

=

R n

LX,Y dx.

The geodesic equation on T Land N ( R n ) .Elements of the cotangent bundle

N N N


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T Land N ( R n ) = Land N ( R n )

(( R n ) N )

are denoted by

( q, ) = ( q1 , . . . , q N ) , 1...

N

=q11 . . . q

1N

. . .qn1 . . . q

nN

, 11 . . .

1n

. . . N 1 . . .

N n

and we shall use this as global coordinates.The metric looks like

( gL )1q (, ) =i,j

K ( q) ij i , j ,

K ( q) ij = K ( qi q j ) .

We consider the the energy function


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gy

E ( q, ) = 12 ( gL )1q (, ) = 12 i,j K ( q) ij i , j

= 12i,j

K ( q) ij i , j

and its Hamiltonian vector eld (using R n -valuedderivatives to save notation)

H E ( q, ) = 1

2

N

i,j,k =1

K ( q) ij i , j

k

qk

K ( q) ij i , j

qk

k .

=N

i,k =1

K ( qk qi) i

qk+ grad K ( qi qk) i , k

k

.

So the geodesic equation is the ow of this vector

eld:


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qk = i K ( qi qk) i

k =i

k , i grad K ( qi qk)

A covariant formula for curvature and its rela-tions to ONeills curvature formulas.Mario Micheli in his 2008 thesis derived the thecoordinate version of the following formula for thesectional curvature expression, which is valid forclosed 1-forms , on a Riemannian manifold ( M, g ),

where we view g : T M T M and so g1

is the


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dual inner product on T M . Here

= g1

( ).gL R ( , ) , = 14 d( g1 ( , )) 2

+ 14 g1 d( 2 ) , d( 2 )+ 34 g [ , ], [ , ]

12 ( 2 ) 12 ( 2 )+ 12 (

+ ) g1 ( , )


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Notation for the coordinate formula:A = indices of landmark points in R n


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A indices of landmark points in

a,b,c, = elements of A, = { a |a A}, { a |a A}, cotangent vectors to L , = the dual tangent vectors, e.g.

a =

b

K ( P a P b) bK ( x) = k( x ) , the kernel dening the metric

note: K ( x) = k( x ) xx

dab =

P a

P b , uab = ( P a

P b) /d ab ,

the unit vector between landmarksK ab = k( dab) ,K ab = DK ( P a P b) = k( dab) uabK ab =

k( dab)k2 ( dab)

1dabk( dab)

Four expressions in the skew form :


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p

ab,cd ( , ) = a ,K cd b a ,K cd bbcd( , ) =a

( K ac K ad ) ab,cd ( , )= c d,K cd b c d,K cd b

(Note that the terms in angle brackets are discrete strains) ab,cd ( , ) = ( a c) ( a c) , ( b d) ( c d) ,

(Bracket in R nR n , points a, b on left, c, d on right)

bd( , ) = (

b d)( b d) ( b d)( b d) , b d b d


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With these notations, we get the following formula:

R ( ,,, ) = 12bd

K bd bbd( , ) , dbd( , )

+ 12bcd

( bcb( , ) bcd( , ) ) , cd,bd( , )

34b

bb( , )2K 1 +

12

cd

k bddab cd( , )

18abcd

( K ab K ad K cb + K cd) K ac ,K bd ab,cd ( , ) .


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The geodesic equation on Diff( R 2 ) is V.Arnoldsequation EPDiff:

t ( t, ) Diff( R 2 )v( t ) = ( t) 1 X( R 2 ) , u ( t ) = L( v( t )) ,u it

+ j

v j .u ix j

+ u j .v j

x i+ div v.u i = 0 .

The quotient Emb( S 1 , R 2 ) .Diff( R 2 ) Emb( S 1 , R 2 )


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i , where i : S

1

R 2.If c = i , the ber through is

.{ : i = i} = { : c = c}..The tangent space to the ber is (right translated

by )

{X X ( R 2 ) : X c = 0 }.The horizontal subspace is the translate by of

{Y :

R

2 LY,X dx = 0 , if X

c = 0

}.

If Y is C then Y = 0. So we needLY = c

( p( ) .ds ) for p C ( S 1 , R 2 ), a distribution

carried by c. Thus

Y ( x) = S 1 K ( x c( )) p( ) ds


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Write h = Y c T c Emb( S 1 , R 2 ) and express thehorizontal lift Y = Y h in terms of h:h Y K ( ( d )) K L h


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h = Y h

c = K

( c

( p.ds )) = K c

p so p = Lc

h

Y = Y h = K ( c(( Lch ) .ds ))Y h ( x ) =

= S 1 K ( x c( )) . S

1 Lc( , 1 ) h ( 1 ) |c( 1 ) |d1 |c( ) |dFinally the metric:Gdiff ,nc (h, k ) =

R 2

LY h , Y k dx

= S 1S 1 Lc( , 1 ) h ( 1 ) , k ( ) ds 1 ds.This formula looks innocent, but there is an in-

version of the (nice) operator K c

in it to getLc = ( K c )1

We can now compute K and H and the geodesicequation. It becomes simpler if written for the 1-


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current Lc

ct = p.

|c

| =: :

t ( 0 ) = S 1 K c( 0 , 1 ) ( 0 ) , ( 1 ) d1where K

c(

1,

2) = grad K ( c(

1)

c(

2)).

Existence of geodesics. Theorem.Let n 1. For each k > 2n 12 the geodesic equa-tion has unique local solutions in the Sobolev space of H k-embeddings. The solutions are C in t and in the initial conditions c(0 , . ) and ct (0 , . ) . The domain of existence (in t) is uniform in k and thus this also holds in Emb( S 1 , R 2 ) .


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Horizontal geodesics


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Horizontal geodesics.A eld h along c is horizontal if Lc h, c = 0.For a horizontal path we have , c = 0, so let = a.n . Then the horizontal geodesic equation is

a t ( ) = t , n ( ) == S 1 K c( , 1 ) , n ( ) a ( ) a ( 1 ) n ( ) , n ( 1 ) d1

Note that also n = Jc/

|c

| appears. It is a strange

equation, but it is well-posed by the theorem above.

Requirements for innite dimensional manifoldsL t ( M ) b k Ri i if ld d


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Let ( M, g ) be a weak Riemannian manifold mod-elled on convenient locally convex vector spaces.For x M the metric gx : T xM T x M is usuallyonly injective (weak metric). The image g( T M ) T M is called the smooth cotangent bundle asso-ciated to g. Now 1g ( M ) := ( g( T M )) and =g1 X( M ) , X = gX are as above. The exteriorderivative restricts tod : 1g ( M ) 2 ( M ) = ( L2skew ( T M ; R ))since the embedding g( T M ) T M is a smoothber linear mapping.


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Some constructions above encountered a secondproblem: they lead to vector elds whose values do


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not lie in T xM , but in the Hilbert space completionT xM with respect to gx . To manipulate theseas in the nite dimensional case, we need to knowthat

x

M T xM forms a smooth vector bundle over

M . In other words, in each coordinate chart on anopen subset U M , T M |U is a trivial bundle U V and all the inner products gx , x U dene innerproducts on one and the same topological vectorspace V . We assume that they are all boundedwith respect to each other, so that the completionV of V with respect to gx does not depend on x

and xU T xM = U V .

This means that xM T xM forms a smooth vectorbundle over M with trivialisations the linear exten-


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sions of the trivialisations of the bundle T M M .These two properties will be sufficient for all theconstructions we need so we make them into a def-inition:

Denition. A convenient weak Riemannian mani-fold ( M, g ) will be called a robust Riemannian man-

ifold if (1) The metric gx admits gradients in the abovetwo senses,(2) The completions T xM form a vector bundle as

above.

Covariant curvature and ONeills formula ininnite dimensions. Let p : ( E, g E ) (B, g B ) be a


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p ( , g E ) ( , g B )Riemann submersion between innite dimensionalrobust Riemann manifolds; i.e., for each b Band x E b := p1 ( b) the tangent mapping T x p :( T xE, g E )

(T

bB, g B ) is a surjective metric quo-

tient map so that

b gB := inf X x T xE : T x p.X x = b .The innimum need not be attained in T xE butwill be in the completion T xE . The orthogonalsubspace {Y x : gE ( Y x , T x ( E b) ) = 0 } will thereforebe taken in T x ( E b) in T xE .

If b = gB (

b, ) gB ( T bB ) T b B is an elementi th th d l


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in the gB

-smooth dual,then p b := ( T x p)( b) = gB ( b, T x p ) : T xE R is in T x M but in general it is not an elementin the smooth dual gE ( T xE ). It is, however, an

element of the Hilbert space completion gE ( T xE )of the gE -smooth dual gE ( T xE ) with respect to thenorm g1E , and the elementg

1E ( p b) = : ( p b)

is in the gE -completionT xE of T xE . We can call g1E ( p b) =: ( p b) thehorizontal lift of b = g1B ( b) T bB .


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The metric ( gE ) x can be evaluated at elements inthe completion T xE . Moreover, for any smoothsections X, Y ( T E ) the mapping

gE ( X, Y ) : M Ris still smooth, by the smooth uniform boundednesstheorem.

Lemma. If is a smooth 1-form on an open subset U of B with values in the gB -smooth dual gB ( T B ) ,then p is a smooth 1-form on p1 ( U ) E with


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then p is a smooth 1-form on p ( U )

E with

values in the g1E -completion of the gE -smoothdual gE ( T E ) . Thus also ( p ) is smooth from E into the gE -completion of T E , and it has values in

the gE -orthogonal subbundle to the vertical bundle in the gE -completion. We may continuously ex-tend T x p to the gE -completion, and then we have T p

( p ) =

p. Moreover, the Lie bracket

of two such forms, [( p ) , ( p ) ], is dened. The exterior derivative d( p ) is dened and is applica-ble to vector elds with values in the completion

like ( p )

.That the Lie bracket is dened, is also a non-trivial


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statement: We have to differentiate in directionswhich are not tangent to the manifold.

Theorem Let p : ( E g E ) (B g B ) be a Riemann


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Theorem. Let p : ( E, g E )

(B, g B ) be a Riemann

submersion between innite dimensional robust Riemann manifolds. Then for 1-forms , 1g ( B ) ONeills formula holds in the form:

gB R B ( , ) , =

= gE R E (( p ) , ( p ) )( p ) , ( p )

+ 34

[( p

) , ( p

) ]ver 2gE


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4 Gdiff ,n R ( , ) , =

d ( 1 ) , ( 2 ) ( 1 ) , ( 2 )


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=

( S 1 ) 4

det ( 1 ) , ( 2 )

( 1 ) , ( 2 )

( 3 ) , ( 4 ) ( 3 ) , ( 4 )grad K ( c( 1 ) c( 2 )) , grad K ( c( 3 ) c( 4 )) K c( 1 , 3 ) 2 K c( 1 , 4 ) + K c( 2 , 4 )

+ 3

(S 1 ) 2 L c( 3 , 4 )

S 1 grad K ( c( 3 ) c( 1 )) , ( 3 ) ( 1 ) ( 1 ) grad K ( c( 3 ) c( 1 )) , ( 3 ) ( 1 ) ( 1 ) ,

S 1 grad K ( c( 4 ) c( 2 )) , ( 4 ) ( 2 ) ( 2 )

grad K ( c( 3 ) c( 1 )) , ( 3 ) ( 1 ) ( 1 )+

( S 1 ) 2

2 ( 1 ) , ( 2 ) d2 K ( c( 1 )

c( 2 ))

( 1 ) ( 2 ) , ( 1 ) ( 2 )


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2 ( 1 ) , ( 2 ) d2 K ( c( 1 ) c( 2 )) ( 1 ) ( 2 ) , ( 1 ) ( 2+ 4 ( 1 ) , ( 2 ) d2 K ( c( 1 ) c( 2 )) ( 1 ) ( 2 ) , ( 1 ) (

High dimensional shape spaceImm( M, N ) / Diff( M ) .


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M , a compact smooth connected manifold of di-mension m 1.( N, g ) a connected Riemannian manifold of dimen-sion n > m .

Diff( M ), the regular Lie group of all diffeomor-phisms of M .

Diff x0 ( M ), the subgroup of diffeomorphisms xingx0 M .

Emb = Emb( M, N ), the manifold of all smoothembeddings M N .


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Imm free = Imm free ( M, N ), the manifold of all smoothfree immersions M N (those with trivialisotropy group for the right action of Diff( M )


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isotropy group for the right action of Diff( M )on Imm( M, N )).

B e = Be( M, N ) = Emb( M, N ) / Diff( M ), the mani-

fold of submanifolds of type M in N , base of a smooth principal bundle.

Bi = B

i( M, N ) = Imm( M, N ) / Diff( M ), an innite

dimensional orbifold.

B i,f = B i, ( M, N ) = Imm f ( M, R 2 ) / Diff( M ), a man-

ifold, the base of a principal ber bundle.

Free immersionsAn immersion f : M N is called free if Diff( M )acts freely on it, i.e., f = f for Diff( M )


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y , . ., f

f

( M )

implies = Id. We have the following results:

If Diff( M ) has a xed point and if f = f for any immersion f then = Id .

If for f Imm( M, N ) there is a point x c( M )with only one preimage then f is a free immersion.There exist free immersions without such points.

We might view Imm f ( M, N ) as the nonlinear Stiefelmanifold of parametrized submanifolds of type M in N and consequently B i,f ( M, N ) as the nonlinear

Grassmannian of unparametrized submanifolds of type M .

Non free immersions. Since M is compact, theorbit space B i( M, N ) = Imm( M, N ) / Diff( M ) is Haus-dorff. For any immersion f the isotropy group


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Diff( M ) f is a nite group which acts as groupof covering transformations for a nite coveringqf : M M such that f factors over qf to a freeimmersion f : M

N with f

qf = f .

For each f Imm there exist a slice Q( f ) i n astrong sense: Q( f ) is invariant under the isotropy group Diff( M ) f .

If (Q ( f )

)

Q( f )

=

for

Diff( M ) then is

in the isotropy group Diff( M ) f . Q( f ) Diff( M ) is an invariant open neighbour-hood of the orbit f Diff( M ) in Imm( M, N ) ad-mitting a smooth retraction r onto the orbit. Theber r1 ( f ) equals Q( f ).


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Volumes of an immersion. For an immersionf Imm( M, N ), we consider the volume density


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vol g( f ) = vol( f g) Vol( M ) on M given byvol g( f ) |U = det(( f g) ij ) |du 1 du m |for any chart ( U, u : U R m ) of M .Lemma. The derivative of vol g : Imm( M, N ) Vol( M ) is

d vol g( f )( h ) =

Tr f g( g( S f , h

)) vol( f

g)++ div f g( h)( f g)) vol( f g) .

The second summand vanishes when integrated over M .

The metric on Imm . Let h, k C f (M,TN ) betangent vectors with foot point f Imm( M, N ),


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i.e., vector elds along f . We consider the follow-ing weak Riemannian metric onImm( M, N ), for a constant A 0:

GAf ( h, k ) :=

= M (1 + A Tr f g( S f ) 2gN ( f ) ) g( h, k ) vol( f g)where

Tr f g( S f ) gN ( f ) is the norm of the mean

curvature. The metric GA is invariant for the actionof Diff( M ). This makes the map : Imm( M, N ) B i( M, N ) into a Riemannian submersion (off the

singularities of B i( M, N )).

The tangent vectors to the orbits areT f ( f Diff( M )) = {Tf. : X( M )}. The bundle


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N Imm( M, N ) of tangent vectors normal to theDiff( M )-orbits is independent of A: N f = {h C ( M,TN ) : g( h,Tf ) = 0 }

= ( f ( T N |M /Tf.TM )) = ( f TN/TM ) ,the space of sections of the normal bundle.A tangent vectorh

T

f Imm( M, N ) = C

f (M,TN ) = ( f

T N ) hasan orthonormal decomposition

h = h + h T f ( f Diff + ( M )) N f into smooth tangential and normal components.

The metric GA on Imm( M, N ) is invariant underDiff( M ) and induces a metric on the quotient B i( M, N For any F 0 , F 1 B i , consider all liftings f 0 , f 1 Imm


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such that ( f 0 ) = F 0 , ( f 1 ) = F 1 and all smooth curves t f ( t, ) in Imm( M, N ) with f (0 , ) = f 0 and f (1 , ) =f 1 . The length of t

( f ( t,

)) in B i( M, N ) is given

by

LhorGA ( f ) := LGA( ( f ( t, ))) ==

1

0 GA

( f )( T

f .f

t, T

f .f

t) dt =

1

0 GA

f ( f t , f t ) dt

= 10 M (1 + A Tr f g( S f ) 2g ) g( f t , f t ) vol( f g)12dt

In fact the last computation only makes sense onB i,f ( M, N ) but we take it as a motivation.

The metric on B i( M, N ) is dened by taking theinmum of this over all paths f (and all lifts f 0 , f 1 ):


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dist B iGA( F 1 , F 2 ) = inf f LhorGA ( f ) .

Theorem. For f 0 , f 1 Imm( M, N ) there exists al-ways a path t f ( t, ) in Imm( M, N ) with f (0 , ) =f 0 and ( f (1 , )) = ( f 1 ) such that Lhor G0 ( f ) is arbi-trarily small.

So the lowest order metric is not suitable for vision.Sketch the proof!


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Area swept out bound. If f is any path from F 0to F then


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to F 1

, then

( m + 1) volume of the region swept out by the variation f

maxt Volg( f ( t, )) L

hor GA ( f ) .

Together with Lipschitz continuity this shows thatthe geodesic distance LB iGA separates points at leaston Be( M, N ), if A > 0.

Horizontal energy of a path as anisotropic vol-ume We consider a path t f ( t, ) in Imm( M, N ).


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It projects to a path f in B i whose energy is:E GA( f ) = 12 ba GA ( f ) ( T.f t ,T.f t ) dt =

= 12 ba GAf ( f t , f t ) dt == 12 ba M (1+ A Tr f g( S f ) 2g ) g( f t , f t ) vol( f g) dt.

We now consider the graph f : [a, b ]M ( t, x ) ( t, f ( t, x )) [a, b ]N of the path f and its image f ,an immersed submanifold with boundary of R N .

hor


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E horGA ( f ) == 12 [a,b ]M (1 + A Tr f g( S f ) 2gN ( f ) )

f t

2

1 + f t 2gvol(

f (dt

2 +g

))

This is intrinsic for the graph f and the brationpr 1 : R N R . To nd a geodesic between theshapes ( f ( a, )) and ( f ( b, )) we look for animmersed surface which is critical for E horGA . This isa Plateau-problem with anisotropic volume.


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The geodesic equation of G0 in Imm( M, N )

g t f t + div

f g( f t ) f t g( f t , Tr f g( S f )) f t++ 12 T f. grad f g( f t 2g ) + 12 f t 2g Tr f g( S f ) = 0


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The sectional curvature for G0 in B i( M, N )

kf ( P ( m, h )) =G0f ( R ( m, h ) m, h )

2 2 0 2 .


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m2

h2

G0a ( m, h )

2

We get then for x, y ( N f ):R f ( x,y,x,y ) = G

0f ( R f ( x, y ) x, y ) =

= M vol( f g) 12 Tr( L f L f )( xy) 0

1

4Tr( L f

x) y

Tr( L f

y) x 2

g 0

+ 14 xy2 Tr g( S f ) 2 0

+ g( R g( x, y ) x, y )+ xy

2 Ric( T M, span( x, y ))

12 ( g( x,y) g( y,x) 2 1M 0

1 2


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+ 2 x

y yx 1M 2 N ( f ) 0 .

Corollary. If M has codimension 1 in N then

all sectional curvatures are non-negative. For any codimension, sectional curvature in the plane spanned by x and y is non-negative if x and y are parallel,i.e., x

y = 0 in

2 T N .


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The right invariant H 0 -metric on Diff 0 ( N ) is thengiven as follows, where h, k : N T N are vector


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elds with compact support along and where X =h 1 , Y = k 1 Xc( N ):G0( h, k ) =

N g( h, k ) vol( g)

= N g( X , Y )vol( g)= N g( X, Y ) vol( g)

Theorem. Geodesic distance on Diff 0 ( N ) withrespect to the H 0 -metric vanishes.

x 2retr etr


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t

ph

3

ph(x)

ph+(x)ph(x)

x

slopea

ph

slopela

ph+

3.5Particle trajectories under , = 0.6


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0.5 0 0.5 1 1.5 2 2.5 30.5

0

0.5

1

1.5

2

2.5

3

Time t

S p a c e x


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In our case, for Diff 0 ( N ), we havead( X ) Y = [X, Y ]


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G0 ( X, Y ) = N g( X, Y ) vol( g)G0 (ad( Y )X, Z ) = G0 ( X, [Y, Z ]) == N g LY X + ( g1LY g) X + div g( Y ) X, Z vol( g)ad( Y ) = LY + g1LY ( g)+d iv g( Y ) = LY + ( Y ) ,

where the tensor eld ( Y ) = g1LY ( g)+d iv g( Y ) : T N T N is self adjoint with respect to g.

Thus the geodesic equation for G0 is

u t = ( g1Lu ( g))( u ) div g( u ) u = ( u ) u,


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u = t 1 .The main part of the sectional curvature is givenby:

4 G ( R ( X, Y ) X, Y ) =

= N ( X ) Y ( Y ) X + [ X, Y ] 2g4 g([ ( X ) , ( Y )] X, Y ) vol( g)

So sectional curvature consists of a part which isvisibly non-negative, and another part which is dif-

cult to decompose further.

Example. For ( N, g ) = ( R , can) or ( S 1 , can) the


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p ( , g ) ( , ) ( , )geodesic equation is Burgers equation , a com-pletely integrable innite dimensional system,

ut =

3 ux u, u =

t

1 ,

to which corresponds vanishing geodesic distance.and we get G0 ( R ( X, Y ) X, Y ) = [X, Y ]2 dx sothat all sectional curvatures are non-negative.

Example. For ( N, g ) = ( R n , can) or (( S 1 ) n , can):

(ad( X ) Y ) k =i

(( iX k) Y i X i( iY k))


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G0 (ad( X ) Y, Z ) = R n dX.Y dY.X,Z dx= R n i,k Y k ( kX i) Z i + ( iX i) Z k + X i( iZ k) dx(ad( X )Z )

k=

=i

( kX i) Z i + ( iX i) Z k + X i( iZ k) ,

so that the geodesic equation is given by

tuk = (ad( u )u ) k == i

( kui) u i + ( iu i) uk + u i( iuk) ,

the n-dimensional analogon of Burgers equation,called the basic Euler-Poincare equation (EPDiff)


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by Holm. Also here we have vanishing geodesicdistance.

Stronger metrics on Diff 0 ( N ) .A very small strengthening of the weak Riemannian


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H 0 -metric on Diff 0 ( N ) makes it into a true met-ric. We dene the stronger right invariant weakRiemannian metric by the formula:

GA ( h, k ) = N ( g( X, Y ) + A div g( X ) . div g( Y )) vol( g) .Theorem. For any distinct diffeomorphisms 0 ,1 ,the inmum of the lengths of all paths from 0 to 1 with respect to GA is positive.

Example We consider the groups Diff c( R ) or Diff( S 1 )with Lie algebras Xc( R ) or X( S 1 ) with Lie bracketad( X ) Y = [X, Y ] = X Y XY . The GA-metric


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equals the H 1 -metric on Xc( R ), and we have:GA( X, Y ) = R ( XY + AX Y ) dx

= R X (1 2x ) Y dx,

ad( X ) = (1 2x )1 (2 X + X x )(1 A 2x )so that the geodesic equation in Eulerian represen-

tation u = ( t) 1 Xc( R ) or X( S 1 ) is tu = ad( u )u

=

(1

2x ) 1 (3 uu

2 Au u

Au u ) ,

u t u txx = Au xxx .u + 2 Au xx .u x 3 ux .u,


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which for A = 1 is the Camassa-Holm equation ,another completely integrable innite dimensionalHamiltonian system. Here geodesic distance is ametric.

Virasoro-Bott group. Let Diff denote any of thegroups Diff( S 1 ), Diff c( R ) (diffeomorphisms with


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compact support), or Diff S ( R ). Thenc : Diff Diff R

c(, ) : = 1

2 log((

) ) d log( )

= 12 log( ) d log( )

satises c(,1 ) = 0, c(Id , ) = 0, c(, Id) = 0,and is a smooth Hochschild group cocycle, i.e.,c(2 ,3 )c(12 ,3 )+ c(1 ,23 )c(1 ,2 ) = 0 ,called the Bott cocycle.


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The corresponding central extension group R cDiff, called the Virasoro-Bott group, is a regularLie group with operations

= + + c(, ) , 1 = 1

1for , Diff and , R .

The Lie algebra of the Virasoro-Bott Lie group isthe central extension R X of X, called the Vira-soro Lie algebra, with bracket:


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X a

, Y b

= [X, Y ]( X, Y )

= X Y XY ( X, Y )

( X, Y ) = ( X ) Y =

X dY =

X Y dx =

= 12 det X Y X Y dx,is the Gelfand-Fuchs Lie algebra cocycle : X

X

R , which is a bounded skew-symmetric

bilinear mapping satisfying the cocycle condition

([ X, Y ], Z ) + ([ Y, Z ], X ) + ([ Z, X ], Y ) = 0 .

It is a generator of the 1-dimensional bounded Cheval-ley cohomology H 2 ( X, R ) for any of the Lie algebras


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X = X( S 1 ), Xc( R ), or S ( R ) x .

We shall use the L2 -inner product on R X, whereX = X( S 1 ) , Xc( R ) , S ( R ) x :X , Y

b 0 := XY dx + ab.


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a b 0 Integrating by parts we getad

X a

Y b

,Z c 0

=X Y XY

( X, Y ),

Z c 0

= (X Y Z XY Z + cX Y ) dx= (2 X Z + XZ + cX ) Y dx= Y

b, ad X

a Z c 0

, where

adX aZ

c =

2 X Z + XZ + cX 0

.

The H 0 geodesic equation on the Virasoro-Bottgroup (Ovsienko-Khesin):

d 3 h


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u ta t

= ad ua ua = 3 uxu au xxx0 whereu ( t )a ( t )

= s( s) ( s)

.( t ) 1

( t ) s= t

= s ( s ) ( t ) 1

( s ) ( t ) + c(( s) ,( t ) 1 ) s= t=

t 1 t

txxx22x dx

Thus a is a constant in time and the geodesic equa-tion is hence the Korteweg-de Vries equation

u t + 3 uxu + au xxx = 0 .


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with its natural companions

t = u , t = a +

txxx

22x

dx.

I d k h h h i h i i L2 i


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I do not know whether the right invariant L2 -metricon the Virasoro-Bott group has vanishing geodesicdistance?On Mondays I think: YESOn Tuesdays I think: NO. . .


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At the end of last main lecture:Many thanks to the organizers (except one of them)for a great conference, and for the ne weather andgreat snow.

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