CHAPTER IV

DIFFERENTIAL GEOMETRY OF

DIFFEOMORPHISM GROUPS

In EN Lorenz stated that a twoweek forecast would b e the theoretical

b ound for predicting the future state of the atmosphere using largescale numerical

mo dels Lor Mo dern meteorology has currently reached a go o d correlation of

the observed versus predicted for roughly seven days in the northern hemisphere

whereas this p erio d is shortened by half in the southern hemisphere and by two

thirds in the tropics for the same level of correlation Kri These dierences are

due to a very simple factor the available data density

The mathematical reason for the dierences and for the overall longterm unre

liability of weather forecasts is the exp onential scattering of ideal uid or atmo

spheric ows with close initial conditions which in turn is related to the negative

curvatures of the corresp onding groups of dieomorphisms as Riemannian mani

folds We will see that the conguration space of an ideal incompressible uid is

highly nonat and has very p eculiar interior and exterior dierential geom

etry

Interior or intrinsic characteristics of a Riemannian manifold are those

p ersisting under any isometry of the manifold For instance one can b end ie

map isometrically a sheet of pap er into a cone or a cylinder but never without

stretching or cutting into a piece of a sphere The invariant that distinguishes

Riemannian metrics is called Riemannian curvature The Riemannian curvature of

a plane is zero and the curvature of a sphere of radius R is equal to R If one

Riemannian manifold can b e isometrically mapp ed to another then the Riemannian

curvature at corresp onding p oints is the same

The Riemannian curvature of a manifold has a profound impact on the b ehavior

of geo desics on it If the Riemannian curvature of a manifold is p ositive as for

a sphere or for an ellipsoid then nearby geo desics oscillate ab out one another

in most cases and if the curvature is negative as on the surface of a onesheet

hyperb oloid geo desics rapidly diverge from one another

Typeset by A ST X

M E

IV DIFFERENTIAL GEOMETRY OF DIFFEOMORPHISM GROUPS

It turns out that dieomorphism groups equipp ed with a onesided invariant

metric lo ok very much like negatively curved manifolds In Lagrangian mechanics a

motion of a natural mechanical system is a geo desic line on a manifoldconguration

space in the metric given by the dierence of kinetic and p otential energy In the

case at hand the geo desics are motions of an ideal uid Therefore calculation of

the curvature of the dieomorphism group provides a great deal of information on

instability of ideal uid ows

In this chapter we discuss in detail curvatures and metric prop erties of the groups

of volumepreserving and symplectic dieomorphisms and present the applications

of the curvature calculations to reliability estimates for weather forecasts

x The Lobachevsky plane and preliminaries in dierential geometry

A The Lobachevsky plane of ane transformations We start with an

oversimplied mo del for a dieomorphism group the twodimensional group G of

all ane transformations x a bx of a real line or more generally consider the

n dimensional group G of all dilations and translations of the ndimensional

n n

space R x a R b R

Regard elements of the group G as pairs a b with p ositive b or as p oints of the

upp er halfplane halfspace resp ectively The comp osition of ane transforma

tions of the line denes the group multiplication of the corresp onding pairs

a b a b a a b b b

To dene a onesided say left invariant metric on G and the corresp onding Euler

equation of the geo desic ow on G see Chapter I one needs to sp ecify a quadratic

form on the tangent space of G at the identity

Fix the quadratic form da db on the tangent space to G at the identity a b

Extend it to the tangent spaces at other p oints of G by left translations

Proposition The leftinvariant metric on G obtained by the procedure

above has the form

da db

ds

b

b

Proof The left shift by an element a b on G has the Jacobian matrix

b

Hence the quadratic form da db on the tangent space T G at the identity is

x PRELIMINARIES IN DIFFERENTIAL GEOMETRY

the pullback of the quadratic form da db b on T G at the p oint a b

ab



Note that starting with any p ositive denite quadratic form one obtains an

isometric manifold up to a scalar factor in the metric

Definition The Riemannian manifold G equipp ed with the metric ds

da db b is called the Lobachevsky plane resp ectively the Lobachevsky

n n

space for x a R

Proposition Geodesic lines ie extremals of the length functional are

the vertical half lines a const b and the semicircles orthogonal to the aaxis

or ahyperplane respectively Here vertical half lines can b e viewed as semicircles of innite radius see Fig

b

a

Figure The Lobachevsky plane and geo desics on it

Proof Reection of in a vertical line or in a circle centered at the aaxis is

an isometry 

Note that two geo desics on with close initial conditions diverge exp onentially

from each other in the Lobachevsky metric On a path only few units long a

deviation in initial conditions grows times larger The reason for practical

indeterminacy of geo desics is the negative curvature of the Lobachevsky plane it

is constant and equal to in the metric ab ove The curvature of the sphere of

radius R is equal to R The Lobachevsky plane might b e regarded as a sphere

p

of imaginary radius R

Problem BYa Zeldovich Prove that medians of every geo desic triangle

in the Lobachevsky plane meet at one p oint Hint Prove it for the sphere then

regard the Lobachevsky plane as the analytic continuation of the sphere to the

imaginary values of the radius

IV DIFFERENTIAL GEOMETRY OF DIFFEOMORPHISM GROUPS

B Curvature and parallel translation This section recalls some basic no

tions of dierential geometry necessary in the sequel For more extended treatment

see Mil Arn DFN

n

Let M b e a Riemannian manifold one can keep in mind the Euclidean space R

n n

the sphere S or the Lobachevsky space of the previous section as an example

let x M b e a p oint of M and T M a vector tangent to M at x

x

Denote by any of f x tg f tg f tg the geodesic line on M

with the initial velocity vector at the p oint x Geo desic lines

R

can b e dened as extremals of the action functional dt It is called the

principle of least action

Parallel translation along a geodesic segment is a sp ecial isometry mapping the

tangent space at the initial p oint onto the tangent space at the nal p oint dep end

ing smo othly on the geo desic segment and ob eying the following prop erties

Translation along two consecutive segments coincides with translation along

the rst segment comp osed with translation along the second

Parallel translation along a segment of length zero is the identity map

The unit tangent vector of the geo desic line at the initial p oint is taken to

the unit tangent vector of the geo desic at the nal p oint

Example The usual parallel translation in Euclidean space satises the

prop erties

The isometry prop erty along with the prop erty implies that the angle formed

by the transp orted vector with the geo desic is preserved under translation This

observation alone determines parallel translation in the twodimensional case ie on surfaces see Fig

P

Figure Parallel translation along a geo desic line

In the higherdimensional case parallel translation is not determined uniquely

by the condition of preserving the angle One has to sp ecify the plane containing

the transp orted vector

x PRELIMINARIES IN DIFFERENTIAL GEOMETRY

Definition Riemannian parallel translation along a geo desic is a family

of isometries ob eying prop erties ab ove and for which the translation of a

vector along a short segment of length t remains tangent mo dulo O t small

correction as t to the following twodimensional surface This surface is formed

by the geo desics issuing from the initial p oint of the segment with the velocities

spanned by the vector and by the velocity of the initial geo desic

Remark A physical description of parallel translation on a Riemannian

manifold can b e given using the adiabatic slow transp ortation of a p endulum

along a path on the manifold Radon see Kl The plane of oscillations is parallelly

translated

A similar phenomenon in optics is called the inertia of the p olarization plane

along a curved ray see Ryt Vld It is also related to the additional rotation of a

gyroscop e transp orted along a closed path on the surface or in the o ceans of the

Earth prop ortional to the swept area Ish A mo dern version of these relations

b etween adiabatic pro cesses and connections explains among other things the

AharonovBohm eect in quantum mechanics This version is also called the Berry

phase see Berr Arn

Definition The covariant derivative r of a tangent vector eld in a

direction T M is the rate of change of the vector of the eld that is parallel

x

transp orted to the p oint x M along the geo desic line having at this p oint the

velocity the vector observed at x at time t must b e transp orted from the p oint

t

Note that the vector eld along a geo desic line ob eys the equation r

Remark For calculations of parallel translations in the sequel we need the

following explicit formulas Let x t b e a geo desic line in a manifold M and

let P T M T M b e the map that sends any T M to the vector

t t

d

j x t T M P

t t

t d

The mapping P approximates parallel translation along the curve in the fol

t

lowing sense The covariant derivative r of the eld in the direction of the

vector T M is equal to

x

d

r j P t T M

t x

t

dt

Remark The following prop erties uniquely determine the covariant de

rivative on a Riemannian manifold and can b e taken as its axiomatic denition

see eg Mil KN

r v is a bilinear function of the vector and the eld v

IV DIFFERENTIAL GEOMETRY OF DIFFEOMORPHISM GROUPS

r f v L f v f r v where f is a smo oth function and L f is the

derivative of f in the direction of the vector in T M

x

L hv w i hr v w xi hv x r w i and

r w r v fv w gx

v x w x

Here h i is the inner pro duct dened by the Riemannian metric on M and fv w g

is the Poisson bracket of two vector elds v and w In lo cal co ordinates x x

n

on M the Poisson bracket is given by the formula

n

X

w v

j j

v w fv w g

i i j

x x

i i

i

Parallel translation along any curve is dened as the limit of parallel translations

along broken geo desic lines approximating this curve The increment of a vector

after the translation along the b oundary of a small region on a smo oth surface is

in the rst approximation prop ortional to the area of the region

Definition The Riemannian curvature tensor describ es the inni

tesimal transformation in a tangent space obtained by parallel translation around

an innitely small parallelogram Given vectors T M consider a curvilinear

x

parallelogram on M spanned by and The main bilinear in part of the

increment of any vector in the tangent space T M after parallel translation around

x

this parallelogram is given by a linear op erator T M T M The action

x x

of on a vector T M can b e expressed in terms of covariant dierentiation

x

as follows

r r r r r j

xx

f g 0

where are any vector elds whose values at the p oint x are and The

value of the righthand side do es not dep end on the extensions of the vectors

and

The sectional curvature of M in the direction of the plane spanned by two

orthogonal unit vectors T M in the tangent space to M at the p oint x is the

x

value

C h i

For a pair of arbitrary not necessarily orthonormal vectors the sectional cur

vature C is

h i

C

h ih i h i

x PRELIMINARIES IN DIFFERENTIAL GEOMETRY

Example The sectional curvature at every p oint of the Lobachevsky plane

Section is equal to Hint use the explicit description of the geo desics in

the plane See also Section for general formulas for sectional curvatures on Lie

groups

Definitions The normalized Ricci curvature at a p oint x in the direc

tion of a unit vector is the average of the sectional curvatures of all tangential

planes containing It is equal to r n where the Ricci curvature r

P P

h e e i calculated in any orthonormal basis C is the value r

i i e

i

i i

e e of the tangent space T M

n x

The normalized scalar curvature at a p oint x is the average of all sectional cur

vatures at the p oint It is equal to nn the scalar curvature b eing the sum

P

see eg Mil C r e r e

e e i n

i j

ij

C Behavior of geo desics on curved manifolds From the denition of

curvature one easily deduces the following

Proposition The distance y t between two innitely close geodesics on

a surface satises the dierential equation

y C y

where C C t is the Riemannian curvature of the surface along the geodesic

In order to describ e in general how the curvature tensor aects the b ehavior

of geo desics we lo ok at a variation t of a geo desic t For each

suciently close to the curve is a geo desic whose initial condition is close to

d

that of The eld t j t dened along is called the vector eld

d

of geodesic variation

Lemmadefinition The vector eld of geodesic variation satises the

secondorder linear dierential equation called the Jacobi equation

r

Proof Dene the vector eld of geo desic variation t for all geo desics of

d

the family t with small as the derivative t t Then the elds

d

and commute f g as partial derivatives of the map t t

Using the prop erties of covariant dierentiation listed ab ove and the denition of

the curvature tensor we get

r r r r r

IV DIFFERENTIAL GEOMETRY OF DIFFEOMORPHISM GROUPS



Assume for a moment that the curvature is p ositive in all twodimensional direc

tions containing the velocity vector of the geo desic A closer analysis of the Jacobi

equation or its analogy with the standard p endulum see Prop osition shows

that the normal comp onent of the vector eld of geo desic variation oscillates with

t This means that geo desics with close initial velocities on a manifold of p ositive

curvature oscillate around each other or lo cally converge see Figa On the

other hand negativity in sectional curvatures prompts analogy with the upside

down p endulum and implies the exp onential divergence of nearby geo desics from

the given one see Figb

C > 0 C < 0

(a) (b)

Figure Geo desics on manifolds of a p ositive and b negative

curvature

Remark For numerical estimates of the instability it is useful to dene

the characteristic path length as the average path length on which small errors in the

initial conditions are increased by the factor of e If the curvature of our manifold

in all twodimensional directions is b ounded away from zero by the number b

then the characteristic path length is not greater than b cf Prop osition

In view of the exp onential character of the growth of error the course of a geo desic

line on a manifold of negative curvature is practically imp ossible to predict

D Relation of the covariant and Lie derivatives Every vector eld on

a Riemannian manifold denes a dierential form the p ointwise inner pro duct

with vectors of the eld For a vector eld v we denote by v the form whose

value on a tangent vector at a p oint x is the inner pro duct of the tangent vector

with the vector v x

Every vector eld also denes a ow which transp orts dierential forms For

instance one might transp ort the form corresp onding to some vector eld by

x PRELIMINARIES IN DIFFERENTIAL GEOMETRY

means of the ow of this eld and get a new dierential form Innitesimally

this transp ort is describ ed by the Lie derivative of the form corresp onding to the

eld along the eld itself and the result is again a form This natural derivative

of a form is related to the covariant derivative of the corresp onding vector eld

along itself by the following remarkable formula

Theorem The Lie derivative of the oneform corresponding to a vector

eld on a Riemannian manifold diers from the oneform corresponding to the

covariant derivative of the eld along itself by a complete dierential

L v r v dhv v i

v v

Here hv v i is the function on the manifold equal at each point x to the Riemannian

square of the vector v x

Note that this statement do es not require the vector eld to b e divergence free

Proof Let w b e a vector eld commuting with the eld v ie fv w g

First since parallel translation is an isometry

L hb ci hr b ci hb r ci

a a a

for any vector elds a b and c Applying this to the elds a w b c v gives

L hv v i hr v v i hv r v i From this we nd that

w w w

dhv v iw hr v v i

w

Applying the isometry prop erty once more to a c v b w we get

L hw v i hr w v i hw r v i

v v v

On the other hand for commuting elds v and w prop erty of Remark

implies

hr w v i hr v v i

v w

Substituting this into we obtain that

L hw v i hr v v i hw r v i

v w v

Using we rewrite the ab ove in the form

L hw v i hr v w i dhv v iw

v v

IV DIFFERENTIAL GEOMETRY OF DIFFEOMORPHISM GROUPS

Next we use the identity

L v w L v w v L w

which expresses the naturalness of the Lie derivative The ow of transp orts the

form v the vector w and the value of the form on this vector This is the

reason why vector elds are transp orted in the opp osite direction in the denition

of the Lie derivative

Applying to v we obtain

L v w L v w

v v

since L w fv w g Here we use the commutativity of v and w for the

v

second time

Note that by the denition of the map v v one has

L v w L hv w i

v v

Combining this with and we nd that for any vector of a eld w

commuting with the eld v

L v w r v w dhv v iw

v v

At every p oint x where the vector eld v is nonzero it is easy to construct a

eld w commuting with v and having at this p oint any value Hence the identity

implies

L v r v dhv v i

v v

which proves the theorem

At the singular p oints v x there is nothing to prove since b oth sides of the

relation are equal to zero 

Remark One can take an arbitrary eld w instead of the one commuting

with v and then the formulas are slightly longer Two commutator terms have to b e

introduced at the two places where commutativity was used The additional term

hfw v g v i on the righthand side of cancels with the extra term hv fv w gi

on the righthand side of

This theorem explains the form of the Euler equation of an incompressible uid

on an arbitrary Riemannian manifold M presented in Sections I and I

x SECTIONAL CURVATURES OF LIE GROUPS

Corollary The Euler equation

v

r v rp

v

t

on the Lie algebra g S VectM of divergencefree vector elds is mapped by the

inertia operator A g g to the Euler equation

u

L u

v

t

on the dual space g M d M of this algebra Here the eld v and the

form u are related by means of the Riemannian metric u v and u d

is the coset of the form u

Proof The inertia op erator A S VectM d sends a divergencefree

eld v to the form u v considered up to the dierential of a function By

the ab ove theorem it also sends the covariant derivative r v to the Lie derivative

v

L u mo dulo the dierential of another function Hence the Euler equation for the

v

form u assumes the form

u

L u df

v

t

hv v i It is equivalent to Equation for the coset with the function f p

u 

x Sectional curvatures of Lie groups

equipp ed with a onesided invariant metric

Let G b e a Lie group whose leftinvariant metric is given by a scalar pro duct

h i in the Lie algebra The sectional curvature of the group G at any p oint is

determined by the curvature at the identity since by denition left translations

map the group to itself isometrically Hence it suces to describ e the curvatures

for the twodimensional planes lying in the Lie algebra g T G

e

Theorem Arn The curvature of a Lie group G in the direction deter

mined by an orthonormal pair of vectors in the Lie algebra g is given by the

formula

C h i h i h i hB B i

IV DIFFERENTIAL GEOMETRY OF DIFFEOMORPHISM GROUPS

where

B B B B

B B B B

and where is the commutator in g and B is the bilinear operation on g dened

by the relation hB u v w i hu v w i see Chapter I

Remark In the case of a twosided invariant metric the formula for the

curvature has the particularly simple form

h i C

In particular in this case the sectional curvatures are nonnegative in all two

dimensional directions

Remark The formula for the curvature of a group with a rightinvariant

Riemannian metric coincides with the formula in the leftinvariant case In fact a

rightinvariant metric on a group is a leftinvariant metric on the group with the

reverse multiplication law g g g g Passage to the reverse group changes

the signs of b oth the commutator and the op eration B in the algebra But in

every term of the curvature formula there is a pro duct of two op erations changing

the sign Therefore the formula for curvature is the same in the rightinvariant

case The righthand side of the Euler equation changes sign under passage to the

rightinvariant case

The mapping of the group G to itself sending each element g to the inverse

element g is an involution preserving the identity element It sends any left

invariant metric on the group to the corresp onding rightinvariant metric de

ned on the Lie algebra by the same quadratic form Hence the group with the

leftinvariant metric is isometric to the same group with the corresp onding right

invariant metric

To give the co ordinate expression for the curvature choose an orthonormal basis

e e for the leftinvariant vector elds The Lie algebra structure can b e

n

P

describ ed by an n n n array of structure constants where e e e

ij k i j ij k k

k

or equivalently he e e i This array is skewsymmetric in the rst two

ij k i j k

indices Then Theorem claims the following

Theorem see Mil The sectional curvature C is given by the for

e e

1 2

mula

X

C

k k k k e e

1 2

k

k k k k k k k k

x SECTIONAL CURVATURES OF LIE GROUPS

Remark Before proving the theorem we give here an account of notewor

thy facts ab out leftinvariant metrics on Lie groups that can b e formulated in a

co ordinatefree way and some have innitedimensional counterparts We refer to

Mil for all the details

If b elongs to the center of a Lie algebra then for every leftinvariant

metric the inequality C is satised for all cf Section VIA on

the Virasoro algebra

If a connected Lie group G has a leftinvariant metric with all sectional

curvatures C then it is solvable example ane transformations of

the line see Section A

If G is unimo dular ie the op erators ad are traceless for all u g then

u

any such metric with C must actually b e at C cf Remark

I I

Every compact Lie group admits a left invariant and in fact a biinvariant

metric such that all sectional curvatures satisfy C cf Remark

ab ove

If the Lie algebra of G contains linearly indep endent vectors satisfy

ing then there exists a leftinvariant metric such that the Ricci

curvature r is strictly negative while the Ricci curvature r is strictly

p ositive For instance one can dene such a metric on SO the congu

ration space of a rigid b o dy such that a certain Ricci curvature is negative

Wallach If the Lie group G is noncommutative then it p ossesses a left

invariant metric of strictly negative scalar curvature

If G contains a compact noncommutative subgroup then G admits a left

invariant metric of strictly p ositive scalar curvature Mil

Proof of Theorem To obtain explicit formulas for sectional curvatures

of the group G we start by expressing the covariant derivative in terms of the

op eration B or of the array

ij k

Lemma Let and be two leftinvariant vector elds on the group G

Then the vector eld r is also leftinvariant and at the point e G is given by

the fol lowing formula

B B r j

e

where on the righthand side and are vectors in g T G dening the corre

e

sponding leftinvariant elds on G

IV DIFFERENTIAL GEOMETRY OF DIFFEOMORPHISM GROUPS

In co ordinates

X

e r e

ij k j k i k ij k e j

i

k

Proof of Lemma Parallel transp ort on a Riemannian manifold preserves the

inner pro duct ha bi On the other hand the leftinvariant pro duct ha bi of any

leftinvariant elds a and b is constant Therefore for any c the op erator r is

c

antisymmetric on leftinvariant elds

hr a bi hr b ai

c c

Furthermore for the covariant derivative on a Riemannian manifold the following

symmetry condition holds see Remark

r a r c fc ag

c a

Recall that on a Lie group the Poisson bracket f g of the leftinvariant vector

elds a and c coincides at e G with the commutator in the Lie algebra

g

fa cg j a c

e g

The Poisson bracket of two rightinvariant vector elds has the opp osite sign see

Remark I or Arn

Combining the ab ove identities we obtain the formula

h i h i h i hr i

easily seen to b e equivalent to For the orthonormal basis e e it imme

n

diately implies

hr e e i

e j k ij k j k i k ij

i

This completes the pro of of Lemma 

Finally the co ordinate expression for sectional curvature is a straightfor

ward consequence of formulas and Theorem is proved 

Remark Lemma is deduced in Arn from the Euler equation

B see Chapter I

Consider a neighborho o d of the p oint in the Lie algebra g as a chart of a

neighborho o d of the unit element e in the group using the exp onential map exp

g G It sends t to the element expt of the oneparameter group starting at

x SECTIONAL CURVATURES OF LIE GROUPS

t from e with initial velocity We leave aside the diculties of this approach

in the innitedimensional case where the image of the exp onential mapping do es

not contain the neighborho o d of the group unit element The exp onential mapping

identies the tangent spaces of the group T G with the Lie algebra g

g

The Euler equation implies that the geo desic line on the group has in our co or

dinates the following expansion in t

t

B O t t t t

Then the approximate translation

d

P j t T g g

t t

t d

of a vector T g g is explicitly given by

t

B B O t P

t

By denition the covariant derivative in the direction g of the leftinvariant

vector eld on the group G determined by the vector g is

d

j P 1 L r

t

t t

dt

where on the lefthand side stands for the corresp onding leftinvariant vector

eld on G

Note that for any Lie group G the op erator of left translation by group elements

exp a close to the identity ie as jaj acts on the Lie algebra g considered as

a chart of the group as follows

L a O a

a

Indeed the general case of any Lie group follows from the calculus on matrix groups

exp a exp b exp a b a b O a O b

for any Lie algebra elements a b Setting b t t jaj we get

exp a exp t exp a a O a t O t

which is equivalent to

IV DIFFERENTIAL GEOMETRY OF DIFFEOMORPHISM GROUPS

Now substituting into the expressions for P and for the left

translation L we get the following

t d

j P 1 O t r

t

t

dt

d t

j B B O t

t

dt



Theorem can now b e proven in the following co ordinatefree way see Arn

Proof of Theorem Let b e leftinvariant vector elds on the group

G Then the elds f g r and r are leftinvariant as well The formula

combined with the notations gives the following values of these vector

elds at the identity Id G

r B r

r B r

Now in order to evaluate the terms in we use these expressions along with

the skew symmetry of r to obtain

hr r i hr r i h i

hr r i hr r i hB B i

Moreover by virtue of

hr i hr i

f g

h i hB i hB i

h i h i

Finally incorp orating into the denition of sectional curvature

we get

C h i hB B i h i h i

which is equivalent to This completes the pro of 

x GEOMETRY OF THE GROUP OF TORUS DIFFEOMORPHISMS

x Riemannian geometry of the group of

areapreserving dieomorphisms of the twotorus

A The curvature tensor for the group of torus dieomorphisms The

co ordinatefree formulas for the sectional curvature allow one to apply them to the

innitedimensional case of groups of dieomorphisms The numbers that we obtain

by applying the formula for the curvature of a Lie group to these innitedimensional

groups are naturally called the curvatures of the dieomorphism groups We de

scrib e in detail the case of areapreserving dieomorphisms of the twodimensional

torus and review the results for the twosphere the S case is of sp ecial interest

b ecause of its relation to atmospheric ows and weather predictions for the n

n

dimensional torus T the T case is imp ortant for stability analysis of the AB C

ows for a compact twodimensional surface and for any at manifold

We start with the twodimensional torus T equipp ed with the Euclidean metric

T R where is a lattice a discrete subgroup in the plane eg the set of

p oints with integral co ordinates

Consider the Lie algebra of divergencefree vector elds on the torus with a

singlevalued stream function or a singlevalued Hamiltonian function with resp ect

to the standard symplectic structure on T given by the area form The corre

sp onding group S DiT consists of areapreserving dieomorphisms isotopic to

the identity that leave the center of mass of the torus xed

Remark The subgroup S DiT is a total ly geodesic submanifold of the

group S DiT of all areapreserving dieomorphisms that is any geo desic on

S DiT is a geo desic in the ambient group This follows from the momentum

conservation law If at the initial moment the velocity eld of an ideal uid has

a singlevalued stream function then at all other moments of time the stream

function will also b e singlevalued Note that a similar statement holds in the

more general case of a manifold M with b oundary The two Lie subgroups in

S DiM corresp onding to the Lie subalgebras of exact g and semiexact g

se

vector elds see Remark I form totally geo desic submanifolds in the Lie group

of all volumepreserving dieomorphisms of M

The rightinvariant metric on the group S DiT is dened by the doubled

kinetic energy Its value at the identity of the group on a divergencefree vec

R

v v d x We will describ e the sectional tor eld v S VectT is hv v i

2

T

curvatures of the group S DiT in various twodimensional directions passing

through the identity of the group The curvatures of S DiT and S DiT in

these directions are the same since the submanifold S DiT is totally geo desic

IV DIFFERENTIAL GEOMETRY OF DIFFEOMORPHISM GROUPS

The divergencefree vector elds that constitute the Lie algebra S VectT of

the group S DiT can b e describ ed by their stream ie Hamiltonian functions

with zero mean v H x H y Thus in the sequel the set S VectT

y x

will b e identied with the space of real functions on the torus having average value

zero It is convenient to dene such functions by their Fourier co ecients and to

carry out all calculations over C

We now complexify our Lie algebra inner pro duct h i commutator and

the op eration B in the algebra as well as the Riemannian connection and curvature

tensor so that all these op erations b ecome multilinear in the complex vector

space of the complexied Lie algebra

To construct a basis of this vector space we let e where k called a wave vector

k

is a p oint of the Euclidean plane denote the function whose value at a p oint x of

ik x

our plane is equal to e

This determines a function on the torus if the inner pro duct k x is a multiple

of for all x All such vectors k b elong to a lattice in R and the functions

fe j k k g form a basis of the complexied Lie algebra

k

Theorem Arn The explicit formulas for the inner product commu

tator operation B connection and curvature of the rightinvariant metric on the

group S DiT have the fol lowing form

a he e i for k

k

he e i k S

k k

b e e k e where k k k

k k

k

c B e e b e where b k

k k k k

k

v uu v

e d e where d r d

k k uv e

k

v

e he e e e i if k m n

k mn k l m n

a a a a S if k m n

k mn n k m m k n

u v

f where a

uv

ju v j

In these formulas S is the area of the torus and u v the oriented area of the

parallelogram spanned by u and v The parentheses u v denote the Euclidean

scalar pro duct in the plane and angled brackets denote the scalar pro duct in the

Lie algebra

We p ostp one the pro of of this theorem as well as of the corollaries b elow until

the next section The formulas ab ove allow one to calculate the sectional curvature in any twodimensional direction

x GEOMETRY OF THE GROUP OF TORUS DIFFEOMORPHISMS

Example Consider the parallel sinusoidal steady uid ow given by the

stream function cosk x e e Let b e any other real vector of the

k k

algebra S VectT ie x e with x x

Theorem The curvature of the group S DiT in any twodimensional

plane containing the direction is

X

S

a jx x j C

k

k

and therefore nonpositive

Corollary The curvature is equal to zero only for those twodimensional

planes that consist of parallel ows in the same direction as such that

Corollary The curvature in the plane dened by the stream functions

cosk x and cos x is

C k sin sin S

where S is the area of the torus is the angle between k and and is the angle

between k and k

Corollary The curvature of the areapreserving dieomorphism group of

the torus fx y mo d g in the twodimensional directions spanned by the velocity

elds sin y and sin x is equal to C

Remark These calculations show that in many directions the sectional

curvature is negative but in a few it is p ositive The stability of the geo desic

is determined by the curvatures in the directions of al l p ossible twodimensional

planes passing through the velocity vector of the geo desic at each of its p oints the

Jacobi equation

Any uid ow on the torus is a geo desic of our group However calculations

simplify noticeably for a stationary ow In this case the geo desic is a oneparameter

subgroup of our group Then the curvatures in the directions of all planes passing

through velocity vectors of the geo desic at all of its p oints are equal to the curvatures

in the corresp onding planes going through the velocity vector of this geo desic at

the initial moment of time To prove it right translate the plane to the identity

element of the group Thus stability of a stationary ow dep ends only on the

curvatures in those twodimensional planes in the Lie algebra that contain the

velocity eld of the steady ow

IV DIFFERENTIAL GEOMETRY OF DIFFEOMORPHISM GROUPS

B Curvature calculations

Proof of Theorem Formula a is an immediate consequence of the

denition Statement b follows from the version of for rightinvariant

elds fa cg j a c Moreover combining b with the denition of B we

e g

come to the relation

hB e e e i mhe e i

k m m k

Further application of a shows that B e e is orthogonal to e for k m

x m

Thus B e e b e Expression c follows from a and for

k k k

m k

Now formulas bc along with expression for the covariant derivative

imply that

k

e r e k

k e

k

k

This implies d after the evident simplication

k k

k k

In order to nd the curvature tensor we rst compute from d

r r e d d e

e e m mk m mm k m

k

r e k d e

m mk m k m

e e

k

]

Along with a this implies that for k m n and

k mn

d d d d k d n S

k mn k mn nk n mn mn mn

for k m n note that d is symmetric in u and v due to d We

uv

leave to the reader the reduction of this identity to the form e 

e e

k k

Proof of Theorem To derive formula we substitute

P

x e into and

C h i

X

x x x x

k k k k k k k k

x x x x

k k k k

x GEOMETRY OF THE GROUP OF TORUS DIFFEOMORPHISMS

Taking into account the relations ef of Theorem one obtains the co ef

cients of this quadratic form

a S

k k k k k

k

a S

k k k k k

k

Then the form C for the orthonormal vectors and b ecomes

X

S

a x x x x a x x x x h i

k k

k k

X

S

a x x x x x x x x

k k k k

k

where the last identity is due to a a see f Finally from the

k k k

reality condition x x we get

j j

X

S

C x x x x x x x x a

k k k k

k

which is equivalent to 

Proof of Corollary By denition the sectional curvature in the plane

spanned by a pair of orthogonal vectors and is

h i

C

h ih i

For our choice of and we have h i k S h i l S and x x

Therefore by virtue of Theorem and f one obtains

S

h i a a

k k

Moreover the explicit expression f gives the identity

a a k

k k

h h

with h k This in turn can b e written as

k h h

a a k

k k

h h

where we made use of the evident relations h h k and h h

k Putting all the ab ove together and substituting k k sin

sin we obtain formula of the corollary  h h h h

IV DIFFERENTIAL GEOMETRY OF DIFFEOMORPHISM GROUPS

x Dieomorphism groups and unreliable forecasts

A Curvatures of various dieomorphism groups For an arbitrary com

pact ndimensional closed Riemannian manifold M the curvatures of the dieo

morphism group S DiM were calculated by Lukatsky Luk see also Smo

We refer to Luk for computations of the curvature tensor for the dieomorphism

group of any compact twodimensional surface the case of the twodimensional

sphere S imp ortant for meteorological applications can b e found in Luk Yo

and to Luk for those of a lo cally at manifold the case of the at ndimensional

torus was treated in Luk see also explicit formulas for T in KNH

Riemannian geometry and curvatures of the semidirect pro duct groups relevant

for ideal magnetohydrodynamics are considered in Ono In Mis the curvatures

of the group of all dieomorphisms of a circle are discussed The latter group

as well as its extension called the Virasoro group is the conguration space of

the Kortewegde Vries equation see Section VIA The geometry of biinvariant

metrics and geo desics on the symplectomorphism groups was studied in Don

Another way of investigating the Riemannian geometry of the group of volume

preserving dieomorphisms is to embed it as a submanifold in the group of all

dieomorphisms of the manifold and then to study the exterior geometry of the

corresp onding submanifold see Section b elow

Keeping in mind applications to weather forecasting we lo ok rst at the group

S DiS of dieomorphisms preserving the area on a standard sphere S R

fx y z g Consider the following two steady ows on S the rotation eld u

y x and the nonrealistic tradewind current v z y x Fig the

real tradewind current has the same direction in the northern and southern hemi

spheres

It was proved in Luk that the sectional curvatures C in all twoplanes con

uw

taining u are nonnegative for every eld w S VectS while for twoplanes con

taining the eld v the curvatures C are negative for most directions w Notice

v w

that the nonrealistic tradewind current v is a spherical counterpart of the parallel

sinusoidal ow on the torus sin y see statements ab ove

In the case of volumepreserving dieomorphisms of a threedimensional domain

M R of Euclidean space most imp ortant for hydrodynamics the Jacobi equa

tion was used by Rouchon to obtain the following information on the sectional

curvatures of the dieomorphism group

Definition Rou A divergencefree vector eld on M is said to b e a

perfect eddy if it is equal to the velocity eld of a solid rotating with a constant

angular velocity around a xed axis in particular the vorticity is constant Such

x DIFFEOMORPHISM GROUPS AND UNRELIABLE FORECASTS

z

y

x

Figure The velocity prole of the tradewind current on the sphere

a uid motion is p ossible if and only if the domain M admits an axis of symmetry

Theorem Rou If the velocity eld v t of a ow of an ideal incompress

ible uid l ling a domain M is a perfect eddy with constant vorticity then the

sectional curvature in every twodimensional direction in S VectM containing v

is nonnegative

If the velocity v t of an ideal uid ow is not a perfect eddy then for each time

t there always exist plane sections containing v t the velocity along the geodesic

where the sectional curvature is strictly negative

The result on nonnegativity of all the sectional curvatures holds also for rotations

of spheres of arbitrary dimension Mis

Remark One can exp ect that the negative curvature of the dieomorphism

group causes exp onential instability of geo desics ie ows of the ideal uid in the

same way as for a nitedimensional Lie group see eg computer simulations in

KHZ For instance on an ndimensional compact manifold with nonp ositive sec

tional curvatures the Jacobi equation along the uid motion with constant pressure

function always has an unbounded solution Mis

We emphasize that the instability discussed here is the exp onential instability

also called the Lagrangian instability of the motion of the uid not of its velocity

eld compare with Section I I The ab ove result shows that from a Lagrangian

p oint of view all solutions of the Euler equation in M R with the exception of

the p erfect eddy are unstable

IV DIFFERENTIAL GEOMETRY OF DIFFEOMORPHISM GROUPS

On the other hand a stationary ow can b e a Lyapunov stable solution of

the Euler equation while the corresp onding motion of the uid is exp onentially

unstable The reason is that a small p erturbation of the uid velocity eld can

induce an exp onential divergence of uid particles Then even for a wellpredicted

velocity eld the case of a stable solution of the Euler equation we cannot predict

the motion of the uid mass without a great loss of accuracy

Remark The curvature formulas for the dieomorphism groups S DiM

simplify drastically for a locally at or Euclidean manifold M ie for a Rie

mannian manifold allowing lo cal charts in which the Riemannian metric b ecomes

Euclidean Let p VectM S VectM b e the orthogonal pro jection of the

space of all smo oth vector elds onto their divergencefree parts where the or

thogonality is considered with resp ect to the L inner pro duct on VectM Let

q Id p VectM GradM b e the orthogonal pro jection onto the space of

the gradient vector elds

Consider some lo cal Euclidean co ordinates fx x g on M and assign to a

n

pair of vector elds u and v their covariant derivative in M

X X

v

i

A

u x r v

j u

x x

j i

j i

Theorem Luk Let M be locally Euclidean Then the sectional curvature

C for an orthonormal pair of divergencefree vector elds u v S VectM is

uv

C hq r v q r v i hq r u r v i

uv u u u v

Corollary Luk If the vector eld r u is divergence free ie q r u

u u

n

then the curvature is nonpositive for instance u is a simple harmonic on T

C hq r v q r v i

uv u u

Remark It is natural to describ e the curvature tensor for the threedimensional

ik x

torus T in the basis e of S DiT where e e k Z nfg An arbitrary

k k

P

velocity eld ux is represented as ux u e where the Fourier comp o

k k

k

nents satisfy k u divergence free and u u reality condition Then

k k k

according to NHK one has

hu e v e w e z e i

k k m m n n

v nz v mw u nz k u mw k

n m k n k m

jk mj j nj j mj jn k j

x DIFFEOMORPHISM GROUPS AND UNRELIABLE FORECASTS

All the sectional curvatures in the threedimensional subspace of the AB C ows

in S DiT see Section I I are equal to one and the same negative constant

ie the curvatures do not dep end on A B and C KNH

k

Fix a divergencefree vector eld v S VectT on the k dimensional at torus

k k

T The average of the sectional curvatures of all tangential planes in S VectT

containing v is characterized by an innitedimensional analogue of the normalized

Ricci curvature

Definition Let b e the LaplaceBeltrami op erator on vector elds from

k

S VectT and fe j i g b e its orthonormal ordered eigen

i i i

basis e e Dene the normalized Ricci curvature in the direction v by

i i i

N

X

C Ricv lim

v e

i

N

N

i

The normalized Ricci curvature in a given direction on a nitedimensional manifold

is the average of the sectional curvatures of all tangential planes containing the

direction Denition It diers from the classical Ricci curvature by the factor

of dimension of manifold and it makes sense as the dimension go es to innity

k

Theorem Luk For a divergencefree vector eld v S VectT on the

k

at torus T

p

k

Ricv k v k

2 k

L T

k

volT k k k

p

where is the positive square root of the minus Laplace operator on vector

elds

B Unreliabili ty of longterm weather predictions To apply the cur

vature calculations ab ove we make the following simplifying assumption The

atmosphere is a twodimensional homogeneous incompressible uid over the two

torus and the motion of the atmosphere is approximately a tradewind current

parallel to the equator of the torus and having a sinusoidal velocity prole

Though the twosphere is a b etter approximation for the earth than the two

torus the calculations carried out for a tradewind current over S in Luk

show the same order of magnitude for curvatures in b oth groups S DiT and

S DiS Hence the same conclusions on the characteristic path length and in

stability of ows hold in b oth cases

To estimate the curvature we consider the tradewind current with velocity

eld x y sin y on the torus T fx y mo d g Then Theorem

IV DIFFERENTIAL GEOMETRY OF DIFFEOMORPHISM GROUPS

shows that the curvature of the group S DiT in the planes containing with

the wave vector k varies within the limits S C where S

is the area of the torus However the lower limit here is obtained by a rather crude

estimate To make a rough estimate of the characteristic path length we take a

quarter of this limit as the value of the mean curvature C S There

exist many twodimensional directions with curvatures of approximately this size

Having agreed to start with this value C for the curvatures we obtain the

p

p

C S see Remark Recall that characteristic path length s

the characteristic path length is the average path length on which a small er

ror in the initial condition grows by the factor of e Note that along the group

S DiT the velocity of motion corresp onding to the tradewind current is

p

2 2

equal to k k S since the average square value of the sine is

L T

Hence the time it takes for our ow to travel the characteristic path length is equal

to

Now the fastest uid particles go a distance of after this time ie part of

the entire orbit around the torus Thus if we take our value of the mean curvature

then the error grows by e after the time of one orbit of the fastest particle

Taking kmhour as the maximal velocity of the tradewind current we get

hours for the time of orbit ie less than three weeks

Thus if at the initial moment the state of the weather was known with small

error then the order of magnitude of the error of prediction after n months would

b e

k n

where k log e

For example to predict the weather two months in advance we must have ve

more digits of accuracy than the prediction accuracy In practice this implies that

calculating the weather for such a p erio d is imp ossible

x Exterior geometry of the group of

volumepreserving dieomorphisms

The group S DiM of volumepreserving dieomorphisms of a Riemannian

n

manifold M can b e thought of as a subgroup of a larger ob ject the group DiM

of all dieomorphisms of M cf EM Just like its subgroup the larger group

is also equipp ed with a weak Riemannian metric which is however no longer

x EXTERIOR GEOMETRY OF THE GROUP OF DIFFEOMORPHISMS

rightinvariant

Z

hg g i x

g x

M

where VectM is the inner pro duct of and with resp ect to the

a

metric on M at the p oint a and g DiM

Viewing the group of volumepreserving dieomorphisms as a subgroup in the

group of all dieomorphisms of the manifold happ ens to b e quite fruitful for various

applications To some extent the bigger group is always atter than the subgroup

The source of many simplications lies in the following fact

Theorem Mis The components of the curvature tensor of the bigger

group Di M are the mean values of the curvature tensor components for the

Riemannian manifold M itself

Z

M

ux v xw x z xi x h hu v w z i

x

x

M

M

where is the curvature tensor of M at x M the volume form is dened by

x

the metric and u v w z VectM

Below we derive following Mis Shn Tod the second fundamental form of

the embedding of the curved subgroup S DiM DiM into the atter am

bient group Though not intrinsic in nature it gives a nice shortcut to calculations

of the curvatures

n

For simplicity let the manifold M b e the at ntorus T Represent a dieo

n

morphism g DiT close to the identity in the form g x x x

Proposition In the coordinates f g a C smal l neighborhood of the iden

n n

tity Id Di T of the group Di T equipped with the metric is isometri

n n

cally embedded in the Hilbert space H f L T R g

Proof of Proposition is a straightforward calculation 

Abusing notation we will denote by H the preHilb ert space of smo oth maps

n n

from the torus T to R equipp ed with the L inner pro duct Then a neighborho o d

n

of the identity of the group DiT is isometric with a neighborho o d of the origin

n n

in H The group S DiT of volumepreserving dieomorphisms of T will b e

viewed as a submanifold D of H Fig

n n n

g H Id D S DiT f L T R j det x

IV DIFFERENTIAL GEOMETRY OF DIFFEOMORPHISM GROUPS T

T0

pr (tv)

t2w

tv T0

H

Figure The embedding of the volumepreserving dieomorphisms

D S DiM into the group of all dieomorphisms H DiM

Definition The second fundamental form L at D of the embedding

D from the tangent space T D H to D H is a quadratic map L T D T

D H The value of the second fundamental form its orthogonal complement T

Lv v at a vector v T D is equal to the acceleration of a p oint moving by inertia

along D with initial velocity v see KN

In other words L measures the second derivative of the distance in H b e

tween the p oint tv moving in the tangent space T D with constant velocity v and

the orthogonal pro jection pr of this p oint to D

D

t

Lv v O t as t pr tv tv

D

The spaces T D and T D are more explicitly describ ed as follows

n n

T D S VectT fv VectT j div v g

n n n

T D GradT fw VectT j w rp for some p C T g

we have included in T D the divergencefree elds shifting the center of mass of

n

D consists of the gradients of all univalued functions T and hence T

n

Observe that for a vector eld v S VectT the transformation x x

n

tv x means that every p oint x T moves uniformly with velocity v x along

the straight line passing through x Such transformations are dieomorphisms for

smo oth v x and suciently small t

To demonstrate the machinery we conne ourselves for now to the case n

and give an alternative pro of of Theorem on curvatures S Di T in two

dimensional directions containing the sinusoidal ow on the torus

x EXTERIOR GEOMETRY OF THE GROUP OF DIFFEOMORPHISMS

Proof of Theorem A vector eld v S VectT can b e describ ed by

the corresp onding univalued stream or Hamiltonian function v sgrad

that is v x and v x

Theorem The value of the second fundamental form L at a vector eld v

is

Lv v r det Hess

where Hess is the Hessian matrix of the stream function of the eld v and

is the Green operator for the Laplace operator in the class of functions with

zero mean on T

Proof of Theorem The following evident relation shows how far the

transformation Id tv is from D ie from b eing volumepreserving

v

t div v t v v v v t v v v v det Id t

x

where v v x and the last equality is due to div v The transformation

ij i j

Id tv do es not b elong to D and it changes the standard volume element on T by

a term quadratic in t

Hence we have to adjust tv by adding to it a vector eld t w T D to suppress

the divergence of Id tv To compute the second fundamental form see

observe that its value at the vector v is Lv v w where w T D is dened

by the condition that the transform x x tv t w is volumepreserving mo dulo

O t

The dening relation on the eld w div w v v v v follows

immediately from the expansion

w v

t v v v v div w O t t det Id t

x x

D the vector eld w is a gradient w r Therefore From the denition of T

div w r and

Lv v w r v v v v

The introduction of the stream function for the eld v reduces the latter formula

to the required form 

The symmetric fundamental form Lu v can now b e obtained from the qua

dratic form Lv v via p olarization

Lu v Lu v u v Lu u Lv v

IV DIFFERENTIAL GEOMETRY OF DIFFEOMORPHISM GROUPS

Finally the sectional curvature C of the group D in the twodimensional direc

uv

tion spanned by any two orthonormal vectors u and v is according to the Gauss

Co dazzi formula Prop osition VI I in KN given by

C hLu u Lv v i hLu v Lu v i

uv

where h i is the inner pro duct in the Hilb ert space H cf also Theorem

ab ove For nonorthonormal vectors one has to normalize the curvature according

to formula

Example We will now calculate using the second fundamental form the

sectional curvature in the direction spanned by the vector elds u and v with the

stream functions cos ay and cos bx where the wave vectors of Corollary

are k a and b

One easily obtains that det Hess detHess while detHess

a b cos ay cos bx Then the application of is equivalent to the multiplication

of the ab ove function by a b Passing to the gradient one sees that Lu v

is the vector eld

a b

b sin bx cos ay a cos bx sin ay Lu v

a b x y

while b oth Lu u and Lv v vanish Note also that hu ui a S hv v i

b S where S is the area of the torus Finally evaluating the L norm of the

vector eld Lu v over the torus we come to the following formula for the sectional

curvature

hLu v Lu v i a b

C

uv

hu ui hv v i a b S

which is in a p erfect matching with

We leave it to the reader to check that the substitution of the vector elds u and

e e

k k

and x e with x x into the v with the stream functions

curvature formula gives an alternative pro of of Theorem in full generality 

Remark Bao and Ratiu BR have studied the total ly geodesic or asymp

totic directions on the Riemannian submanifold S DiM DiM ie those

directions in the tangent spaces T S DiM alternatively divergencefree vector

g

elds on M for which the second fundamental form L of S DiM relative to

n

DiM vanishes For an arbitrary manifold M they obtained an explicit de

scription of such directions g v T S DiM in the form of a certain rstorder

g

nonlinear partial dierential equation on v For the twodimensional case this equa

tion can b e rewritten as an equation for the stream function Assume for simplicity

that M and H M

x CONJUGATE POINTS IN DIFFEOMORPHISM GROUPS

Theorem BR For a twodimensional Riemannian manifold M the divergence

free vector eld g v T S Di M is a total ly geodesic direction on S Di M if

g

and only if the stream function of the eld v satises the degenerate Monge

Ampere equation

det Hess K m kr k

where m is the determinant of the metric on M in given coordinates fx x g Hess

is the Hessian matrix of in these coordinates and K is the Gaussian curvature

function on M

Their pap er also contains examples of manifolds for which the MongeAmpere

equation has or has no solutions see also BLR for a characterization of all

manifolds M for which the asymptotic directions are harmonic vector elds

Asymptotic directions on S DiM arise intrinsically in the context of a discrete

version of the Euler equation of an incompressible uid MVe A solution of the

discretized Euler equation is a recursive sequence of dieomorphisms The Monge

Ampere equation is the constraint on the initial condition ensuring that all the

dieomorphisms of the sequence preserve the area element on M

Remark Consider an equivalence relation on the dieomorphism group

Di M where two dieomorphisms are in the same class if they dier by a volume

preserving transformation We obtain a bration of Di M over the space of

densities ie the space of p ositive functions on M with the b er isomorphic

n

to the set of volumepreserving dieomorphisms S DiM Let the manifold M

n n

b e one of the following an ndimensional sphere S Lobachevsky space or

n

Euclidean space R According to SM GuseinZade there is no section of this

S DiM bundle over the space of densities that is invariant with resp ect to motions

n

of the corresp onding space M However there exists a unique connection in

n

this bundle that is invariant with resp ect to motions on M Parallel translation

in this connection is essentially describ ed in the pro of of the Moser theorem cf

Lemma I I I The case of a twodimensional sphere has interesting applications

in cartography

x Conjugate p oints in dieomorphism groups

Although in most of the twodimensional directions the sectional curvatures of

the dieomorphism group S DiT are negative in some directions the curvature is p ositive

IV DIFFERENTIAL GEOMETRY OF DIFFEOMORPHISM GROUPS

Example Arn In the algebra S VectT of all divergencefree vector

elds on the torus T fx y mo d g consider the plane spanned by the two

stream functions cospx y cospx y and cospx y cospx y

Then for the sectional curvature one has

h i

C as p

h ih i

It is tempting to conjecture by analogy with the nitedimensional case that

p ositivity of curvatures is related to the existence of the conjugate p oints on the

group S DiT

Definition A conjugate point of the initial p oint along a geo desic line

t t on a Riemannian manifold M is the p oint where the geo desic line

hits an innitely close geo desic starting from the same p oint The conjugate

p oints are ordered along a geo desic and the rst p oint is the place where the

geo desic line ceases to b e a lo cal minimum of the length functional

Strictly sp eaking one considers zero es of the rst variation rather than the actual

intersection of geo desics see eg KN

Theorem Mis Conjugate points exist on the geodesic in the group S Di T

emanating from the identity with velocity v sgrad dened by the stream function

cos x cos y

A segment of a geo desic line is no longer the shortest curve connecting its ends

if the segment contains an interior p oint conjugate to the initial p oint Fig

Indeed the dierence of lengths of a geo desic curve segment and of any C close

curve joining the same endp oints is of order the geo desic b eing an extremal

The length of a geodesic close to the initial one and connecting the initial p oint A

with its conjugate p oint C diers from the length of the initial geo desic b etween A

and C by a quantity of higher order The dierence b etween BD and BC CD

is of order Hence AD B is shorter than AC B

D

A C B

Figure A geo desic ceases to b e the length global minimum after

the rst conjugate p oint

x FINITENESS OF THE DIAMETER OF THE DIFFEOMORPHISM GROUP

If a segment of a geo desic contains k interior p oints conjugate to the initial p oint

then the quadratic form of the length second variation has k negative squares

n

Remark Mis Conjugate p oints can b e found on the group S DiT

n

where T is a at torus of arbitrary dimension n this is a simple corollary of the

twodimensional example ab ove

More examples are provided by geo desics on the group of areapreserving dif

feomorphisms of a surface of p ositive curvature example uniform rotation of the

standard twodimensional sphere see Section A

On the other hand those geo desics in S DiM that are also geo desics in DiM

have no conjugate p oints whenever M is a Riemannian manifold of nonp ositive

sectional curvature Mis Such geo desics have asymptotic directions on S DiM

and corresp ond to the solutions of the Euler equation in M with constant pressure

functions

Remark It has b een neither proved nor disproved that the Morse index of

a geo desic line corresp onding to a smo oth stationary ow is nite for any nite

p ortion of the geo desic It is interesting to consider whether the conjugate p oints

might accumulate in this situation

Note that the existence of a small geo desic segment near the initial p oint that is

free of conjugate p oints follows from the nondegeneracy of the geo desic exp onential

map at the initial p oint and in its neighborho o d One might also ask what the

shortest path is to a p oint on the geo desic that is separated from the initial p oint

by a p oint conjugate to it One hop es that the overall picture in this classical

situation is not sp oiled by the pathology of the absence of the shortest path b etween

sp ecial dieomorphisms discovered by Shnirelman in Shn see Section D

x Getting around the niteness of the diameter of

the group of volumepreserving dieomorphisms

Consider a volumepreserving dieomorphism of a b ounded domain In order

to reach the p osition prescrib ed by the dieomorphism every uid particle has to

move along some path in the domain The distance of this dieomorphism from

the identity is the averaged characteristic of the path lengths of the particles

It turns out that the diameter of the group of volumepreserving dieomorphisms

of a threedimensional ball is nite while for a twodimensional domain it is innite

This dierence is due to the fact that in three and more dimensions there is enough

1 This section was written by A Shnirelman

IV DIFFERENTIAL GEOMETRY OF DIFFEOMORPHISM GROUPS

space for particles to move to their nal places without hitting each other On the

other hand the motion of the particles in the plane might necessitate their rotation

ab out one another The latter phenomenon of braiding makes the system of

paths of particles necessarily long in spite of the b oundedness of the domain and

hence of the distances b etween the initial and nal p ositions of each particle

In this section we describ e some principal prop erties of the group of volume

preserving dieomorphisms D M S DiM as a metric space along with their

dynamical implications

A Interplay b etween the internal and external geometry of the dif

feomorphism group

n

Definition Let M b e a Riemannian manifold with volume element dx

n

Introduce a metric on the group D M S DiM as follows To any path

g t t t on the group D M we asso ciate its length

t

Z Z Z

t t

2 2

g x

t

t

2

2

dt dx fg g kg k dt

t t

L M

t

1

t

n

t t M

1 1

For two uid congurations f h D M we dene the distance b etween them

on D M as the inmum of the lengths of all paths connecting f and h

dist f h inf fg g

t

D M

fg gD M

t

g f g h

0 1

This denition makes D M into a metric space Now the diameter of D M is the

supremum of distances b etween its elements

diam D M sup dist f h

D M

f hD M

The metric on the group of volumepreserving dieomorphisms D M dened

here is induced by the rightinvariant metric on the group dened at the identity

by the kinetic energy of vector elds compare formula with Example I

We start with the study of the following three intimately related problems

A Diameter problem Is the diameter of the group D M of volumepreserving

dieomorphisms innite or nite In the latter case how can it b e estimated for a

n

given manifold M

n

Let M b e a b ounded domain in the Euclidean space R with the Euclidean

volume element dx In this case the dieomorphism group D M is naturally

n n n

embedded into the Hilb ert space L M R of vector functions on M This

embedding denes an isometry of D M with its weak Riemannian structure onto

its image equipp ed with the Riemannian metric induced from the Hilb ert space

x FINITENESS OF THE DIAMETER OF THE DIFFEOMORPHISM GROUP

2

Definition The standard distance dist b etween two dieomorphisms

L

n

f h D M L M R is the distance b etween them in the ambient Hilb ert

n

space L M R

2 2 n

dist f h kf hk

L L M R

B Relation of metrics What is the relation b etween the distance dist

D M

2

in the group D M dened ab ove and the standard distance dist pulled back to

L

n

D M directly from the space L M R

2

Evidently dist dist But do es there exist an estimate of dist g h

D M L D M

2

through dist g h In particular is it true that if two volumepreserving dieo

L

morphisms are close in the Hilb ert space then they can b e joined by a short path

within the group D M

C Shortest path Given two volumepreserving dieomorphisms do es there

exist a path connecting them in the group D M that has minimal length If so

it is a geo desic ie after an appropriate reparametrization it b ecomes a solution

n

of the Euler equation for an ideal uid in M Finding the shortest path b etween

two arbitrary uid congurations promises to b e a go o d metho d for constructing

uid ows

Remark Similar problems can b e p osed for dieomorphisms of an arbitrary

n n q

Riemannian manifold M if we rst isometrically embed M in R for some q

q

Such an embedding for which the Euclidean metric in R descends to the given

n

Riemannian metric on M exists by virtue of the Nash theorem Nash where one

n

can take q nn for a compact M and q nn n for a

n

noncompact M

B Diameter of the dieomorphism groups In what follows we con

n n

ne ourselves to the simplest domain M namely to a unit cub e M fx

n

x x R j x g We thus avoid the top ological complications due

n i

to the top ology of M

n

Theorem Shn For a unit ndimensional cube M where n the

diameter of the group of smooth volumepreserving dieomorphisms D M is nite

in the rightinvariant metric dist

D M

p

n

n

Theorem Shn diam D M

These theorems generalize to the case of an arbitrary simply connected manifold

M However the diameter can b ecome innite if the fundamental group of M is

not trivial ER The twodimensional case is completely dierent

IV DIFFERENTIAL GEOMETRY OF DIFFEOMORPHISM GROUPS

Theorem ER For an arbitrary manifold M of dimension n the

diameter of the group D M is innite

One can strengthen the latter result in the following direction

Definition A dieomorphism g M M of an arbitrary domain or a

Riemannian manifold M is called attainable if it can b e connected with the identity

dieomorphism Id by a piecewisesmo oth path g D M of nite length

t

n

Theorem Shn Let M be an ndimensional cube and n Then every

element of the group D M is attainable In the case n there are unattainable

dieomorphisms of the square M Moreover the unattainable dieomorphisms can

be chosen to be continuous up to the boundary M and identical on M

n

A dieomorphism g may b e unattainable if its b ehavior near the b oundary M

of M is complicated enough We will give an example of an unattainable dieo

morphism in Section A Note that only attainable dieomorphisms are physically

reasonable since the uid cannot reach an unattainable conguration in a nite

time

The statement ab ove allows one to sp ecify Theorem

Theorem For twodimensional M the subset of attainable dieo

morphisms in D M is of innite diameter

It is not known whether all the attainable dieomorphisms form a subgroup in

D M ie whether the inverse of an attainable dieomorphism is attainable It

is not always true that if a path fg g D M has nite length then the length

t

of the path fg g D M is nite The group D M splits into a continuum

t

of equivalence classes according to the following relation Two dieomorphisms are

in the same class if they can b e connected to each other by a path of nite length

Every equivalence class has innite diameter

The pro ofs of the twodimensional results are rather transparent and are dis

cussed in Sections AB Various approaches to the three and higher dimen

sional case are on the contrary all quite intricate and only the ideas are discussed

b elow

C Comparison of the metrics and completion of the group of dieo

morphisms The main dierence b etween the geometries of the groups of dieo

morphisms in two and three dimensions is based on the observation that for a long

path on D M which twists the particles in space there always exists a short

cut untwisting them by making use of the third co ordinate More precisely the

following estimate holds

x FINITENESS OF THE DIAMETER OF THE DIFFEOMORPHISM GROUP

Theorem Shn Given dimension n there exist constants C and

n

such that for every pair of volumepreserving dieomorphisms f h D M

of the unit cube

2

dist f h C dist f h

D M L

Theorem Shn The exponent in this inequality is not less than n

n n

This prop erty means that the embedding of D M into L M R n is

Holderregular Apparently it is far from b eing smo oth ie Certainly

the Holderprop erty Theorem implies the niteness of the diameter Theorem

No such estimate is true for n Namely for every pair of p ositive constants

c C there exists a dieomorphism g D M such that dist g Id C but

D M

2

dist g Id c This complements Theorem but requires of course a separate

L

pro of

Theorems and imply the following simple description of the completion

n

of the metric space D M in the case n

n

Corollary For n the completion of the group D M in the metric

n n n

dist coincides with the closure D M of the group in L M R

D M

n

Proof of Corollary Each Cauchy sequence fg g in D M with resp ect

i

n n

n

to the metrics dist is a Cauchy sequence in L M R and therefore it

D M

n n

converges to some element g L M R

n n n

Conversely if g b elongs to the closure of D M in L M R then there exists

n n n

a sequence of dieomorphisms fg g D M that converges to it in L M R

i

n

Therefore by virtue of Theorem fg g is a Cauchy sequence in D M and

i

n

thus g lies in the completion of D M 

n n

Theorem Shn The completion D M n of the group D M

consists of al l measurepreserving endomorphisms on M ie of such Lebesgue

n n n

measurable maps f M M that for every measurable subset M

mes f mes

n

The idea for a pro of of this theorem is as follows Divide the unit cub e M into

n

N equal small cub es having linear size N Consider the class D of piecewise

N

continuous mappings translating in a parallel way each small cub e into another

small cub e where is some p ermutation of the set of small cub es First one

proves that every measurepreserving map f M M may b e approximated with

n n

arbitrary accuracy in L M R by a p ermutation of small cub es for suciently

IV DIFFERENTIAL GEOMETRY OF DIFFEOMORPHISM GROUPS

large N In turn every such p ermutation may b e approximated in L by a smo oth

volumepreserving dieomorphism Conversely if a map g b elongs to the closure

n n

of the dieomorphism group g D M then g lim g g D M almost

i i

n

everywhere in M and hence g is a measurepreserving endomorphism see Shn

for more detail

In the twodimensional case the completion of D M in the metric dist is

D M

a prop er subset of the L closure D M no go o d description of this completion is

known

D The absence of the shortest path We will see b elow how the facts

presented so far in this section imply a negative answer to the question of existence

of the shortest path in the dieomorphism group

n

Theorem Shn For a unit cube M of dimension n there exist

a pair of volumepreserving dieomorphisms that cannot be connected within the

group D M by a shortest path ie for every path connecting the dieomorphisms

there always exists a shorter path

Thus the attractive variational approach to constructing solutions of the Euler

equations is not directly available in the hydrodynamical situation This is not to

say that the variational approach is wrong but merely that our understanding is

still incomplete and further work is required

Remark The pro of of Theorem is close in spirit to the Weierstrass

example of a variational problem having no solution Weierstrass prop osed his

example in his criticism of the use of the Dirichlet principle for proving the existence

of a solution of the Dirichlet problem for the Laplace equation

A B

Figure There is no smo oth shortest curve b etween A and B that

would b e orthogonal to the segment AB at the endp oints

This example illustrates that in some cases a functional cannot attain its in

mum Consider two p oints A and B in the plane We are lo oking for a smo oth

x FINITENESS OF THE DIAMETER OF THE DIFFEOMORPHISM GROUP

curve of minimal length connecting A and B such that its tangents at the p oints

A and B are orthogonal to the line AB Fig It is clear that if is dierent

from the segment AB then it may b e squeezed toward the line AB say by the

factor and this transformation reduces its length However if coincides with

the segment AB it do es not satisfy the b oundary conditions Thus the inmum

cannot b e attained within the class of admissible curves

Weierstrasss criticism encouraged Hilb ert to establish a solid foundation for the

Dirichlet variational principle

We pro ceed to describ e an example of a dieomorphism g D M of the three

dimensional cub e M that cannot b e connected to the identity dieomorphism Id

by a shortest path

Let x x z b e Cartesian co ordinates in R and let M f x x z g

Consider an arbitrary dieomorphism g D M of the form

g x z hx z

where h is an areapreserving dieomorphism of the square M and x x x

Theorem If

dist 3 Id g dist 2 Id h

D M D M

then Id D M cannot be connected with the dieomorphism g by a shortest path

in D M

Proof Rather than the length we shall estimate an equivalent quantity the

action along the paths

Definition The action along a path g t t t on the group of

t

n n

dieomorphisms D M of a Riemannian manifold M is the quantity

Z Z Z

t t

2 2

g x

t

t

2

kg k dt jfg g dy dt

2

t t

t

L M

1

t

n

t t M

1 1

The action and the length are related via the inequality

j t t

t t

2 2

Unlike the length fg g the action jfg g dep ends on the parametrization and

t t

t t

1 1

the equality in holds if and only if the parametrization is such that

Z

g x

t

dx const kg k

2

t

L M

t M

IV DIFFERENTIAL GEOMETRY OF DIFFEOMORPHISM GROUPS

This allows us in the sequel to pass freely from one notion to the other

Supp ose there exists a shortest path g connecting Id and g we shall construct

t

another path that has smaller length

First squeeze the ow g by a factor of along the z direction Instead of a family

t

of volumepreserving dieomorphisms of the threedimensional unit cub e M we

now have volumepreserving dieomorphisms of the parallelepip ed

P fx z M j z g

Now consider the new discontinuous ow g in the cub e M P P that is

t

the ab ove squeezed ow on each of the halves P and P see Fig Here P

is sp ecied by the condition z It is easy to see that the ow g is

t

incompressible in M and is in general discontinuous on the invariant plane

z Notice also that the ow g satises the same b oundary conditions as

t

g g Id g g

t

gt

P2

P1 gt

M3

Figure Each of the two parallelepip eds contains the former ow

in the cub e squeezed by a factor of

Compare the actions along the paths g and g Dene the horizontal j and

t t H

vertical j comp onents of the action j j j for g as follows

V H V t

Z Z

x x z t

j fg g k d x dz k dt

H t

t

3

M

Z Z

z x z t

dt k k d x dz j fg g

V t

t

3

M

and similarly for the path g Here g x z x z From the denition of g

t t t

we easily obtain

j fg g j fg g while j fg g j fg g

H t H t V t V t

x FINITENESS OF THE DIAMETER OF THE DIFFEOMORPHISM GROUP

Therefore the action along the pathg is smaller than that along the path g jfg g

t t t

jfg g if the vertical comp onent of the action is p ositive j fg g

t V t

The last condition j fg g follows from the assumption of the theo

V t

rem Indeed if the vertical comp onent of the action vanishes j fg g then

V t

z x z t t and the map x x x z t for any xed z and t is an area

preserving dieomorphism of the square M In this case the action along the ow

g is

t

Z Z Z

x x z t

dt dz d x jfg g

t

t

2

M

which implies that there is a value z such that

Z Z

x x z t

dt jfg g d x

t

t

2

M

However this is imp ossible since by the assumption the distance in D M from

Id to the dieomorphism x x x z t j hx is greater than the distance

t

in D M from Id to g the length of the shortest path g in D M connecting Id

t

and g

and we have constructed a discontinuous owg connecting Hence j fg g

t V t

Id and the dieomorphism g whose action is less than that of g jfg g jfg g

t t t

Now Theorem follows from the following lemma on a smo oth approximation

Lemma For every there exists a smooth ow D M t

t

that starts at the identity Id reaches an vicinity of g in the standard

2 3

L metric dist g and whose action approximates the action along

L M

j jf g the discontinuous path g jjfg g

t t t

To complete the pro of we replace the short but discontinuous path g by its

t

smo oth approximation It starts at the identity and ends up at L close to

t

g By Theorem there exists a path f in D M t connecting and

t

g and such that

ff g C

t

and hence the length ff g tends to together with It follows that the comp osite

t

path f has length not exceeding fg g C Finally observe that for

t t t

suciently small the comp osite path is shorter than g b ecause fg g fg g

t t t

This completes the pro of of Theorem mo dulo Lemma 

Proof of Theorem By virtue of Theorem for every C there is

2

a dieomorphism h of the square M such that dist Id h C On the other

D M

hand if g is a dieomorphism of the dimensional cub e M having the form

g x z hx z x M z

IV DIFFERENTIAL GEOMETRY OF DIFFEOMORPHISM GROUPS

3

then by Theorem dist Id g Hence if C this dieomorphism

D M

g cannot b e connected with Id by a shortest path This completes the pro of of

Theorem 

Proof of Lemma For a discontinuous owg we dene almost every

t

where its Eulerian velocity eld by v x t g g x t For a small let

t

t

M b e the set M with the neighborho o d of the b oundary M removed and

the dilation mappings M M Denote by v x t v x t the image of the

eld v under the dilations By setting v outside M we obtain an L vector

eld in the whole of R that is incompressible in the generalized sense ie the

vector eld v is L orthogonal to every gradient vector eld

Now dene the smo oth eld

Z

w x t v y t x y dy

3

R

as a convolution of the eld v with a mollier x x where x

R

C R and xdx The eld w x t has compact supp ort w x t

M for all t and as it converges to the eld v x t uniformly on every C

Moreover w v in compact set in M outside M and outside the plane z

L M This implies that for suciently small the smo oth ow obtained by

t

integrating the vector eld w satises the conditions of Lemma 

E Discrete ows The pro ofs of Theorems and are similar and

n

are based on the following discrete approximation of the group D M cf also

n n n

Lax Mos Split the cub e M R into N identical sub cub es Let M b e the

N

set of all these cub es Denote by D the group of all p ermutations of the set M

N N

this is a discrete analogue of the group D M of volumepreserving dieomorphisms

Two sub cub es M are called neighboring if they have a common n

N

dimensional face A p ermutation D is called elementary if each sub cub e

N

M is either not aected by or is a neighbor of A sequence of

N

elementary p ermutations is called a discrete ow the number k is called

k

its duration We say that the discrete ow connects the congurations

k

D if

N k k

The following purely combinatorial theorem is the cornerstone of the study It

can b e regarded as a discrete version of Theorem

Theorem For every dimension n there exists a constant C such that

n

for every N every two congurations D can be connected by a discrete ow

N

whose duration k is less than C N

k n

The pro of is tedious yet elementary see Shn

x FINITENESS OF THE DIAMETER OF THE DIFFEOMORPHISM GROUP

To formulate the discrete analogue of Theorem we dene the length of a

discrete ow not to b e confused with its duration

Definition Let b e a discrete ow in M Let each p ermutation

k N

n

take m sub cub es into neighboring ones and leave N m sub cub es in place

j j j

k

The length f g of the discrete ow is cf

j

k

n

X

m N

j

k

f g

j

N

j

The reasoning is transparent A p ermutation is approximated by a vector eld

j

n

of magnitude N supp orted on a set of volume m N Then the summands are

j

L norms of the p ermutations

The distance in D b etween congurations is dened as

N

k

min f g dist

j D

N

where min is taken over all discrete ows of arbitrary duration connecting and

The L distance b etween D is dened as

N

1

2

X

k k

2

dist

L

n

N

M

N

where k k is the distance in R b etween the centers of the corresp onding

small cub es

The following is the analogue of Theorem on the relation of metrics

Theorem For every dimension n there are constants C and

n n

such that for every N and every pair of permutations D

N

n

2

dist C dist

D n

L

N

The pro of is an inductive multistep construction of a short discrete ow con

necting two given L close discrete congurations see Shn The explicit con

struction in the threedimensional case n yields

F Outline of the pro ofs The pro of of Theorems and pro ceeds as

follows

n n

Let g D M with n We construct a path g D M t that

t

connects the identity and g g Id g g and such that

2

Id g fg g C dist

t

L

IV DIFFERENTIAL GEOMETRY OF DIFFEOMORPHISM GROUPS

First of all we prove that g can b e approximated by some p ermutation D

N

of small cub es for suciently large N Furthermore for every there exists a

discontinuous piecewisesmo oth ow connecting and g and such

n n

that Lf g More precisely for every the mapping M M is smo oth

in every small cub e M discontinuous on interfaces b etween neighboring cub es

N

and measurepreserving

Construct the short discrete ow connecting Id and in D and

k N

satisfying the conclusion of Theorem One can show that there exists a dis

continuous ow t that interpolates the ow at the moments

t k

t j k for all j k and has the same order

j j

tj k

of length

k

f g const f g

t j

Therefore the comp osition of the paths and is a discontinuous ow con

t t

necting Id and g and having a controllable length

The nal step is the smo othening of the latter ow provided by the following

n n n

Lemma Let g D M and let M M be a discontinuous

t

measurepreserving ow such that Id g and is smooth in every

t

smal l cube M If n then for every there exists a smooth ow

N

n n

g M M t with the same boundary conditions g Id g g as

t

and such that fg g f g

t t t

n

The ow g coincides with in every cub e fx j dist x g

t t R

for some small The most subtle p oint in extending the ow is to dene it on the

S

neighborho o d of all sub cub e faces the frothlike domain K n

M

N

Here we use the fact that the fundamental group of the domain K is trivial

K which is true if n Theorem and Theorem as a particular

case follows see Shn and analogous arguments in the pro of of Theorem I I I

in Section I I I for the details

The pro of of Theorem is similar but more complicated we refer to Shn

G Generalized ows We now return to the problem of nding the shortest

n

paths in the group of volumepreserving dieomorphisms D M We already know

that there exist pairs of dieomorphisms that cannot b e connected by a smo oth

ow of minimal length Is there however some wider class of ows say discon

tinuous or measurable where the minimum is always attainable This problem

has b een resolved by Y Brenier Bre He found a natural class of generalized

incompressible ows for which the variational problem is always solvable

x FINITENESS OF THE DIAMETER OF THE DIFFEOMORPHISM GROUP

The generalized ows GF are a farreaching generalization of the classical ows

where uid particles are not only allowed to move indep endently of each other but

also their tra jectories may meet each other The particles may split and collide

The only restrictions are that the density of particles remains constant all the time

and that the mean kinetic energy is nite The formal denition of the GF is

presented b elow

n

Let X C M b e the space of all parametrized continuous paths xt in

n n

M Fix a dieomorphism g D M

n

Definition A generalized ow GF in M connecting the dieomor

phisms Id and g is a probabilistic measure fdxg in the space X satisfying the

fol lowing conditions

n

i For every Lebesguemeasurable set A M and every t

fxt j xt Ag mes A

this may be called incompressibility

ii For almost al l paths xt the action along each of them is nite

Z

xt

k k dt jfxg

t

and so is the total action

Z

jfg jfxgfdxg

X

niteness of action

iii For almost al l paths xt the endpoints x and x are related by means

of the dieomorphism g x g x boundary condition

Thus a generalized ow fdxg can b e thought of as a random pro cess In general

this pro cess is neither Markov nor stationary This notion is very similar to the

notion of a p olymorphism app earing in the work of Neretin Ner Polymorphisms

arise as a natural domain for the extensions of representations of dieomorphism

groups

Every smo oth ow g D M may b e regarded as a generalized ow if we

t

asso ciate to g the measure fdxg such that for every measurable set Y X its

t

g

t

measure is equal to the measure of p oints whose tra jectories b elong to Y

g

t

n

Y mesfa M fg ag Y g

t

g t

IV DIFFERENTIAL GEOMETRY OF DIFFEOMORPHISM GROUPS

This measure is concentrated on the ndimensional set of tra jectories g a of the

t

ow g

t

Another example of a GF is a multiow that is a convex combination of GF s

corresp onding to smo oth ows In other words in the multiows dierent p ortions

of uid move p enetrating each other see Fig in dierent directions within the

same volume Generic GF s are much more complicated than multiows

n

Theorem Bre For every dieomorphism g D M there always ex

ists a generalized ow with the boundary conditions Id and g that realizes the

minimum of action

jfg min jf g

0

where the minimum is taken over generalized ows connecting the identity Id

and the dieomorphism g

Thus although in our example the inmum cannot b e assumed among smo oth

ows there exists a generalized ow minimizing the action as well as the length

see also Ro e

(a) (b)

Figure Trajectories of particles in the GF corresp onding to a

the ip of the interval and b the intervalexchange map

In fact Theorem is even more general since it applies equally to discontin

uous and to orientationchanging maps g In the latter case the minimizing GF is

esp ecially interesting b ecause the uid is b eing turned inside out No measurable

ow or even multiow can pro duce such a transformation These problems are

nontrivial even in the onedimensional case

x FINITENESS OF THE DIAMETER OF THE DIFFEOMORPHISM GROUP

Here are two b eautiful examples found by Brenier Let g x and g x b e

the transformations of the segment dened resp ectively as a ipop map

g x x x and as an intervalexchange map g

Figures a and b display the tra jectories of uid particles for the minimal

ows connecting Id with g and g resp ectively In the rst case each uid particle

splits at t into a continuum of tra jectories of smaller particles they move

indep endently pass through all the p oints of the segment and then coalesce at

t In the second case the GF is a multiow more precisely a ow For

more examples of exotic minimal GF s see Bre

An imp ortant question is to what extent these minimal GF s may b e regarded

as generalized solutions of the Euler equation The similarity b etween these gen

eralized ows and the true solutions extends very far For example for every

GF there exists a scalar function px t playing the role of pressure Bre such

that for almost every uid particle its acceleration at almost every x t is equal to

r p The minimal GF s are generalized solutions of the mass transp ort problem

x

of the socalled Kantorovich problem However their hydrodynamical meaning

is not yet completely understo o d

H Approximation of uid ows by generalized ones Generalized ows

have proved to b e a p owerful and exible to ol for studying the structure of the space

n

D M of volumepreserving dieomorphisms The key role here is played by the

following approximation theorem

n

Theorem Shn Let g D M n be a smooth volumepreserving

dieomorphism Then each generalized ow fdxg connecting Id and g can be ap

proximated together with the action by smooth incompressible ows There exists

k

connecting Id and g such that as a sequence of smooth incompressible ows g

t

k

i the measures weakconverge in X to the measure

(k)

g

t

k

ii the actions along g converge to the total action along fdxg

t

k

jfg jfg g

t

Here weakconvergence means that for every b ounded continuous functional

fxg on X

hfxg fdxgi hx fdxgi as k

(k)

g

t

We shall not present here the lengthy pro of of this theorem referring to Shn

instead An immediate consequence of this theorem and formula is the fol

n

lowing estimate on the distances in D M n

IV DIFFERENTIAL GEOMETRY OF DIFFEOMORPHISM GROUPS

n

Corollary If n then for every dieomorphism g D M

distId g inf jfg

where the inmum is taken over al l generalized ows connecting Id and g

n

Thus to estimate the distance b etween Id and g D M we may try to

construct a GF connecting Id and g and having the smallest p ossible action Then

Lemma guarantees a ma jorant for the distance

n

Example Theorem Let us estimate the diameter diam D M

of the group of volumepreserving dieomorphisms of the ndimensional unit cub e

An accurate computation of all the intermediate constants in the pro of of Theorem

for n yields diam D M which is very far from reality Here we

prove

p

n

n Theorem Shn If the dimension n then diam D M

Proof We use a construction close to that of Y Brenier who proved that all

uid congurations on the torus are attainable by GF s Bre

The required GF is constructed as follows At t every uid particle in

the cub e splits into a continuum of particles moving in all directions Having

originated at a p oint y this cloud of cubical form expands and at t it

n

lls out the whole cub e M with a constant density During the second half of the

motion t the cloud shrinks and collapses at t to the p oint g y

All clouds expand and shrink simultaneously and the overall density remains

constant for all t

n

More accurately supp ose that the cub e M is given by the inequalities jx j

i

i n Let b e a discrete group of motions generated by the reections

n n n

in the faces of M For each initial p oint y M and a velocity vector v M

n

we dene the corresp onding billiard tra jectory in M for the time t ie

the path x t M where

y v

n

x t y v t M

y v

n

Given a p oint y the endp oint mapping v x is a fold covering

y y v

n

of M and moreover is volumepreserving These billiard tra jectories are

y

tra jectories of the microparticles into which every initial p oint y splits At t

n

the microparticles ll M uniformly and after this moment they move along other

billiard tra jectories gathering at the p oint g y at the end All particles split and move indep endently in the same manner incompressibility is fullled automatically

x FINITENESS OF THE DIAMETER OF THE DIFFEOMORPHISM GROUP

n

Let M M M M b e copies of the cub e M with co ordinates y v z u re

y v z u

sp ectively Dene a set M M M M that consists of all fourtuples

y v z u

y v z u such that z g y and such that the endp oints of the corre

sp onding tra jectories coincide x x

y v z u

n

Denote by d dy dv the normed volume element on Then the required

n

GF is the following random pro cess in M with probability space d

x t t

y v

xt

x t t

g y u

where y v g y u The action of this GF is

Z Z

n n

v x dv dx jfg

M

v

By virtue of Corollary this implies that the distance b etween the identity

Id and the dieomorphism g which has b een chosen arbitrarily is ma jorated as

follows

q

p

distId g jfg n

n

Hence the diameter of the group D M has the same upp er b ound 

Analogous though much longer reasoning proves Theorem which mi

n

norates the Holder exp onent n for the embedding of D M into

n

n

L M R n see Shn Given g we construct explicitly the

n

GF satisfying that estimate Our constructions are p ossibly not optimal and a

natural question is to nd the b est p ossible estimate for the diameter and for the

exp onent Is the latter equal to or less than Both p ossibilities are interesting

n

I Existence of cut and conjugate p oints on dieomorphism groups One

more application of the techniques of generalized ows is the pro of of the existence

n

of cut p oints on the space D M cf Section

n

Definition Let g D M b e a geo desic tra jectory on the group of

t

dieomorphisms We call a p oint g on the tra jectory the rst cut of the initial

t

c

p oint g along g if the geo desic g has minimal length among all curves connecting

t t

g and g for all t and it ceases to minimize the length as so on as t

c c

ie for every t there exists a curve g connecting g and g whose length is

c

t

less than the length of the segment fg j t g We call a p oint g the rst

t t

c

local cut if it is the rst cut and the curve g may b e chosen arbitrarily close to the

t

geo desic segment fg j t g for every t

t c

IV DIFFERENTIAL GEOMETRY OF DIFFEOMORPHISM GROUPS

On a complete nitedimensional Riemannian manifold the cut p oint of a p oint

g comes no later than the rst conjugate p oint of g The example of a at torus

shows that one can have cut p oints but no conjugate p oints But in the nite

dimensional case the rst lo cal cut p oint is always a conjugate p oint so all cut

p oints on the torus are nonlo cal which is evident In the case of dieomorphism

groups the precise relationship b etween cut p oints and conjugate p oints has yet to

b e claried

In the twodimensional case n the conjugate p oints on certain geo desics

in D T were found by G Misiolek Mis see Section It is curious that for

n there are lo cal cut and probably conjugate p oints on every suciently

long geo desic curve This is a consequence of the following result

n

Theorem Shn Let fg j t T g D M be an arbitrary path on

t

the group and n If the length of the path exceeds the diameter of the group

T n n

fg g diam D M then there exists a path fg j t T g D M with the

t

t

same endpoints g and g that

T

such that i is uniformly close to g ie for every there exists a path g

t

t

for every t T and dist g g

t

D M

t

T T

ii has a smal ler length fg g fg g

t

t

n T

diam D M In other words if the geo desic segment g is long enough fg g

t t

with t T A shorter path can b e cho then there exists a lo cal cut p oint g

c t

c

sen arbitrarily close to the initial geo desic which on a complete nitedimensional

manifold would imply the existence of a conjugate p oint

n

Proof Let h t T b e a path in D M connecting g and g and such

t T

that

T T n

fg g fh g diam D M

t t

for some small Such a path exists by the denition of diameter

Assume that the parametrization of the paths g and h is chosen in such a

t t

manner that kg k const kh k const and hence we have the inequality

t t

T T

for the actions as well jfg g jfh g

t t

Let b e the GF s corresp onding to the classical ows g h Consider the

g h t t

t t

convex combination of the measures in the space X

g h g

t t t

for some Then the total action for the generalized ow is

h

t

T T T T

jfg jfg g jfh g jfg g

t t t

To return to the classical ows we use approximation Theorem It guar

antees that there exists a smo oth ow f t T connecting g and g and

t T

x INFINITE DIAMETER OF THE GROUP OF SYMPLECTOMORPHISMS

T T

weaklyapproximating the GF together with its action so that jff g jfg g

t t

The ow f certainly dep ends on and it is easy to see that for small the ow

t

f is L close to g

t t

2

dist f g C

t t

L

Hence for suciently small these two ows are close on the group by virtue of

n

Theorem dist f g for all t T This completes the pro of of

t t

D M

Theorem 

For other applications of the generalized ows and for more detail we refer to

Shn Shn

x Innite diameter of the group of Hamiltonian

dieomorphisms and symplectohydrodynamics

The picture changes drastically when we turn from the group of volumepreserving

dieomorphisms of three and higher dimensional manifolds to areapreserving

dieomorphisms of surfaces Practically none of the asp ects under consideration

in the preceding section such as metric prop erties and diameter of the group

existence of solutions for the variational problem of Cauchy and Dirichlet types

or completion of the group and description of attainable dieomorphisms can b e

literally transferred to this case It is natural to describ e the prop erties of the

groups of areapreserving dieomorphisms of surfaces in the more general setting

of dieomorphisms of arbitrary symplectic manifolds

Definition A symplectic manifold M is an evendimensional manifold

n

endowed with a nondegenerate closed dierential twoform M

A group of symplectomorphisms consists of all dieomorphisms g M M that

preserve the twoform ie g We will b e considering symplectomor

phisms b elonging to the identity connected comp onent in the symplectomorphism

group and particularly those symplectomorphisms that can b e obtained as the

timeone map of a Hamiltonian ow By the Hamiltonian ow we mean the ow

of a timedep endent Hamiltonian vector eld having a singlevalued Hamiltonian

function Such symplectic dieomorphisms of M are called Hamiltonian Denote

the group of Hamiltonian dieomorphisms by HamM and the corresp onding Lie

algebra of Hamiltonian vector elds by hamM

For a twodimensional manifold the symplectic twoform is an area form and

the group of area and mass centerpreserving dieomorphisms S Di M coin

IV DIFFERENTIAL GEOMETRY OF DIFFEOMORPHISM GROUPS

cides with the group HamM The role of the group HamM in plasma dynamics

is similar to that of the group S DiM in ideal uid dynamics

The study of geo desics of rightinvariant metrics on symplectomorphism groups

is an interesting and almost unexplored domain It might b e called symplecto

hydrodynamics and it is a rather natural generalization of twodimensional hy

dro dynamics The relation b ecomes even more transparent for complex or almost

complex manifolds where the metric h i is related to the symplectic structure

by means of the relation h i i

The symplectohydrodynamics in higher dimensions diers drastically from that

in dimension two For instance every b ounded domain on the plane can b e em

b edded in any other domain of larger area by a symplectomorphism ie by a

dieomorphism preserving the areas Already in dimension four this is not always

the case Even some ellipsoids in a symplectic space cannot b e embedded in a

ball of larger volume by a symplectomorphism Gro For example the ellipsoid

p q p q cannot b e sent into a ball p q p q R

2 2

a b

of bigger volume if R maxa b Moreover a symplectic camel a b ounded

domain in the symplectic fourdimensional space cannot go through the eye of a

needle a small hole in the threedimensional wall while in volumepreserving hy

dro dynamics such a p ercolation through an arbitrarily small hole is always p ossible

in any dimension

Thus the preservation of the symplectic structure of the phase space M by

the Hamiltonian phase ow implies some p eculiar restrictions on the resulting dif

feomorphisms making symplectomorphisms scarce among the volumepreserving

maps in dimensions Moreover the group of symplectomorphisms of a sym

plectic manifold is C closed in the group of all dieomorphisms of the manifold

ie in general a volumepreserving dieomorphism cannot b e approximated by

symplectic ones El Gro These restrictions might even imply some unexp ected

phenomena in statistical mechanics where in spite of the symplectic nature of the

problem one usually takes into account the rst integrals and volume preserva

tion only and freely p ermutes the particles of the phase space One may also hop e

that symplecto contacto conformo hydrodynamics will nd other physically

interesting applications In this section we will describ e a few results known in

symplectohydrodynamics

We will concentrate mostly on two main metrics with which the group HamM

can b e equipp ed The rst one is the rightinvariant metric which arises from the

kinetic energy and is resp onsible for hydrodynamic applications we follow ER

The second one is the biinvariant metric introduced in Hof and studied in EP

x INFINITE DIAMETER OF THE GROUP OF SYMPLECTOMORPHISMS

LaM which has turned out to b e a p owerful to ol in symplectic geometry and

top ology

n

A Rightinvariant metrics on symplectomorphisms Let M b e

a compact exact symplectic manifold This means that the symplectic form is a

dierential of a form d

Such a manifold neccessarily has a nonempty b oundary Otherwise the integral

Z Z

n n

d

M

n

would vanish which is imp ossible since the nform is a volume form on

M We x a Riemannian metric on M with the same volume element

p

Definition The rightinvariant L metric on HamM is determined by

p

the L norm p on Hamiltonian vector elds hamM at the identity of the

group for hydrodynamics one needs the L case corresp onding to the kinetic en

p

ergy of a uid Given a path fg j t g HamM we dene its L length

t

fg g by the formula

p t

p

Z Z Z

p

dg dg

t t

A

k fg g k k k dt dt

p t

p

dt dt

L

M

The length functional gives rise to the distance function dist on HamM by

p p

dist f g inf fg g

p p t

where the inmum is taken over all paths g joining g f and g g Finally

t

dene the diameter of the group by

diam HamM sup dist f g

p p

f g HamM

Theorem ER The diameter diam Ham M of the group of

p

p

Hamiltonian dieomorphisms HamM is innite in any rightinvariant L metric

Note that the strongest result is that for the L norm since

C M p

p

Remark Contrary to the volumepreserving case for dim M the

inniteness of the diameter of the symplectomorphism group has a lo cal nature

and it is not related to the top ology of the underlying manifold The source of the

IV DIFFERENTIAL GEOMETRY OF DIFFEOMORPHISM GROUPS

Figure Prole of the Hamiltonian function and the tra jectories of

the corresp onding ow which is a long path on the group HamB

distinction b etween these two cases is in the dierent top ologies of the corresp onding

groups of linear transformations The fundamental group of the group of linear

symplectic transformations Hamn is innite while it is nite in the volume

preserving case of SLn for n

To give an example of a long path in a group of Hamiltonian dieomorphisms

we consider the unit disk B R with the standard volume form Then such a

path on HamB is given for instance by the Hamiltonian ow with Hamiltonian

function H x y x y for a long enough p erio d of time Fig The nal

symplectomorphism is suciently far away from the identity dieomorphism Id

HamB since twodimensionality prevents highly twisted clusters of particles

to untwist via a short path

This allows one to present the following example of an unattainable dieomor

phism of the square Shn It corresp onds to the timeone map of the ow whose

Hamiltonian function is depicted in Fig It has hills of innitely increasing

height and with supp orts on a sequence of disks convergent to the b oundary of the

square

We will prove Theorem for the case of the L norm and the group of sym

n

plectomorphisms of the ball B that are xed on the b oundary B ER The

main ingredient of the pro of is the notion of the Calabi invariant

B Calabi invariant Consider the group Ham B of the Hamiltonian dif

x INFINITE DIAMETER OF THE GROUP OF SYMPLECTOMORPHISMS

Figure An unattainable dieomorphism of the square

n

feomorphisms of the ball B stationary on the sphere B b eing the dif

P

ferential of a form say the standard symplectic structure dp dq in

i i

n

R

Proposition Given a form on the ball B and a Hamiltonian dieo

morphism g Ham B xed on the sphere B there exists a unique function

n

h B R vanishing together with its gradient on B and such that

g dh

Proof The form g is closed d g d g d g

n

and hence exact in the ball B Therefore it is the dierential of some function

h The vanishing prop erty for h is provided by the condition that g is steady on

the b oundary 

Lemmadefinition The integral of the function h over the ball B does not

depend on the choice of the form satisfying d The Calabi invariant of the

Hamiltonian dieomorphism g is this integral divided by n

Z

n

h Cal g

n B

IV DIFFERENTIAL GEOMETRY OF DIFFEOMORPHISM GROUPS

Proof The form is dened mo dulo the dierential of a function Under the

change df the function h b ecomes h h f g f since the

n n

dierential commutes with pullbacks The forms g f and f have the same

integrals since the map g preserves the symplectic structure Then the integral

of h is preserved

Z Z Z Z

n n n n

h h f g f h Cal g

B B B B



Lemmadenition also holds for an arbitrary symplectomorphism g of the

ball xed on the b oundary However for Hamiltonian dieomorphisms there is

the following alternative description of the Calabi invariant

Let ham B b e the Lie algebra of the group Ham B of Hamiltonian dieo

morphisms of the ball It consists of the Hamiltonian vector elds vanishing on the

b oundary sphere B We shall identify it with the space of Hamiltonian functions

H normalized by the condition that H and its dierential b oth vanish on B

Notice that the denition of the Hamiltonian function corresp onding to a vector

eld from ham B is the innitesimal version of relation

Let g Ham B b e a Hamiltonian dieomorphism of the ndimensional ball

B Consider any path fg j t T g Id g T g g on the group Ham B

t

connecting the identity element with g The path may b e regarded as the ow

of a timedep endent Hamiltonian vector eld on B whose normalized Hamiltonian

n

function H dened on B T vanishes on B along with its dierential

t

n

Theorem Ca The integral of the Hamiltonian function H over B

t

T is equal to the Calabi invariant of the symplectomorphism g

T

Z Z

n

A

H dt Cal g

t

B

In particular this integral does not depend on the connecting path that is on the

choice of timedependent Hamiltonian H provided that the timeone map g T g

t

is xed

Geometrically the Calabi invariant is the volume in the n dimensional

n

space fx t z g B T R under the graph of the function x t z

H x t

x INFINITE DIAMETER OF THE GROUP OF SYMPLECTOMORPHISMS

Lemma The Calabi invariant Cal Ham B R is the group homomor

phism of the group of Hamiltonian dieomorphisms of B xed on the boundary

onto the real line

Proof of Lemma Let g g Ham B R b e two Hamiltonian dieomor

phisms and g g g We have to show that the corresp onding functions h and

h i vanishing on the b oundary B and determined by the condition

i

satisfy the relation

Z Z Z

n n n

h h h

The latter holds since

dh g g g g g dh g dh

and b ecause g preserves the symplectic form 

Remark The kernel of the Calabi homomorphism formed by the Hamil

tonian dieomorphisms whose Calabi invariant vanishes is a simple group the

commutant of Ham B see Ban The commutant of a group consists of the

pro ducts of commutators of the group elements

The fact that the Hamiltonian dieomorphisms whose Calabi invariant vanishes

form a connected normal subgroup is evident one can multiply the timedep endent

Hamiltonian by a constant The Lie algebra of this subgroup consists of the Hamil

tonian vector elds whose normalized Hamiltonian functions have zero integral

The fact that the subgroup consists of the pro ducts of commutators in Ham B

is similar to the following The Hamiltonian functions with vanishing integral are

representable as nite sums of Poisson brackets of functions from ham B

The subgroup of Hamiltonian dieomorphisms with vanishing Calabi invariant

has an innite diameter just as the ambient group of all Hamiltonian dieomor

phisms of the ball ER

More generally the Calabi invariant is the homomorphism of the group Symp B

of all symplectomorphisms of the ball xed on the b oundary to R We do not

know whether the group of symplectomorphisms Symp B of a ndimensional

ball xed on the b oundary and this normal subgroup fCalg g are simply

connected The group Symp B is known to b e contractible for n Mos and

for n Gro

Proof of Theorem Let fg Ham B j t T g Id g T

t

g g b e a path of Hamiltonian dieomorphisms with the timedep endent Hamiltonian

function H t

IV DIFFERENTIAL GEOMETRY OF DIFFEOMORPHISM GROUPS

Owing to the homomorphism prop erty of Cal it is enough to prove the relation

T

Z Z Z

n n

A

H dt h n

t

B B

for an innitesimally short p erio d of time T In other words we dierentiate

this relation in t at t and will prove the identity

Z Z

d

n n

h n H

dt

B B

Note that the time derivative at t of the lefthand side of formula for

the dieomorphism g is by denition minus the Lie derivative of the form

t

d

along the Hamiltonian vector eld v j g generated by the function H H

t t

dt

d

L d h

v

dt

We apply the homotopy formula L i d di see Section IB to the Lie

v v v

derivative L and use the denition of the Hamiltonian function dH i

v v

where d

L i d di dH i

v v v v

From formulas one nds the derivative ddt h

d

h H i

v

dt

Actually formulas allow one to reconstruct the derivative up to an additive

constant only which turns out to b e zero by virtue of the vanishing b oundary

conditions for all H v and h

Now Theorem follows from the following lemma

Lemma

Z Z

n n

i n H

v

B B

Proof of Lemma Owing to the prop erties of the inner derivative op erator i

v

we have

Z Z

n n

i i

v v

B B

Z Z

n n

i n dH n

v

B B

x INFINITE DIAMETER OF THE GROUP OF SYMPLECTOMORPHISMS

Moving the exterior derivative d to the form gives

Z Z Z

n n n

n d H n H n H

B B

B

since the function H vanishes on B This completes the pro of of Lemma and

Theorem 

Remark Theorem can b e reformulated as follows Dene the Cal

R

n

abi integral of a Hamiltonian function H as H This formula denes a linear

B

function or an exterior form on the Lie algebra ham B

The Calabi form on the corresp onding group Ham B is the rightinvariant dif

ferential form coinciding with the Calabi integral on the Lie algebra ham B The

Calabi form is actually a biinvariant ie b oth left and rightinvariant form

on the group of Hamiltonian dieomorphisms Ham B It immediately follows

from the fact that the Calabi integral dened on ham B is invariant under the

adjoint representation of this group Ham B in the corresp onding Lie algebra

ham B In turn the latter holds b ecause a symplectomorphism sends the Hamil

tonian vector eld of H to the Hamiltonian vector eld of the transp orted function

n

while preserving the form Hence the symplectomorphism action preserves the

integral

The Calabi invariant Cal g of a Hamiltonian dieomorphism g in Ham B is

the integral of the Calabi form along a path g in Ham B joining the identity

t

dieomorphism with g

Theorem The Calabi form is exact The integral depends only on the nal

point g and not on the connecting path

Although we have already proved the exactness of the Calabi form in slightly

dierent terms we present here a shortcut to prove its closedness It would im

ply exactness if we knew that the group of Hamiltonian dieomorphisms is simply

connected ie that every path connecting the identity with g in Ham B is ho

motopical or at least homological to any other Unfortunately we do not know

whether this is the case in all dimensions see Remark

Proof of closedness We start with a wellknown general fact

Lemma For any rightinvariant dierential form on a Lie group

d

IV DIFFERENTIAL GEOMETRY OF DIFFEOMORPHISM GROUPS

for every pair of vectors in the Lie algebra

This formula follows from the denition of the exterior dierential d see Section

IB Therefore the dierential of the Calabi form is minus the integral of the

Poisson bracket of two Hamiltonian functions

Lemma Let H be a Hamiltonian function dened on the ball B and con

stant on the boundary B Then the Poisson bracket of H with another Hamiltonian

function F has zero integral over B

Example For two smo oth functions F and H in a b ounded domain D of

the plane p q

Z

fF H g dp dq

D

provided that H is constant on D ie that the Hamiltonian eld of H is tangent

to D Indeed

Z Z Z Z

fF H g dp dq dF dH H dF H dF

D D D D

since H is constant on the b oundary D

Proof of Lemma The Poisson bracket fF H g is minus the derivative

of F along the Hamiltonian vector eld of H Consider the nform in B that is

n

the result of transp orting the nform F by the ow of H This ow leaves the

ball B invariant since H is constant on the b oundary B and the corresp onding

Hamiltonian ow is tangent to B

Then the integral of the Poisson bracket fF H g is equal to minus the time

derivative of the integral over B of this resulting form But this Hamiltonian ow

n

and hence preserves the integral Thus the time derivative of the preserves

integral of the transp orted form vanishes and so do es the integral of the Poisson

bracket 

Remark Similarly for any two smo oth functions on a closed compact

symplectic manifold the integral of their Poisson bracket vanishes Here one might

replace one of the functions by a closed nonexact dierential form it do es

not change the pro of Moreover every function on a connected closed symplectic

manifold whose integral vanishes can b e represented as a sum of Poisson brackets

of functions on this manifold Arn

The closedness of the Calabi form the invariance of integral under the

deformations of the path follows immediately from Lemmas and The

x INFINITE DIAMETER OF THE GROUP OF SYMPLECTOMORPHISMS

dierential of the Calabi form is the integral of minus the Poisson bracket of any

two functions from Symp B which always vanishes 

Now we are ready to prove the inniteness of the diameter of the symplectomor

phism group

n n

Proof of Theorem for Ham B Let the ball B b e equipp ed with

n

the standard symplectic structure and let denote the corresp onding volume

form The length of a path fg g in the rightinvariant metric on Ham B

t

g b eing the ow of a timedep endent Hamiltonian function H x joining the

t t

endp oints g Id and g g is given by

Z Z

dg x

t

A

k fg g k dt

t

dt

B

Z Z Z

A

krH g k dt krH k dt

t t t

L B

2

B

Then the desired estimate follows from the Poincareand Schwarz inequalities

Z Z

2 2

kH k dt krH k dt c fg g

t t t

L B L B

Z Z Z

1

H dt c Cal g kH k dt c c

t t

L B

B

Owing to the surjectivity of the map Cal Ham B R one can nd a Hamil

tonian dieomorphism see Remark with an arbitrarily large Calabi invariant

and therefore arbitrarily remote from the identity 

Analogous statements for the rightinvariant metric generated by the L norm

on vector elds hamM and for nonexact symplectomorphisms Theorem in

full generality require noticeably more work ER

C Biinvariant metrics and pseudometrics on the group of Hamilton

n

ian dieomorphisms The group Ham R of compactly supp orted Hamil

n

tonian dieomorphisms of the standard space R or the group of symplectomor

phisms of a ball that are stationary in a neighborho o d of its b oundary admits

interesting biinvariant metrics see Hof EP HZ LaM Plt Don The right

invariant metrics discussed ab ove are dened in terms of the norm of vector elds

n

which requires an additional ingredient a metric on R On the contrary the

biinvariant metrics are dened solely in terms of the Hamiltonian functions

IV DIFFERENTIAL GEOMETRY OF DIFFEOMORPHISM GROUPS

p n

Definition EP Any L norm p on the space C R of

compactly supp orted Hamiltonian functions assigns the length l to any smo oth

p

n

curve on the group HamM Given the Hamiltonian function H C R of a

t

ow from f to g we dene

Z

p 2n

kH k dt l f g

t p

L R

The length functional generates a pseudometric on the group HamM ie

p

a symmetric nonnegative function on HamM HamM ob eying the triangle

inequality

The pseudometrics are biinvariant This immediately follows from invariance

p

p

of the L norm under the adjoint group action The integral

Z

p

p

n

j H x j kH k

p 2n

L R

2n

R

p ersists under symplectic changes of the variable x More generally one can start

with an arbitrary symplectically invariant norm on the algebra ham M

In particular the distance Id f in the L pseudometric b etween any

n

Hamiltonian dieomorphism f HamR and the identity element Id reads

Z

sup j H x t j dt Id f inf

H

x

where the inmum is taken over all Hamiltonian functions H x t corresp onding

to ows starting at Id and ending at f By denition f g Id f g

This biinvariant pseudometric is equivalent to the one introduced by Hofer

Hof

Z

sup H x t inf H x t dt Id f inf

x

H

x

We use the notation in the sequel for b oth and

Theorem Hof The pseudometric is a genuine biinvariant metric

on HamM ie in addition to positivity and the triangle inequality the relation

f g implies that f g

Lalonde and McDu LaM showed that dened by the same formula

for any symplectic manifold M is a true metric on the group HamM They

x INFINITE DIAMETER OF THE GROUP OF SYMPLECTOMORPHISMS

used it to prove Gromovs nonsqueezing theorem in full generality for maps of

arbitrary symplectic manifolds into a symplectic cylinder

p

It turns out however that the limit case p is the only L norm on Hamil

tonians that generates a metric For p there are distinct symplecto

morphisms with vanishing distance b etween them EP The features of the

p

pseudometrics ab ove are deduced from sp ecial prop erties of the following sym

n

plectic invariant rst introduced by Hofer for subsets of R and called the dis

placement energy

Let b e a biinvariant pseudometric on the group of Hamiltonian dieomor

phisms HamM of an op en symplectic manifold M

Definition The displacement energy eA of a subset A M is the

pseudodistance from the identity map to the set of all symplectomorphisms that

push A away from itself eA inf f Id f g where the inmum is taken over

all f HamM such that f A A If there is no such f we set eA

Theorem EP Let be a biinvariant metric on HamM Then the

displacement energy of every open bounded subset A M is nonzero e A

For instance for a disk B R of radius R the displacement energy in Hofers

metric is R Hof Furthermore the displacement energy is nonzero for every

compact Lagrangian submanifold of M Che A submanifold L of a symplectic

n

manifold M is called Lagrangian if dim L n and the restriction of the

form to L vanishes

Proof of Theorem First notice that for the group commutator

of any two elements HamM one has

Id minId Id

This follows from the biinvariance of the metric and the triangle inequality

Cho ose arbitrary dieomorphisms HamM such that their supp orts are

in A and Id Then Theorem will b e proved with the following lemma

Lemma Id e A

Proof of Lemma Assume that a Hamiltonian dieomorphism h HamM

displaces A hA A Then the dieomorphism h h has the same

restriction to A as Hence Utilizing the biinvariance and the

inequality we have

Id Id Id h

Minimization over h completes the pro of 

IV DIFFERENTIAL GEOMETRY OF DIFFEOMORPHISM GROUPS

Corollary EP The pseudometric on HamM generated by the

p

p

L norm on C M is not a metric for p

H

Proof of Corollary Let B M b e an embedded ball and fg g a com

t

H

pactly supp orted Hamiltonian ow that pushes B away from itself g B B

M Introduce a new Hamilton This ow is generated by a function H C

ian function K t by smo othly cutting o H t outside a neighborho o d U M

t

H

of the moving b oundary g B see Fig t

K B g 1(B)

K( . ,t)

Figure Displacement of a ball with rotation

K

B The ows of K and H coincide when restricted to the b oundary B g

t

K H

B B B for every t and therefore g g

t

p

Shrinking the neighborho o ds U one can make the L norm of K t and hence

t

K

arbitrarily small for every p Thus the displacement the distance Id g

p

energy of B asso ciated to p vanishes and Theorem is applicable

p



Informally one can push a ball away from itself with an arbitrarily low energy

but the tradeo is an extremely fast rotation of the shifted ball near the b oundary

The function K t has steep slop es and hence a large gradient in a neighborho o d

of B

Let us conne ourselves to the case of compactly supp orted Hamiltonian dieo

n

morphisms in R

n

Remark EP For every dieomorphism HamR and p

one has Id If p then Id jCalj

p

The situation is completely dierent for the biinvariant metrics of L type We

refer the reader to HZ for an account of other p eculiar prop erties of symplecto morphism groups

x INFINITE DIAMETER OF THE GROUP OF SYMPLECTOMORPHISMS

In particular consider the embedding of the group of Hamiltonian dieomor

n

phisms say of the ball B R into the group of all compactly supp orted Hamil

n

tonian dieomorphisms of R

Theorem Sik The subgroup of al l Hamiltonian dieomorphisms Ham B

of a unit ball steady near the boundary has a nite diameter in Hofers metric

n

in the group of al l compactly supported Hamiltonian dieomorphisms of R

For the dieomorphisms with supp ort in the ball of radius R the diameter is

ma jorated by R HZ Furthermore for Hofers metric the following analogue

of the S DiM and L metric estimates holds cf Theorem

Theorem Hof The metric is continuous in the C topology for

n

every HamR

0

Id diameter of supp j Id j

C

where is given by

The diameter result changes drastically if we consider the group Ham B by

itself For every symplectic manifold M with b oundary its group of Hamiltonian

dieomorphisms Ham M stationary at the b oundary has an innite diameter in

Hofers metric compare with Theorem asserting the inniteness of the diameter

in the rightinvariant L metric This follows from the existence of symplectomor

phisms with arbitrarily large Calabi invariant which b ounds b elow Hofers metric

see eg ER A more subtle statement is that the diameter of the commutant

subgroup of Ham M the group of Hamiltonian dieomorphisms with zero Calabi

invariant is also innite see LaM

Problem Is the diameter of the group of Hamiltonian dieomorphisms

of the twodimensional sphere nite in Hofers metric

D Biinvariant indenite metric and action functional on the group

of volumepreserving dieomorphisms of a threefold Though divergence

free vector elds do not have analogues of Hamiltonian functions if the dimension

of the manifold is at least the group of volumepreserving dieomorphisms of a

simply connected threedimensional manifold can b e equipp ed with a biinvariant

yet indenite metric

Let M b e a simply connected compact threedimensional manifold equipp ed with

a volume form To dene a biinvariant metric one needs to x on the Lie algebra

S VectM of divergencefree vector elds a quadratic form that is invariant under

the adjoint action of the group S DiM ie under a change of variables preserving

IV DIFFERENTIAL GEOMETRY OF DIFFEOMORPHISM GROUPS

the volume form Such a form has already b een introduced in Chapter I I I Section

I I ID as the Hopf invariant or the helicity functional or the asymptotic linking

number of a divergencefree vector eld

Recall that we start with a divergencefree vector eld v on M and dene the

dierential twoform i which is exact on M The Hopf invariant Hv is the

v

indenite quadratic form

Z

Hv d

M

The group S DiM can b e equipp ed with a rightinvariant indenite nite sig

nature metric by right translations of H into every tangent space on the group

The quadratic form Hv is invariant under volumepreserving changes of vari

ables by virtue of the co ordinatefree denition of H It follows that the corre

sp onding indenite metric on the group S DiM is biinvariant This metric

has innite inertia indices due to the sp ectrum of the d or curl op

erator see Arn Smo The prop erties of this metric apart from those discussed

in Chapter I I I are still obscure

A similar phenomenon is encountered in symplectic top ology or symplectic

Morse theory see eg AG Arn Cha Vit Gro The action functional on the

space of contractible lo ops in a symplectic manifold also has inertia indices

Remark For a nonsimply connected threemanifold M equipp ed with

a volume form the denition of the helicity invariant can b e extended to null

homologous vector elds ie the elds b elonging to the image of the curl op erator

Z

i d i Hv w

v w

M

Proposition The nul lhomologous vector elds form a subalgebra of the

Lie algebra of divergencefree vector elds on M

Proof For any two divergencefree vector elds v and w on M their commu

tator fv w g is nullhomologous i d i i 

v w

fv w g

Corollary The subgroup of volumepreserving dieomorphisms of M

corresponding to the subalgebra of nul lhomologous vector elds is endowed with a

biinvariant nite signature metric

The subalgebra of nullhomologous vector elds is also a Lie ideal in the ambient

Lie algebra of divergencefree elds Moreover the nullhomologous vector elds

form the commutant ie the space spanned by all nite sums of commutators

of elements of the Lie algebra of all divergencefree vector elds on an arbitrary

x INFINITE DIAMETER OF THE GROUP OF SYMPLECTOMORPHISMS

n

compact connected manifold M with a volume form Arn see also Ban for the

symplectic case

Remark Consider a divergencefree vector eld on a threedimensional

manifold that is exact and has a vectorpotential We can asso ciate to this eld

some kind of Morse complex by the following construction Asso ciate to a closed

curve in the manifold the integral of the vectorpotential along this curve if the

curve is homologous to zero it is the ux of the initial eld through a Seifert surface

b ounded by our curve

We have dened a function on the space of curves The critical p oints of this

function are the closed tra jectories of the initial eld Indeed if the eld is not

tangent to the curve somewhere its ux through the small transverse area would

b e prop ortional to the area and the rst variation cannot vanish

The p ositive and negative inertia indices of the second variation of this functional

are b oth innite Indeed in the particular case of a vertical eld in a manifold

b ered into circles over a surface our functional is the oriented area of the pro jection

curve The latter is exactly the nonp erturb ed functional of the RabinowitzConley

Zehnder theory see HZ

From this theory we know that the inniteness of b oth indices is not an obstacle

to the application of variational principles We may therefore hop e that the study

of the Morse theory of our functional might provide some interesting invariants of

the divergencefree vector eld In hydrodynamical terms these would b e invariants

of the class of isovorticed elds that is of coadjoint orbits of the volumepreserving

dieomorphism group

The Morse index of a closed tra jectory changes when the tra jectory collides with

another one that is when a Flo quet multiplier is equal to For the nfold covering

of the tra jectory the index changes when the Flo quet multiplier traveling along the

th

unit circle crosses an n ro ot of unity Thus one may hop e to have a rather full

picture of the Morse complex at least for the curves in the total space of a circle

bundle that are suciently close to the b ers