数理解析研究所講究録
第 769 巻 1991 年 62-73

63

Geodesics and Curvature of a Group of Diffeomorphisms
and Motion of an Ideal Fluid
F. Nakamura, Y. Hattori and T. Kambe

Depar.tment of Physics, University of Tokyo, Bunkyo-ku, Tokyo 113, Japan

Abstract

Motion of an ideal fluid is represented as geodesics on the group of all volume
preserving diffeomorphisms. Explicit form of the geodesic equation is presented forthe fluid flow on a three-torus. Riemannian connection, cornmutator and curvaturetensor are given explicitly and applied to a couple of simple flows with Beltrami
property. It is found that the curvature is non-positive for the section of two ABC
flows with different values of the constants ( $A,$

$B$

and C). The present study is anextension of the Arnold’s results in two dimensional case to three dimensional fluid.
motions.

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We are concerned with a method to connect the problem of hydrodynamics ofan ideal (incompressible and inviscid) fluid with a problem of finding geodesics on
the group of $aU$ volume preserving diffeomorphisms. The fact that this group is

the appropriate configuration space for the hydrodynamics of an ideal fluid was firstremarked by Arnold.’ Under the restriction of two dimensional flows on the torus$T^{2}=R^{2}/(2\pi Z)^{2}$( : space of real numbers;
$R$

$Z$

: set of integers), explicit formulas forthe commutator, inner product, riemannian connection and geodesic equation arepresented. These formulas $aUow$ us to calculate the riemannian curvature tensors on
the geodesics of any two-dimensional cross-section. By the Jacobi equation in the
differential geometry, the stability of the geodesics is determined by the curvatureson them. By the stability it is meant here kinematically how the difference of particle
positions between two flows of different initial conditions develops with time. Neg-

ative curvature leads to enhanced growth of the deviation in initial conditions withtime. (This problem is different from that of the dynamical stability of the velocity
field.)In this paper, we present an explicit form of the geodesic equation for the motionof an ideal fluid on a three-torus$T^{3}$. It is possible to determine an expression of

the riemannian curvature of the group of volume preserving diffeomorphisms. It

is interesting to find that the curvature tuns out to be non-positive for the two-

dimensional section consisting of a particular vector (velocity) field ( $i.e$ . a simple
flow with Beltrami property) and a general vector field.

Consider a flow of an ideal fluid of uniform density in a
$M$

$n$

dimensional flow domain(a bounded riemannian manifold without boundary). The present analysis is

aimed at the the flows in three dimensional space although the formulation is valid
for

$n$

dimensional flows. The manifold$M$

is provided with the metric given by the

inner product $(X\cdot Y)$ and the covariant derivative$\nabla_{X}Y$(or riemannian connection forthis metric) at any point $x\in M$ for any two vector fields $X$ an $dY$ on M. The covariant

65

derivative is a differentiation of the vector field
velocity$Y$

in the direction of $X$ . When theof fluid motion is given for $y\in M$ and $t\in R$ , the particle motion $y(t)$ is$v_{t}(y)$described by the ordinary differential equation, $dy/dt=v_{t}(y)$ with $y(0)=x\in M$ ,whereis the initial position of the particle. The solution is written in the form:$x$$y=g_{t}(x)$, which describes a smooth curve in$M$starting from$x$

, usually called

Lagrangian particle path.
For each , the mapping$t$

particle of the fluid in$g_{t}$: $Marrow M$ is an auto-diffeomorphism carrying every

from the place it was at time

$M$

$0$

to the place it is at time ,$t$and in the other sense for the time parameter in an open interval$1\subset R,$$t$$g_{t}$

describes

time development of the configuration of particles, caUed a flow. In case that$v_{t}(x)$is divergence-free, the diffeomorphism is volume preserving. The volume preserving

diffeomorphisms fromas$\mathcal{D}_{v}(M)$(shortly$\mathcal{D}_{v}$$M$

to itself form an infinite dimensional group, which is denoted), and the flow

$g_{t}$

is a curve in the group

Here we make some remarks about the infinite Lie group&Marsden.2 (i) For two elements

$f$

and

$g$

$f$

tangent space$T_{e}\mathcal{D}_{v}$$g$according to Ebin$e$is the identity map of M.

. (ii) Rightis defined by $R_{9}f=fog$ . (iii) The Lie algebra, . theis the inverse map oftranslation by an element$\mathcal{D}_{v}$.

in the group, the operation $fg$ is definedby $fog,$ $i.e$ . $fog(x)=f(g(x))$ for $x\in M$ . Unit element
Inverse element of$\mathcal{D}_{v}$$f$itself, so we denote it as$f^{-1}$$i.e$of the group at the identity , is a set of all divergence-free vector.

$e$fields on M. (iv) Tangent space $T_{f}D$ is a set of $aU$ divergence-free vector fields along
$f,$

$i.e$

. consisting of every vector fieldright translation of a vector$V_{f}\in T_{f}D_{v}$$V$such that $Vof^{-1}$ is divergence-free. The

by $g\in D_{v}$ is defined by $V_{f}og\in T_{fog}D_{v}$ . We

use the same notation of the right translation of a vector as that of an element of$\mathcal{D}_{v}$

.

$(v)$

The initial position$x$of all the particles is usuaUy taken as the Lagrangian(particle) coordinates. The flowat each$t$

is a tangent vector

fixed point$y$in the space$M$$g_{t}$

starts at$\dot{g}_{t}\in T_{9t}D_{v}$$e$. The fluid vclocity with respect to. To get the Eulerian velocity field$v_{t}(y)$

$x$

at a

, we must carry this vector to the tangent space at the

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identity . This can be done by the right translation$e$$\dot{g}_{t}(x)=v_{t}(g_{t}(x))$. (vi) The group$\mathcal{D}_{v}$$v_{t}=\dot{g}_{t}og_{t^{-1}}$, or equivalently

admits a riemannian metric

$<$

,

$>$

and

riemannian (Levi-Civita) connection V.
The metric for two tangent vectors $X,$

$Y\in T_{f}\mathcal{D}_{v}$is defiried by(1)$<X,$ $Y>|_{f}=\int(X(x)\cdot Y(x))|_{f(x)}dx$ ,where $dx$ is the (riemannian) volume element on
evaluated at a point

$Q$

$f$

$M$

, and

denotes the quantity$Q|_{f}$. Due to the volume preserving nature of the mapping, wehave the right invariant property: $<Xog,$ $Yog>|_{fog}=<X,$ $Y>|_{f}$ , for $f,$

Concerning the connection$\mathcal{D}_{v},$$\tilde{\nabla}_{X}Y$$g\in \mathcal{D}_{v}$.

, first we consider right invariant vector fields $on$

$i.e.\tilde{X}(f)=Xof,\tilde{Y}(f)=Yof$ for$f\in \mathcal{D}_{v}$where $X,$

$Y\in T_{\epsilon}\mathcal{D}_{v}$(divergence-free

vector fields on M). The connection V determines a new right invariant vector fieldon$\tilde{\nabla}_{X}\tilde{Y}$$D_{v}$

called covariant derivative of$\tilde{Y}$in the direction ofワ$\tilde{\nabla}_{\tilde{X}}$where$\nabla$is the riemannian connection ofderivation$X,$$\nabla_{X}Y$$M$

defined by,(2), introduced previously. The covariant

is not necessarily divergence-free for two divergence-free vector fields

on M. The operation $P[$

$Y$$|_{f}=P[\nabla_{X}Y]of$$\overline{X}$denotes the projection to divergence-free part on M.

$]$

It is shown2 that the operation (??) gives the right invariant riemannian connectionon$\mathcal{D}_{v}$associated with the riemannian metric (??).

Next we take a curveis, the flow

$g_{t}$$g_{t}$

inis generated by$\mathcal{D}_{v}$$v_{t}$

which satisfies $\dot{g}_{l}\equiv dg_{t}/dt=v_{t}og_{t}=v_{t}(g_{t})$ , that

. Let$\tilde{X}_{t}$be a vector field alongright translation of a time-dependent vector field$X_{t}$

on

$M$

:

$g_{t}$

which is given by$\tilde{X}_{t}=X_{t}og_{t}$. Then theconnection determines a new vector field called the covariant derivative of$g_{t}$$X_{t}$

along,$\ovalbox{\tt\small REJECT}_{X_{t}^{\ovalbox{\tt\small REJECT}}/dt}=(\partial X_{t}/\partial t+P[\nabla_{v_{t}}X_{t}])og_{t}$

Since
term

$\tilde{D}/dt$.

(3)

is the differentiation with respect to the parameter , we have to add the$\partial X_{t}/\partial t$

$t$

.

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The operations {X,$\tilde{Y}$}$arrow\tilde{\nabla}_{\overline{X}}\tilde{Y}$and$\tilde{X}_{t}arrow\tilde{D}X_{t}/dt$

is characterized uniquelyby the axioms in the riemannian geometry. 3,4 In particular, a vector field$g_{t}$$\tilde{X}_{t}$alongis said to be parallel if the covariant derivative is identically zero. The curve

is a geodesic when its tangent vector$\dot{g}_{C}$is parallel along itself (the curve$g_{\ell}$). In$g_{t}$$\mathcal{D}_{v}$endowed with the connection V, the geodesics are defined by the equation,$\tilde{D}\dot{g}_{t}/dt=0$

The principle of least$\cdot action$.

(4)

asserts that the motion of an ideal fluid is a geodesic

with the (weak) riemannian structure (??) and riemannian connection (??). In fact,
the action$I$

is defined as$I= \int<\dot{g}_{t},\dot{g}_{t}>|_{g\ell}dt=\int dt\int(v_{t}(x)\cdot v_{t}(x))|_{x}dx$,

and the variational problem leads to the equation (??).2

Using the right invariant property of the metric $<,$$>$

, and identifying$X_{t}$

as$v_{t}$in (??), we get the following equation,$\tilde{D}\dot{g}_{t}/dt=(\partial v_{t}/\partial t+P[\nabla_{v_{\ell}}v_{t}])og_{t}=0$

For the covariant derivative$\nabla_{v\ell}v_{t}$

$p$

(5)

, we have the orthogonal decomposition,2$\nabla_{v\ell}v_{t}|_{c}=\{P[\nabla_{v\ell}v_{t}]-gradp\}|_{c}$where the function

.

,

(6)

is a smooth scalar function on M. Thus, the right translationof equation (??) leads to,$\partial v_{t}/\partial t+P[\nabla_{v_{t}}v_{t}]=0$

or in view of (??),$\partial v_{t}/\partial t+\nabla_{v_{t}}v_{t}=-gradp$(7). This is a generalized expression of theEuler equation for an ideal fluid on riemannian manifold$M$

without boundary. It isshown that Eq. (??) holds in most general cases with boundary.2 For a flat cartesian

space, the covariant derivative reduces to the form$\nabla_{v}v=(v\cdot grad)v$

,

(8)

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for the velocity fieldwith cartesian components$v$

at $x=(x:)$ . Thus in terms of$(v_{i})$the hydrodynamic notations for $n=3$ , we have recovered the Euler equation,$\partial v/\partial t+(v\cdot grad)v=-gradp$

where$grad=(\partial/\partial x_{i}),$$(i=1,2,3)$ and

$p$

,

(9)

$divv=0$is the pressure divided by the uniform fluid

density.
In the Lie algebra of the group$D_{v}$, the commutator $[, ]$ , is defined for two

divergence-free vector fields $X$ , Yas(10)$[X, Y]_{*}=\tilde{\nabla}_{X}Y-\tilde{\nabla}_{Y}X$The right invariant property of the metric and connection allows us to makecorresponding calculation at $e\in D_{v}$ . From now on we write

Given divergence-free vector fields $X,$

$Y,$

$Z$

$\tilde{\nabla}$

instead of $P[\nabla ]$ .

and $W$ , a new divergence-free vector

field $\tilde{R}(X,Y)Z$ called the curvature tensor is defined by
一\nabla x$\tilde{R}(X, Y)Z=$and then the curvature$\overline{R}_{XYZW}$\tilde コワ$\tilde{\nabla}_{Y}Z+\tilde{\nabla}_{Y}$

,

(11)

is given by$\tilde{R}_{XYZW}=<\tilde{R}(X, Y)Z,$The sectional curvature for the section$\tilde{K}(\sigma)=<\tilde{R}(X, Y)X,$$xZ+\tilde{\nabla}_{[X.Y]}.Z$$\sigma\subset T_{\epsilon}\mathcal{D}_{v}$

.

$W>$

(12)

spanned by $X$ and $Y$ is$Y>/(<X, X><Y, Y>-<X, Y>^{2})$where the denominator is the square of the area of parallelogram spanned by $X$ and
Y. As we mentioned before, the stability of a geodesic
field$A_{t}$

along$g_{t}$. The norm

between two geodesics

origin$e$$g_{t}^{0}$$|A_{t}|\equiv\sqrt{<A_{t},A_{\ell}>}$and$g_{t^{l}}$$g_{t}$

is determined by the Jacobi

denotes the evolution of distance

per unit variation of$s$which start from the samewith different initial tangent vectors $X$ and $X+sY$ respectively:5$|A_{t}|=| Y|t-\frac{1}{6}\tilde{K}(\sigma)t^{3}+O(t^{4})$

(13)

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Thus negativemeans enhanced deviation of two geodesics in the section$\tilde{K}(\sigma)$$\sigma$spanned by $X$ and Y.
We will now investigate the three-dimensional fluid motion on the flat three-torus. $x=\{(x_{1}, x_{2},x_{3});mod 2\pi\}$ for$T^{3}=R^{3}/(2\pi Z)^{3},$$i.e$the group. Note that this$\mathcal{D}_{v}(T^{3})$$T^{3}$$T^{3}$, and the curvature ofis actually bounded manifold without boundary.

The elements of the Lie algebra of the group
vector fields on$x\in T^{3}$$\mathcal{D}_{v}(T^{3})$can be thought of as real periodic

with the divergence-free property. Such a periodic fields are

represented by the real part of corresponding complex Fourier forms.

The Fourier base$e^{ik\cdot x}$

is denoted by$e_{k}$, where $k=(k_{i})$ for $i=1,2,3$ . We$l$now complexify the Lie algebra, inner product, commutator and the riemannian

connection and curvature tensor, so that all these functions become linear (or multilinear) in the complex vector space of the complexified Lie algebra. The functions$e_{k}(k\in Z^{3}, k\neq 0)$

form a basis of this vector space. The velocity field is representedas$v_{t}(x) \equiv u(t,x)=\sum_{k}u_{k}(t)e_{k}$where $u(t, x)$ and$u_{k}(t)$

,

(14)

have three components, the latter being written as$u^{i}(k)$toooccasionally $(i=1,2,3)$ . The Fourier components must satisfy the two properties,$(k\cdot u_{k})=0$

,

(15)

$u_{-k}=u_{k}^{*}$

to represent the solenoidal and real conditions, respectively. Where the asterisk

denotes the complex conjugate. It should be noted that$u_{k}$$*$

has two independent

polarizations consistent with the first condition.
Let us take four vector fields satisfying the conditions (??) :$z_{n}e_{n}$$u_{k}e_{k},$$v_{1}e_{1},$$w_{m}e_{m}$,. Then we have the following expressions. From (??), the inner product is$<u_{k}e_{k},$$v_{1}e_{1}>=(2\pi)^{3}(u_{k}\cdot v_{1})\delta_{0,k+1}$,

which is non-zero only for $k+1=0.$ Using (??) and operating the projection$\cdot$

(16)
$P$

, the

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covariant derivative is$\tilde{\nabla}_{u_{k^{\epsilon}k}}v_{1}e_{1}$$=$$i( u_{k}\cdot 1)\frac{k+1}{|k+1|}\cross(v_{1}\cross\frac{k+1}{|k+1|})e_{k+1}$

.

(17)

The equation (??) gives the commutator:$[u_{k}e_{k}, v_{1}e_{1}]_{*}=i((u_{k}\cdot 1)v_{1}-(v_{1}\cdot k)u_{k})e_{k+1}$

.

(18)

From the definitions (??) and (??), the curvature tensor is$\tilde{R}_{klmn}=<\tilde{R}(u_{k}e_{k}, v_{1}e_{1})w_{m}e_{m},$$z_{n}e_{n}>$. 1)$(u_{k}\cdotn)(z_{n}\cdot k)|n+k|)$$=(2 \pi)^{3}(\frac{(u_{k}\cdot m)(w_{m}\cdot k)(v_{1}\cdot n)(z_{n}\cdot 1)}{|k+m||l+n|}-\frac{(v_{1}\cdotm)(w_{m}}{|1+m|}$

The curvature$k,$ $1,$ $m,$ $n$$\tilde{R}_{klmn}$

takes nonzero value only for $k+1+m+n=O$ and nonzero. (The terms with zero denominator should be deleted.) An analysis forthe distance of each particle convected by two different velocity fields of$T^{2}$, whichis related to the Jacobi field (??), has been carried out by $flattori^{6}$ and found to beconsistent with (??). This analysis can be extended to$T^{3}$

without difficulty.

The geodesic equation (??) reduces to$\sum_{k}\frac{\partial u^{m}(k)}{\partial t}+i\sum_{p+q=k}\sum_{j,l}k_{j}(5_{ml}-\frac{k_{m}k_{l}}{k^{2}})u^{j}(p)u^{l}(q)=0$.

by using (??) where$\delta_{ij}$is Kronecker’s delta. It is interesting to find that this isexactly identical to the Fourier representations of the Euler equation, or the Navier-Stokes equation when the viscosity coefficient$\nu$vanishes.7

In two-dimensional case, we may take $k=(k_{1}, k_{2},0)$ and $u_{k}=i(k_{2}, -k_{1},0)$ ,and similar expressions for the other vectors. Then it can be readily shown that ourformulas reduce to those of Arnold.’

As an application, we consider a flow with Beltrami property, that is, we assume
that the velocity field $U_{p}=u_{p}e_{p}+u_{-p}e_{-p}$ , $i.e$ .

$2{\rm Re}[u_{p}e^{ip\cdot x}]$, satisfies the condition,

71

curl $U_{p}=\lambda U_{P}$ for a parameter$\lambda^{2}=|p|^{2}$$\lambda\in$. It is readily shown that

R. This eigenvalue problem can be solved with

$U_{p}$

is a steady-state solution. Let $X=\sum v_{1}e_{1}$ be

any velocity field satisfying (??). Using the above formulas, we obtain$<\tilde{R}(U_{P}, X)U_{P},$ $X>=-(2\pi)^{3}\sum_{1}\frac{|(u_{p}\cdot 1)|^{2}}{|p+1|^{2}}|\frac{(u_{p}\cdot 1)^{2}}{|(u_{p}\cdot 1)|^{2}}(v_{1} .

p)-(v_{1+2p} .

p)|^{2}\leq 0$.

This non-positive property is considered to be a three dimensional version of theArnold’s result for the curvature of the groupcontaining the direction$\xi$$\mathcal{D}_{v}(T^{2})$in any two-dimensional section

represented by the stream function$\frac{1}{2}(e_{p}+e_{-p})$

The above result can be extended to a two-mode Beltrami flow

defined by the linear combination $U_{p}+U_{q}$ of two Beltrami flows.
which is$U_{p,q}$$U_{p}$

andthough it is not difficult to derive the curvature formula in the section of$U_{q}$. Al-$U_{p,q}$and

the general direction X, we show here only the case in which X is another Beltrami

flow represented by $V_{p,q}=V_{p}+V_{q}$ where $V_{p}=v_{p}e_{p}+v_{-p}e_{-p}$ . The two polar-

ization amplitudes$v_{p}$

and$u_{p}$

are linearly independent and the Beltrami condition issatisfied with $\lambda=|p|$ . Then the sectional curvature is$<\overline{R}(U_{p,q},V_{p,q})U_{p,q},V_{p,q}>/(2\pi)^{3}$$=-N_{+}|(u_{p}\cdot q)(v_{q}^{*} .

p)-(u_{q}^{*}\cdot p)(v_{p}\cdot q)|^{2}$$-4N_{+}{\rm Im}[(u_{p}\cdot q)(v_{p}^{*}\cdot q)]\cdot{\rm Im}[(u_{q}\cdot p)(v_{q}^{*}\cdot p)]$$-N_{-}|(u_{p}\cdot q)(v_{q}\cdot p)-(u_{q}\cdot p)(v_{p}\cdot q)|^{2}$,$-4N_{-}{\rm Im}[(u_{p}\cdot q)(v_{p}^{*} .q)]\cdot{\rm Im}[(u_{q}^{*}\cdot p)(v_{q}\cdot p)]$where $N\pm=2/|p\pm q|^{2}$ .

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Next we show an interesting example, that is an application to the ABC flow8
represented by$U_{ABC}=$

A $[(i, 1,0)e^{ix_{3}}+(-i, 1,0)e^{-ix_{3}}]$$+B[(0,i, 1)e^{ix_{1}}+(0, -i, 1)e^{-ix_{1}}]$$+C[(1,0,i)e^{ix_{2}}+(1,0, -i)e^{-ix_{2}}]$,where A,B, $C\in$ R. This is a three-mode Beltrami flow, $i.e$ . each term on the righthand side satisfies the Beltrami property with $\lambda=-1$ . We have another ABC flow
$U_{A’B’C’}$

for $(A’\sim B’, C’)\neq(A,B, C)$ . It is straightforward to show that$<\tilde{R}(U_{ABC}, U_{A’B’C’})U_{ABC},$ $U_{A’B’C’}>$$=-2\{(AB’-BA’)^{2}+(BC’-CB’)^{2}+(CA’-AC’)^{2}\}$ ,namely the curvature is non-positive. In particular the curvature vanishes when
$A/A’=B/B’=C/C’$ .
$(A, B, C)$

It is interesting to find that even in the case where bothand $(A’, B’, C’)$ are close and are not in the domain of chaos,8 particle

motion by $U_{A’B’C’}$ will not be predicted from the particle motion by $U_{ABC}$ in thecourse of time.
It is expected that the present method may be applied to the analysis of rate of

growth of line-element or surface-element in various flow fields and turbulence.

73

References
1

V. I. Arnold, Ann. Inst. Fourier, Grenoble 16, 1, 319 (1966); Mathematical

Methods of Classical Mechanics, Appendix 2 (Springer-Verlag, New York, 1978).
2

D. G. Ebin and J. Marsden, Ann. Math. 90, 102 (1970).

3

J. Milnor, Morse Theory (Princeton University, New Jersey, 1963).

4

S. Kobayashi and K. Nomizu, Foundations of Differential Geometry Vol. 1 (John

Wiley&Sons, New York, 1969)’

N. J. Hicks, Notes on Differential Geometry (Van Nostrand, Princeton, N.J.,

1965)
6

Y. Hattori, Master thesis, Dept. of Phys., Univ. of Tokyo (1990).

7

R. Kraichnan, J. Fluid Mech. 5, 497 (1959).

8

T. Dombre, U. Frisch, J. M. Greene, M. H\’enon, A. Mehr, &A. M. Soward,

J. Fluid Mech. 167, 353 (1986).