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Differential Geometry of Diffeomorphism Groups

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Differential Geometry of Diffeomorphism Groups 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
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