LorentzFinsler metrics on symplectic and contact transformation groups Alberto Abbondandolo Gabriele Benedetti and Leonid Polterovich

2025-05-02 0 0 3.37MB 103 页 10玖币

侵权投诉

Lorentz–Finsler metrics on symplectic and contact

transformation groups

Alberto Abbondandolo, Gabriele Benedetti and Leonid Polterovich

October 6, 2022

Abstract

It has been noticed a while ago that several fundamental transformation groups

of symplectic and contact geometry carry natural causal structures, i.e., ﬁelds of

tangent convex cones. Our starting point is that quite often the latter come to-

gether with Lorentz–Finsler metrics, a notion originated in relativity theory, which

enable one to do geometric measurements with timelike curves. This includes ﬁnite-

dimensional linear symplectic groups, where these metrics can be seen as Finsler

generalizations of the classical anti-de Sitter spacetime, inﬁnite-dimensional groups

of contact transformations, with the simplest example being the group of circle dif-

feomorphisms, and symplectomorphism groups of convex domains. In the ﬁrst two

cases, the Lorentz–Finsler metrics we introduce are bi-invariant. A Lorentz–Finsler

perspective on these transformation groups turns out to be unexpectedly rich: some

basic questions about distance, geodesics and their conjugate points, and existence

of time functions, are naturally related to the contact systolic problem, group quasi-

morphisms, the Monge–Amp`ere equation, and a subtle interplay between symplectic

rigidity and ﬂexibility. We discuss these interrelations, providing necessary prelimi-

naries, albeit mostly focusing on new results which have not been published before.

Along the way, we formulate a number of open questions.

Contents

Introduction and main results 3

A Lorentz–Finsler structures . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

B A bi-invariant Lorentz–Finsler structure on the linear symplectic group . . 6

C The anti-de Sitter case n=1 ......................... 7

D A bi-invariant Lorentz–Finsler metric on the contactomorphism group . . . 9

E The contactomorphism groups of (S2n−1, ξst) and (RP2n−1, ξst) ....... 12

F Timelike geodesics on Sp(2n) ......................... 13

G Timelike geodesics on Cont(M, ξ) ....................... 15

H A systolic question for non-autonomous Reeb ﬂows . . . . . . . . . . . . . 17

I A time function and a partial order on the universal cover of Sp(2n) . . . . 18

arXiv:2210.02387v1 [math.SG] 5 Oct 2022

J The Lorentz distance on the universal cover of Sp(2n)............ 19

K The Lorentz distance on the universal cover of Cont0(RP2n−1, ξst) . . . . . 23

L Length bounds in Diﬀ1(R)........................... 25

M Length bounds on the universal cover of Cont0(RP2n−1, ξst)......... 27

N Positive paths in the group of symplectomorphisms of uniformly convex

domains ..................................... 30

Acknowledgments................................... 37

1 Proof of Proposition B.1 37

2 Proof of Proposition D.1 39

3 Proof of Proposition D.2 and uniqueness 42

4 Proof of Proposition E.1 44

5 The Morse co-index theorem for timelike geodesics on Sp(2n)47

6 Jacobi ﬁelds along timelike geodesics in Sp(2n)and proof of Theorem F.1 49

7 The second variation of the Lorentz–Finsler length on Cont(M, ξ)52

8 Proof of Theorem H.1 57

9 Krein theory, Maslov quasimorphism and proof of Theorem I.1 59

10 Causality, Lorentz distance and proof of Theorem J.1 on f

Sp(2) 63

11 Proof of Theorem J.1 67

12 Proof of Theorem K.2 71

13 Proof of Theorem L.1 76

14 Proof of Theorem M.3 79

15 Proofs of the results of Section N 82

i Bi-invariant Lorentz–Finsler metrics on Lie groups 86

ii Some facts about the Lie algebra of the symplectic group 93

iii Contact Hamiltonians 96

References 97

Introduction and main results

Endow the vector space R2nwith linear coordinates x1, y1, . . . , xn, ynand with the standard

symplectic form

ω0:=

j=1

dxj∧dyj.

The group of linear automorphisms of R2nthat preserve ω0is the symplectic group Sp(2n).

It is well known that Sp(2n) admits no bi-invariant distance function inducing the Lie group

topology. Here is the simple argument for n= 1: the symplectic automorphisms

Wλ:= 1λ

0 1 , λ > 0,

are all pairwise symplectically conjugate and hence any bi-invariant distance function on

Sp(2) assigns the same positive distance from the identity to each of them. But then the

distance function cannot be continuous with respect to the Lie group topology, as Wλ

converges to the identity for λ→0. The same argument applies in higher dimension

and shows, in particular, that Sp(2n) does not admit bi-invariant Riemannian or Finsler

metrics.

Similarly, the contactomorphism group Cont(M, ξ) of a closed contact manifold (M, ξ)

does not admit any bi-invariant distance function which is continuous with reasonable

Lie group topologies. More precisely, any bi-invariant distance function on Cont(M, ξ) is

discrete, meaning that the distance of any pair of distinct elements has a positive lower

bound, see [FPR18, Theorem 3.1].

In this monograph, we show that Sp(2n) and Cont(M, ξ) admit natural bi-invariant

Lorentz–Finsler structures and initiate a systematic study of their properties. The for-

mer provides yet another multi-dimensional generalization of the classical 3-dimensional

anti-de-Sitter space and yields a new viewpoint at the twist condition in Hamiltonian dy-

namics. The latter (which is related to the former) provides a natural geometric language

for studying a non-autonomous version of the contact systolic problem. While our motiva-

tion comes from symplectic and contact geometry and dynamics, we develop the subject

along the lines which are customary in Lorentzian geometry, and which are inﬂuenced by

its physical interpretation. As we believe that keeping in mind this interpretation may fa-

cilitate the understanding of the (otherwise, purely mathematical) material of the present

monograph, we start with its very brief overview.

Lorentzian (or, more generally, Lorentz–Finsler) metrics are sign-indeﬁnite cousins of

Riemannian (resp. Finsler) metrics. They originated in the relativity theory as a natural

geometric structure on a space-time invariant under Lorentz transformations, modeling

the change of an inertial coordinate system. The necessity to deal with anisotropies of the

space-time [JS14] motivated a passage from sign-indeﬁnite Lorentzian quadratic forms to

more general Lorentz–Finsler functionals having similar convexity/concavity features.

In the Lorentzian world, the space-time Mwhich is modeled by a smooth manifold,

gets equipped with a ﬁeld of cones consisting of vectors of the real (as opposed to the

imaginary) length. Physically admissible causal (resp. timelike) curves are characterized

by the fact that their tangent vectors point into these cones (resp. into their interior).

When there are no closed causal curves, the existence of a causal curve starting at xand

ending at yintroduces a partial order on M, deﬁning the so called causal structure. The

natural parameter along causal curves is neither length nor time, but a so called proper

time, which is modeled by the Lorentzian length. Local extremizers of the Lorentzian

length, i.e., timelike geodesics, are of particular interest: they model the motion of a free

particle in the spacetime.

The behaviour of proper time is far from being intuitive. According to the famous

“twin paradox”, moving close to the light cone consisting of vectors of the zero Lorentzian

length enables a spacetime traveller who takes oﬀ at a point xto arrive at the destination

ywithin an arbitrary small proper time. His twin sibling, simultaneously starting at x,

may pursue a diﬀerent objective - to reach ywithin the maximal possible proper time.

This quantity is sometimes inﬁnite and sometimes ﬁnite, depending on the geometry and

topology of the spacetime, and maybe also on the speciﬁc choice of the points xand y.

When ﬁnite, it deﬁnes the Lorentzian distance dist(x, y), a global geometric invariant.

The main results of the present monograph together with some open questions are

presented in Sections A - N of the Introduction. After recalling the deﬁnition of a Lorentz–

Finsler structure (Section A), we introduce a Lorentz–Finsler structure in parallel on the

ﬁnite dimensional Lie group Sp(2n) (Sections B and C) and on the inﬁnite dimensional

one Cont(M, ξ) (Section D), as the two structures are closely related (Section E). We then

study the local properties of the induced length functionals (i.e., proper time) and of its

geodesics (Sections F and G), which in the inﬁnite dimensional case is related to a contact

systolic question (Section H). The Lorentzian viewpoint enables us to establish “systolic

freedom” for time-dependent contact forms which manifests an interplay between dynamics

and geometry. Loosely speaking, the proper time of our Lorentz–Finsler structures can be

described dynamically, as a certain “magnitude of twisting” of the ﬂow corresponding to

a path on the group, and geometrically, via the contact volume.

Furthermore, we discuss to which extent these structures can be used in order to pro-

duce global bi-invariant measurements on these groups. For Sp(2n), the Lorentz–Finsler

distance dist(x, y) between causally related points xand ycan take both ﬁnite and in-

ﬁnite values, depending on the location of x−1y(Sections I and J). In contrast to this,

for contactomorphisms of the projective space, dist(x, y) is always inﬁnite whenever there

is a timelike curve from xto y(Section K). This can be seen as a manifestation of the

ﬂexibility of contactomorphisms. However, ﬂexibility is expensive: long paths connecting x

and ynecessarily possess a “high complexity”, properly understood. For the simplest con-

tact manifold S1=RP1, our approach to this phenomenon involves a delicate L1-version

of Bernstein’s classical inequality for positive trigonometric polynomials due to Nazarov

(Section L). In higher dimensions, we use an ingredient from “hard” contact topology,

namely Givental’s non-linear Maslov index, in combination with the analysis on Sp(2n)

which turns out to be crucial (Section M).

Finally, in Section N we discuss Lorentz–Finsler phenomena on the group of symplec-

tomorphisms of a uniformly convex domain, which turn out to be related to the Monge–

Amp`ere equation and to a variational problem which is linked to the maximization of the

aﬃne area functional.

A Lorentz–Finsler structures

Let Mbe a (possibly inﬁnite dimensional) manifold. In this monograph, we shall use the

following notion of Lorentz–Finsler structure on M:

Definition A.1.ALorentz–Finsler structure (K, F ) on Mis given by the following data:

(i) An open subset K⊂T M such that for every p∈Mthe intersection K∩TpMis

a non-empty convex cone in the vector space TpM, and K∩ −Kcoincides with the

zero-section of T M. The set Kis called cone distribution on M.

(ii) A smooth function F:K→(0,+∞) which is ﬁberwise positively 1-homogeneous,

ﬁberwise strongly concave in all directions other than the radial one, meaning that

d2F(v)·(w, w)<0∀v∈K∩TpM, ∀w∈TpM\Rv, ∀p∈M,

and extends continuously to Kby setting F|∂K = 0. The function Fis called Lorentz–

Finsler metric on (M, K).

If Mis inﬁnite dimensional, the smoothness of Fcan be understood in several ways,

depending on the class of inﬁnite dimensional objects one is working with. In this mono-

graph, we will work with a Fr´echet manifold which is modeled on the space of smooth real

functions on a closed manifold, and smoothness is to be understood in the diﬀeological

sense: the restriction of Fto any ﬁnite dimensional submanifold of the open set Kis

smooth.

The above deﬁnition generalizes the classical notion of a time-oriented Lorentz structure,

in which the manifold Mis endowed with a non-degenerate symmetric bilinear form g:

T M ×T M →Rof signature (−,+,...,+) and there is a continuous vector ﬁeld Xon M

such that g(X, X)<0: indeed, in this case one chooses as Kthe connected component of

the set {v∈T M |g(v, v)<0}containing the image of Xand sets F(v) := p−g(v, v).

The assumption on the signature of gimplies that Kis convex and Fis ﬁberwise strongly

concave, as required in Deﬁnition A.1.

Apart from regularity and strong convexity issues on the boundary of K, the above

deﬁnition of a Lorentz–Finsler structure agrees with Asanov’s deﬁnition from [Asa85] and

its later reﬁnements, see [Min16], [JS20]. In particular, it agrees with the idea that a

Lorentz–Finsler metric needs to be deﬁned only on the convex cone of causal vectors.

Vectors in Kare called timelike, non-vanishing vectors in ∂K are called lightlike, and

vectors which are either timelike or lightlike are called causal. A C1curve in Mis called

timelike (resp. lightlike, resp. causal) if its derivative is everywhere timelike (resp. lightlike,

文档加载中……请稍候！
如果长时间未打开，您也可以点击刷新试试。

下载文档到电脑，查找使用更方便

10 玖币 0人已下载

立即下载

摘要：

Lorentz{FinslermetricsonsymplecticandcontacttransformationgroupsAlbertoAbbondandolo,GabrieleBenedettiandLeonidPolterovichOctober6,2022AbstractIthasbeennoticedawhileagothatseveralfundamentaltransformationgroupsofsymplecticandcontactgeometrycarrynaturalcausalstructures,i.e.,eldsoftangentconvexcones.O...

展开>> 收起<<

LorentzFinsler metrics on symplectic and contact transformation groups Alberto Abbondandolo Gabriele Benedetti and Leonid Polterovich.pdf

共103页,预览5页

还剩页未读，继续阅读

声明：本站为文档C2C交易模式，即用户上传的文档直接被用户下载，本站只是中间服务平台，本站所有文档下载所得的收益归上传人(含作者)所有。玖贝云文库仅提供信息存储空间，仅对用户上传内容的表现方式做保护处理，对上载内容本身不做任何修改或编辑。若文档所含内容侵犯了您的版权或隐私，请立即通知玖贝云文库，我们立即给予删除！

LorentzFinsler metrics on symplectic and contact transformation groups Alberto Abbondandolo Gabriele Benedetti and Leonid Polterovich

相关推荐

开通VIP享超值会员特权

作者详情

相关内容

热门标签

举报选择: