SEMI-BASIC 1-FORMS AND COURANT STRUCTURE FOR METRIZABILITY PROBLEMS. Mircea Crasmareanu

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1 PUBLICATIONS DE L INSTITUT MATHÉMATIQUE Nouvelle sére, tome , DOI: /PIM C SEMI-BASIC 1-FORMS AND COURANT STRUCTURE FOR METRIZABILITY PROBLEMS Mrcea Crasmareanu Abstract. The metrzablty of sprays, partcularly symmetrc lnear connectons, s studed n terms of sem-basc 1-forms usng the tools developed by Bucataru and Dahl n [2]. We ntroduce a type of metrzablty n relatonshp wth the Fnsler and projectve metrzablty. The Lagrangan correspondng to the Fnsler metrzablty, as well as the Bucataru Dahl characterzaton of Fnsler and projectve metrzablty are expressed by means of the Courant structure on the bg tangent bundle of T M. A byproduct of our computatons s that a flat Remannan metrc, or generally an R-flat Fnsleran spray, yelds two complementary, but not orthogonally, Drac structures on T bg T M. These Drac structures are also Lagrangan subbundles wth respect to the natural almost symplectc structure of T bg T M. 1. Introducton The queston whether a gven symmetrc lnear connecton on the manfold M s the Lev-Cvta connecton of a Remannan metrc s very mportant from both mathematcal where t appears as a recprocal of the fundamental theorem of Remannan geometry and physcal reasons [15]. So, t has a rch hstory whch s possble to begn wth [9] ponted out for example n [8], where the condtons for local metrzablty as well as the global problem are dscussed n detal. Recently, there s a growng nterest for ths queston n control theory [12] and more generally, under the name of nverse problem, n geometrzaton of general and specal Lagrangan systems [14, 16]. Here we ntroduce a type of metrzablty whch nvolves the geometry of the tangent bundle. Namely, we search for a metrc g not on the base manfold M but on the tangent manfold T M by mposng that g s covarant constant wth respect 2010 Mathematcs Subject Classfcaton: Prmary 53C60; Secondary 53C05; 53C15; 53C20; 53C55; 53D18. Key words and phrases: semspray; metrzablty; sem-basc 1-form; symplectc form; homogenety; bg tangent bundle; Courant bracket; Drac structure; sotropc subbundle; Poncaré Cartan 1-form. Communcated by Zoran Rakć. 153

2 154 CRASMAREANU to the dynamcal covarant dervatve of the spray assocated to. A condton for ths knd of metrzablty s then derved by usng the Fnsler metrzablty of sprays characterzed by Bucataru and Dahl n [2] n terms of sem-basc 1-forms. The ntal metrzablty problem s recovered n terms of global smoothness of ths 1-form and also there are consdered several relatonshps wth the symplectc geometry of T M nduced by a 2-homogeneous Lagrangan L of Fnsler type. In fact, the possblty to use the symplectc structures was the motvaton for our choce of [2] as the man pont of begnnng the present study nstead of classcal holonomy methods, [17] or [19]. We express the above Fnsler and also the projectve metrzablty of a general spray S and the correspondng Lagrangan L by means of the Courant structure on the bg tangent bundle of T M. Essental terms n the Courant bracket for several objects are based on the coeffcents of the Jacob operator of S, as well as on the coeffcents of the curvature of the nonlnear connecton nduced by S. For example, even n the Remannan case we derve new results: the flatness of the Remannan metrc s equvalent wth the Courant sotropy of the par the horzontal dstrbuton, vertcal dual of the Poncaré Cartan 1-form. The same flatness yelds a half-drac structure on T bg T M whle the above subbundle together wth the vertcal dstrbuton, the Poncaré Cartan 1-form are sotropc wth respect to the natural almost symplectc structure of the bg tangent bundle of T M. Moreover, we obtan a Drac structure V D S=the vertcal dstrbuton, ts annhlator for every Lagrangan spray S.e., S gves the Euler Lagrange equatons of a regular Lagrangan L; the regularty means that the Hessan of Lx, y wth respect to the vertcal varables y s nondegenerate. Wth respect to the dual pont of vew, the par H D S=the horzontal dstrbuton, ts annhlator s a Drac structure on the T bg T M f and only f the Fnsleran spray S s R-flat whch means the vanshng of the curvature of ts nonlnear connecton; ths means n the partcular case of Remannan geometry that the metrc s flat. In concluson, for a flat Remannnan metrc the bg tangent bundle of T M s the complementary but not orthogonally sum of two Drac structures, both beng also Lagrangan subbundles for the almost symplectc structure of T bg T M as well as Le algebrods over T M. Let us remark that some classes of Drac structures naturally assocated to Lagrangan systems appear also n [20] and the above results are already obtaned for the Remannan case n [8]. 2. Sembasc 1-forms for the metrzablty of lnear connectons Fx M a smooth n 2 dmensonal manfold wth the tangent bundle T M and cotangent bundle T M. Local coordnates on M wll be denoted by x = x, 1 n whle the nduced coordnates on T M wll be denoted by x, y = x, y. Let {0} be the zero-secton of the tangent bundle and X T M the Le algebra of vector felds on T M. A man structure of T M s the tangent structure J = y dx. Fx also a symmetrc lnear connecton on M and let S X T M ts semspray; f has the local coeffcents Γ jk x, then S has the local expresson

3 S = y x 2G x, y y SEMI-BASIC 1-FORMS AND COURANT STRUCTURE wth 2.1 G = 1 2 Γ jky j y k. We have 2.2 JS = C wth C = y y the Louvlle vector feld of T M. Ths vector feld s useful n characterzatons of homogeneous objects on T M {0}, [2, p. 163]; namely let k be an nteger, then a vector feld X s k-homogeneous f L C X = k 1X, a p-form ω s k-homogeneous f L C ω = kω and a 1, 1-tensor feld L s k-homogeneous f L C L = k 1L. Here L Z s the Le dervatve wth respect to the vector feld Z; the vector felds { } y are 0-homogeneous. In fact S s more than a semspray; t s a spray due to the 2-homogenety of the coeffcents G of 2.1. Then S has the reduced form S = y δ δx where wth N j = G y j δ δx = x N j y j = Γ jk yk. We derve a new local bass δx, y n X T M, called Berwald, whch s adapted to our computatons and has the dual bass dx, δy = dy + Nj dxj. Ths decomposton of the terated tangent bundle T T M produces [2, p. 165]: 1 the horzontal projector h = δ δx dx, y 1 the vertcal projector v = δy, and the followng dstrbutons and codstrbutons: 2 the horzontal dstrbuton HT M = span { } δ δx = hx T M and ts annhlator H T M = span{δy }, 2 the vertcal dstrbuton V T M = span { } y = vx T M and ts annhlator V T M = span{dx }. Therefore S s a horzontal vector feld whle C s a vertcal vector feld. Let S be the dynamcal covarant dervatve nduced on T M by S, [2, p. 167]. Inspred by the defnng property of the Lev-Cvta connecton, we ntroduce: Defnton 2.1. The symmetrc lnear connecton s sad to be tangent metrzable f there exsts a Remannan metrc g on T M, of Sasak type g = g j dx dx j + g j δy δy j such that S g = 0 whch means the tangent Chrstoffel formula 2.3 Sg j = N k g kj + N k j g k. Our approach n fndng characterzatons of tangent metrzablty follows closely the technques of Bucataru and Dahl from [2], where, among others, the followng type of metrzablty s studed:

4 156 CRASMAREANU Defnton 2.2. [2, p. 178] The spray S s Fnsler metrzable f there exsts a 2-homogeneous Lagrangan L : T M R such that L S d J L = dl. The spray s called projectve metrzable f the above Lagrangan s 1-homogeneous. Here d J s the exteror dervatve nduced by J on forms of T M; then d J LX = dljx = JXL for any X X T M. The fundamentals of Fnsler geometry are excellently exposed n [1]. The characterzaton of Fnsler metrzablty s obtaned n [2] n terms of sembasc 1-forms.e., 1-forms on T M wth local expresson θ = θ x, ydx : Theorem 2.1. [2, p. 177] The spray S s Fnsler metrzable f and only f there exsts a 1-homogeneous sem-basc 1-form θ on the slt tangent bundle T M {0} such that the followng Helmoltz condtons hold 1 θ s d h -closed: d h θ = 0, 2 θ s d J -closed: d J θ = 0, 3 dθ s covarant constant wth respect to S : S dθ = 0. Then θ s called the Poncaré Cartan 1-form of the Lagrangan L. Let us pont out that our condton 2.3 corresponds to the relaton 3 above. Therefore we apply Theorem 2.1 and to ths am let ω = dθ. Recall also: Defnton 2.3. [2, p. 164] The sembasc 1-form θ s called nondegenerate f ω s a symplectc form on T M {0}. Recall that a par ω, K wth a symplectc form and a compatble almost complex structure yelds a Remannan metrc g through the formula [4, p. 86]: 2.4 g, = ω, K. The requred compatblty s [4, p. 90]: 2.5 ωkx, KY = ωx, Y. But equvalently S yelds an almost complex structure F, [2, p. 166]: 2.6 F δx = y, F y = δ δx or, n the Berwald bass F = δ δx δy y dx. Now, we are able to state the frst man result of ths note. Proposton 2.1. Let be a symmetrc lnear connecton on M such that the assocated spray S s Fnsler metrzable va the sembasc 1-form θ. If θ s nondegenerate.e., the determnant det θ y s nonvanshng, then s tangent j metrzable. Proof. We need to prove two facts.

5 SEMI-BASIC 1-FORMS AND COURANT STRUCTURE I ω and F are compatble, but ths s a consequence of relatons 1 and 2 from Theorem 2.1. Indeed, after [2, p. 172], we have ω = a j dx j dx +2g j δy j dx and d h θ = a j dx j dx, d J θ = g j g j dx j dx wth a j = 1 θ 2 δx j δθ j, δx g j = 1 θ 2 y j. A drect computaton gves ω F δ δx,f δ δx j = 0 = a j = ω δx, δ δx j, ω F y,f y j = a j = 0 = ω y, y j, ω F δ δx,f y j = g j = g j = ω δx, y j. II S g = 0. The dentty 3 reads S ω = 0 and we have S F = 0 from [2, p. 169]; these two relatons yeld S g = 0 because of 2.4. Let us remark that and mean that the dstrbutons HT M and V T M are Lagrangan for ω. The matrx of ω n the dual Berwald bass dx, δy j s 0 gj ω = g j 0 and then ω s non-degenerate f and only f detg 0. The matrx of F s 0 In F = I n 0 and then the correspondng metrc g s of Sasak type 0 gj 0 In gj g = = g j 0 I n g j In ths framework, lftng to the tangent bundle the geodesc equatons of.e., the flow equatons of S we arrve at exactly the Euler Lagrange equatons of the Lagrangan L,θ = gs, S = gc,c = ωc,fc whch can be thought of as the energy of g. A dual relaton to 2.2 s 2.7 FC = S and then L,θ = ωc, S = dθc, S = 1 2{ CθS S θc θ[c, S ] }. Snce θ s sembasc, we have θc = 0 whle the 2-homogenety of S reads and then [C, S ] = S 1 L,θ = C 2 θs 1 2 θs

6 158 CRASMAREANU whch recover the classcal formula for the Hamltonan H assocated to a Lagrangan L H = y L L = CL L. y But agan the 2-homogenety of S : CθS = 2θS gves the fnal expresson L,θ = 1 2 θs = 1 2 θ x, yy whch s 2-homogeneous snce θ s 1-homogeneous. A smlar formula can be derved by usng the F-adjont of θ θ = F θ = 1 2 θ δy F s the verson of F on 1-forms, namely L,θ = θ C. Let us pont out that the Euler characterzaton of 1-homogenety for θ θ = θ y j yj means that θ = 2g j y j, and then L has the well-known expresson L = g j y y j. The fnal answer to the metrzablty problem. Now, we are ready for a return to the ntal problem namely the metrzablty of. If we assume the smoothness of our tools on the whole T M, then the 1-homogenety of θ means that the components θ are homogeneous polynomals of degree 1 n y and then g s n fact a Remannan metrc on the base manfold M.e., g = gx. In concluson, our proposed approach for the metrzablty problem of a lnear connecton conssts n two steps: 1 we lft from M to the tangent bundle T M and study the Fnsler metrzablty of S obtanng the metrc g, 2 we return to the bass manfold M under the homogenety hypothess, obtanng the desred metrc g = gx. Examples Let on M = R 2 the symmetrc lnear connecton wth the only nonzero components, [19, p. 513] Γ 1 11 = x 1 x , Γ2 22 = x 2 x In the cted paper, t s proved that s the Lev-Cvta connecton for a metrc g on the base manfold and ths g s computed. In the present framework we derve that s tangent metrzable wth the sembasc 1-form wth b real parameters. θ 1 = 2b 2 [x ]y 1 + b 1 [x ][x ] y 2, θ 2 = b 1 [x ][x ] y 1 + 2b 3 [x ]y 2 2 Examples of non-fnsler metrzable sprays are n [3]. It s natural to ask about ω = dθ. A straghtforward computaton yelds ω = 1 2 Rk j θ kdx dx j θ jdx δy j

7 SEMI-BASIC 1-FORMS AND COURANT STRUCTURE where R a j are the components of the curvature of nonlnear connecton HT M, [2, p. 166] Rj a = δn a δx j δn j a δx and θ j = δθ j δx N k y j θ k. In Fnsler partcularly Remannan geometry, by usng the Cartan partcularly Lev-Cvta lnear connecton, the coeffcent θ j vansh and then dθ s the unque assocated symplectc form. A Fnsleran spray S wth vanshng curvature R s called R-flat [7]. So, for an R-flat Fnsler metrc partcularly flat Remannan metrc we have that θ s closed. Let us end ths secton by makng several relatonshps wth some prevous works. a Usually, one seeks the soluton of the metrzablty problem bearng n mnd the holonomy groups or algebra accordng to the frst fundamental paper on ths subject namely [17]. So, n [19, p. 514] are studed symmetrc blnear forms G wth 2.8 GAX, Y + GX, AY = 0 for all A n the holonomy group of a gven pont of M. Snce F of 2.6 s an almost complex structure, the compatblty relaton 2.5 reads: 2.9 gfx, Y + gx,fy = 0 and we remark that the formulae 2.8 and 2.9 are formally the same. The essental dfference between these equatons s that formula 2.9 s n a close relaton to the curvature and there s no such condton f the curvature s zero whle 2.8 s just expressng some symmetry of the metrc and survves even n the case when R = 0. In fact, formula 2.8 corresponds rather to one of the Helmholtz condton and ts generalzaton nvolvng the Jacob endomorphsm and ts dervatves. b In [7, p. 100] t s ponted out that F of 2.6 s not homogeneous and a new almost complex structure s consdered F S δx = F y, F S y = 1 δ F δx where F s the Fnsleran fundamental functon correspondng to the Lagrangan L,θ constructed above va L,θ = F 2. Ths new almost complex structure s 0- homogeneous [7, p. 100], and a computaton smlar to that gven n the proof of Proposton 2.1 yelds that ω and F S are stll compatble. The formula correspondng to 2.7 s F S C = 1 F S and then the energy of the new metrc g S = ω,f S s L,θ,S = 1 F L,θ = F = θ x, yy = g j y y j. Ths 1-homogeneous Lagrangan corresponds to the projectve metrzablty of S. c If ω and F are not compatble, then, smlar to relaton 11 of [13, p. 8395], we replace ω wth ω s, = 1 2{ ωf,f + ω, }. d Also, from relaton 29 from [13, p. 8396], we get that S s, at the same tme:

8 160 CRASMAREANU a Kllng vector feld for g, an nfntesmal symplectomorphsm for ω, an nfntesmal automorphsm for F. Accordng to [4, p. 22] for an exact symplectc form ω = dθ, there exsts a unque vector feld ξ such that ξ ω = θ. For our case we have ξ = C so that we can say that all semsprays have an unversal symplectc potental vector feld namelyc. e Agan from [7, p. 100], the almost complex structure F s ntegrable f and only f the spray S s R-flat whch means that s wth vanshng curvature. In ths case, the par g,f s a Kähler structure on T M {0} for whch ω = dθ s the Kähler or fundamental form. 3. The relatonshp wth the Courant structure of bg TM. Assocated Drac structures The nterplay between vector felds and 1-forms of the prevous secton makes necessary the consderaton of the manfold T bg M := T M T M. Ths manfold s the total space of a natural vector bundle π : T bg M M; so T bg M s called the bg tangent bundle of M [18], and s endowed wth the Courant structure,, [, ] [6]: 1. the neutral nner product X, α, Y, β = 1 2 βx + αy 2. the skew-symmetrc Courant bracket [ ] X, α, Y, β = [X, Y ], L X β L Y α 1 2 d βx αy. The same manfold T M T M s called sometmes the Pontryagn bundle of M [11]. Gong now wth the smlar structure by consderng T M nstead of M, we derve some ways to express the Lagrangan assocated to a metrzable spray: Proposton 3.1. Let S be a spray whch s Fnsler metrzable through the sembasc 1-form θ. Then the correspondng 2-homogeneous Lagrangan s L = C, θ, S, θ = C, θ, S, θ = 1 2 C, θ, S, θ. Proof. The above formulae are drect consequences of the followng expressons of L y, θ, S, θ = δx, θ, S, θ = y, θ, C, θ = Another nterestng relaton s y, θ, δx j, θ = 0, δx, θ, C, θ. whch means that the dstrbutons V T M, θ, HT M, θ are, -orthogonal on T bg T M.

9 SEMI-BASIC 1-FORMS AND COURANT STRUCTURE The Courant bracket on T bg T M gves the followng verson of Theorem 5.4 from [2, p. 177]: Proposton 3.2. Let S be a spray on M. Then S s a Lagrangan vector feld, nduced by a k-homogeneous Lagrangan, f and only f there exsts a k 1- homogeneous sembasc 1-form θ Ω 1 T M {0} such that [ C, θ, S, θ ] = S, k 1θ + k 2 1 L S θ It results n partcular that S s: Fnsler metrzable f and only f there exsts a 1-homogeneous sembasc 1- form θ such that [ C, θ, S, θ ] = S, θ, projectve metrzable f and only f there exsts a 0-homogeneous sembasc 1-form θ such that [ C, θ, S, θ ] = S, 1 2 L Sθ. Recall [18], that [, ]-ntegrable,, -sotropc subbundles of T bg M of maxmal rank n are called Drac structures; so, our next am s to derve Drac structures n the present framework. Wth a computaton smlar to that of the frst secton we get L δ θ = R a δx j θ adx j + θ j δy j and then [ δx, θ, δx j, θ ] = Rj k y k, Rja b Ra b θb dx a + θ u θ j u δy u. For a Fnsleran partcularly Remannan spray S the above relaton becomes [ δx, θ, δx j, θ ] = Rj k y k, Rja b Ra b θb dx a. So, for a R-flat Fnsleran metrc partcularly flat Remannan metrc the subbundle HT M, θ can be consdered as a 1 2 -Drac structure on T bg T M. From L δ δx dxj = L y dxj = 0, L δ δx δyj = R j a dxa N j y a δya, L y δyj = N j y a δya we compute [ δx, dxj, δx k, dxl] = = R a k 0, [ y a, 0, y, δyj, N l y a N j k y a δy a. y k, δyl] so HT M, V T M s Courant closed f and only f the Fnsleran spray S partcularly Remannan metrc g s R-flat flat and V T M, H T M s always Courant closed. Another man result of ths secton s: Theorem 3.1. V D S = V T M, V T M s a Drac structure on T bg T M. H D S = HT M, H T M s a Drac structure on T bg T M f and only f the

10 162 CRASMAREANU Fnsleran spray S, n partcular the Remannan metrc g, s R-flat, n partcular s flat. Proof. We have y, dxj, y k, dxl = 0, δx, δyj, δx k, δyl = 0 respectvely [ y, dxj, y k, dxl] = 0, 0, [ δx, δyj, δx k, δyl] = Rk a y a, R j ka N j Rl a dx a k + y a N l y a δy a whch yelds the conclusons. It results that an R-flat Fnsleran spray or a flat Remannan metrc yelds a decomposton of T bg T M n complementary Drac structures: 3.1 T bg T M = V D S H D S n analogy wth the decomposton of the terated tangent bundle T T M = V T M HT M. From y k, dxl, δx, δyj = 1 δ l 2 + δ j k the decomposton 3.1 s not, -orthogonal. Recall, after [10] that a par of complementary Drac subspaces s called a reflector. So, the decomposton 3.1 provdes T bg T M wth a reflector. Recall also that f L s a Drac structure on M, then the trple L, [, ] L, pr : L T M s a Le algebrod on M [6, p. 645]. Therefore, we get two Le algebrods over T M provded by a flat Remannan metrc, or more generally R-flat Fnsler spray. 4. The relatonshp wth the almost symplectc structure of bg T M In the frst secton we have a symplectc structure on T M assocated wth our framework; let us remark that the bg tangent bundle has a nondegenerate skew-symmetrc 2-form [18] Ω X, α, Y, β = 2 1 βx αy. Our am n ths secton s to use ths almost symplectc structure of bg tangent bundle for our computatons. From Ω y, θ, y j, θ = 0, Ω δx, θ, δx j, θ = 0 t results that V T M, θ and HT M, θ are Ω-sotropc subbundles of T bg T M, whle from Ω y, dxj, y k, dxl = 0, Ω δx, δyj, δx k, δyl = 0

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