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Israel Journal of Mathematics 88 (1994), 319–332

arXiv:math/9204223v1 [math.DG] 1 Apr 1992Israel Journal of Mathematics 88 (1994), 319–332GEODESICS ON SPACESOF ALMOST HERMITIAN STRUCTURESOlga Gil-Medrano1Peter W. Michor2Departamento de Geometr´ıa y Topolog´ıaUniversidad de Valencia, Spain.Institut f¨ur Mathematik, Universit¨at Wien, AustriaErwin Schr¨odinger Institute for Mathematical Physics.August 29, 2018Abstract. A natural metric on the space of all almost hermitian structures ona given manifold is investigated.Table of contents0.

Introduction. .

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Almost hermitian structures . .

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The geodesic equation in H. . .

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.53. The variational approach to the geodesic equation.

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.74. Some properties of the geodesics.

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IntroductionIf M is a (not necessarily compact) smooth finite dimensional manifold, thespace M = M(M) of all Riemannian metrics on M can be endowed witha structure of an infinite dimensional smooth manifold modeled on the spaceC∞c (S2T ∗M) of symmetric (0, 2)-tensor fields with compact support.Analo-gously, the space Ω2nd(M) of non degenerate 2-forms on M, is an infinite dimen-sional smooth manifold, modeled on the space Ω2c(M) of 2-forms with compactsupport. See [6] and [7].1991 Mathematics Subject Classification.

58B20, 58D17.Key words and phrases. Metrics on manifolds of structures.1Research partially supported by the CICYT grant n. PB91-0324,2Supported by Project P 7724 PHY of ‘Fonds zur F¨orderung der wissenschaftlichen Forschung’.Typeset by AMS-TEX1

2O. GIL-MEDRANO, P. W. MICHORHere we consider the space of almost Hermitian structures on M, i.e.

the subsetH of M(M) × Ω2nd(M) of those elements (g, ω) such that the (1, 1)-tensor fieldJ = g−1ω is an almost complex structure on M. The aim of this paper is to studythe geometry of H. First we prove in section 1 that H is a splitting submanifoldof the product M(M)×Ω2nd(M). In section 2, after splitting the tangent space ofthe product in a form well adapted to our problem, we derive the equations thata curve in H should satisfy in order to be a geodesic for the metric on H inducedby the product metric on M(M) ×Ω2nd(M).

In section 3 we give an independentvariational derivation of the geodesic equations, by parameterizing elements of Hby automorphisms of TM, i.e. by sections of the bundle GL(TM, TM).

Finally,section 4 is devoted to the study of the geodesic equations found in section 2.We were not able to find an explicit solution, nevertheless we can give someexplicit properties of the geodesics, see 4.1 and 4.2. The subspaces HJ, Hω, andHg of all almost hermitian structures with fixed almost complex structure J,2-form ω, or metric g, respectively, are splitting submanifolds of H. This followsfrom splitting the bundle.

Some other interesting subsets (hermitian structures,K¨ahler structures, with symplectic ω) are much more difficult to treat: we do notknow whether they are submanifolds, since the differential operators describingthem are complicated in the charts we use. Moreover HJ is totally geodesic inM(M), and the other two are totally geodesic in the manifold of all metrics witha fixed volume form.

The geodesics are known explicitly in all three cases.1. Almost hermitian structures1.1.Almost hermitian structures.

Let M be a smooth manifold of evendimension n = 2m. Let g be a Riemannian metric on M and let ω be a nondegenerate 2-form on M; both of them will be regarded as fiber bilinear func-tionals on TM or as fiber linear isomorphisms TM →T ∗M without any changeof notation.

The symmetry of g is then expressed gt = g where the transposedgt is given by TM →T ∗∗Mg∗−→T ∗M, and we also have ωt = −ωThen we consider the endomorphism J := g−1ω: TM →TM which satisfiesJ∗= ωt(gt)−1 = −ωg−1. Then the following conditions are equivalent:(1) J is an isometry for g, i.e.

g(JX, JY ) = g(X, Y ) for all X, Y ∈TxM. (2) g−1ω + ω−1g = 0.

(3) J is an almost complex structure, i. e. J2 = −Id or equivalently J−1 =−J.If these equivalent conditions are satisfied we say that (g, ω) is an almost Her-mitian structure.Remark: For a given almost complex structure J, condition (1) is equivalentto the fact that gJ is skew symmetric. So there is a bijective correspondencebetween the set of almost hermitian structures and the set of Riemannian almostcomplex structures.

GEODESICS ON SPACES OF ALMOST HERMITIAN STRUCTURES31.2. The bundle of almost hermitian structures.

We consider the subspaceHerm := {(g, ω): g−1ω + ω−1g = 0} ⊂S2+T ∗M × Λ2ndT ∗M,where S2+T ∗M is the set of all positive definite symmetric 2-tensors on M, andwe claim that it is a subbundle. For that we consider the following commutativediagram:(1)Hermpr1−−−−→S2+T ∗Mpr2yyπΛ2ndT ∗M −−−−→πM.Lemma.

This is a double fiber bundle, where the standard fiber of pr1 is thehomogeneous space O(2m, R)/U(m) and the standard fiber of pr2 is the homoge-neous space Sp(m, R)/U(m).In fact all the fiber bundles of diagram (1) are associated bundles for the linearframe bundle GL(R2m, TM), where the structure group GL(2m) acts from theleft on the typical fibers given in the diagram(2)GL(2m, R)/U(m)−−−−→GL(2m, R)/O(2m, R)yyGL(2m, R)/Sp(m, R) −−−−→{Id}.Proof. If we fix a metric g, then in an orthonormal frame of TM|U for an opensubset U ⊂M, there is of course a 2-form ω0 such that (g, ω0) is a local sectionof Herm.If (g, ω1) is another local section with the same g then there is ag-isometric local isomorphism f of TM|U with possibly smaller U such thatf ∗ω1 = ω0 which is fiberwise unique up to multiplication from the right by anelement of O(2m, R) ∩Sp(m, R) = U(m).If we fix on the other hand a non degenerate 2-form ω, then there is a frame ofTM|U such that ω takes the usual standard form of a symplectic structure (theframe can be chosen holonomic if and only if dω = 0).

Then obviously there is ametric g0 (constant in that frame) such that (g0, ω) is a local section of Herm. If(g1, ω) is another local section then there is a fiberwise symplectic isomorphismf of TM|U (with possibly smaller U) such that f ∗g1 = g0, which is fiberwiseunique up to right multiplication by an element of Sp(m, R)∩O(2m, R) = U(m).To check the last statement we consider the diagramGL(R2m, TM) × GL(2m, R)/O(2m, R)A−−−−→S2+T ∗MyGL(R2m,T M)×GL(2m,R)/O(2m,R)GL(2m,R)∼=−−−−→S2+T ∗M

4O. GIL-MEDRANO, P. W. MICHORwhere A(u, g.O(2m, R))(X, Y ) = u−1(X)tggtu−1(Y ) for u ∈GL(R2m, TxM) andX, Y ∈TxM.Likewise we haveGL(R2m, TM) × GL(2m, R)/Sp(m, R)B−−−−→Λ2ndT ∗MyGL(R2m,T M)×GL(2m,R)/Sp(m,R)GL(2m,R)∼=−−−−→Λ2ndT ∗Mwhere B(u, g.Sp(m, R))(X, Y ) = u−1(X)tgJgtu−1(Y ).□1.3.

The manifold of almost hermitian structures. The space of almosthermitian structures on M is just the space H := C∞(Herm) of smooth sectionsof the fiber bundle Herm.

Since Herm is a subbundle of S2+T ∗M × Λ2ndT ∗M andsince the latter is an open subbundle of the vector bundle S2T ∗M × Λ2T ∗Mwe see that H is a splitting submanifold of M(M) × Ω2nd(M), by the followinglemma. The splitting property will also follow directly in section 2.1.Lemma.

Let (E, p, M, S) and (E′, p′, M, S′) be two fiber bundles over M andlet i: E →E′ be a fiber respecting embedding. Then the following embeddings ofspaces of sections are splitting smooth submanifolds:C∞(E)i∗−→C∞(E′) ֒→C∞(M, E′).Proof.

This is a variant of the results 10.6 and 10.10 in [10] and the proof issimilar to the ones given there, by direct finite dimensional construction.□1.4. Metrics on H = C∞(Herm).

On the space M(M) × Ω2nd(M) there aremany pseudo Riemannian metrics which are invariant under the diffeomorphismgroup and which are of first order. We shall consider the product metricG(g,ω)((h1, ϕ1), (h2, ϕ2)) ==ZMtr(g−1h1g−1h2) vol(g) +ZMtr(ω−1ϕ1ω−1ϕ2) vol(ω),where (h1, ϕ1), (h2, ϕ2) ∈TgM(M)×TωΩ2nd(M) = C∞c (S2T ∗M)×Ω2c(M).

Notethat vol(g) = vol(ω) if (g, ω) ∈H since Sp(m, R) and O(2m, R) are both sub-groups of SL(2m, R). So the restriction of G to H is given byG(g,ω)((h1, ϕ1), (h2, ϕ2)) :=ZMtr(g−1h1g−1h2) + tr(ω−1ϕ1ω−1ϕ2)vol(g).

GEODESICS ON SPACES OF ALMOST HERMITIAN STRUCTURES52. The geodesic equation in H2.1.

Splitting the tangent space of the space of almost hermitian struc-tures. Let (g, ω) ∈H = C∞(Herm) ⊂M(M) × Ω2nd(M) so that F(g, ω) :=g−1ω + ω−1g = 0.

Then for (h, α) ∈C∞c (S2T ∗M × Λ2T ∗M) we put H := g−1h,A := ω−1α, and of course J = g−1ω. We have (h, α) ∈T(g,ω)H if and only ifdF(g, ω)(h, α) = −g−1hg−1ω + g−1α + ω−1h −ω−1αω−1g = 0,which is easily seen to be equivalent to JH + HJ = JA + AJ.

Note that thisimplies tr(H) = tr(A).For (g, ω) ∈H we have the following G-orthogonal decomposition of the tan-gent space to M(M) × Ω2nd(M) at (g, ω):T(g,ω)(M(M) × Ω2nd(M)) = N 1(g,ω) ⊕N 2(g,ω) ⊕N 3(g,ω) ⊕N 4(g,ω)N 1(g,ω) = {(H, 0): JHJ = H}N 2(g,ω) = {(0, A): JAJ = A}N 3(g,ω) = {(H, A): JHJ = −H and H = A}N 4(g,ω) = {(H, A): JHJ = −H and H = −A}The tangent space to H is given by T(g,ω)H = N 1(g,ω) ⊕N 2(g,ω) ⊕N 3(g,ω) and itsG-orthogonal complement is given by (T(g,ω)H)⊥= N 4(g,ω). The restriction ofthe pseudometric to H is then non degenerate.The projectors on these subspaces can easily be constructed and in particularwe have the orthogonal projectors from T(g,ω)M(M) × Ω2nd(M) to the tangentspace T(g,ω)H and to the G-orthogonal complementPrT(g,ω)(H, A) :=3H + JHJ + A −JAJ4, 3A + JAJ + H −JHJ4Pr⊥(g,ω)(H, A) :=H −JHJ −A + JAJ4, A −JAJ −H + JHJ4.2.2.The geodesic equation.

The space H is a splitting submanifold ofM(M) × Ω2nd(M) and the tangent space splits nicely in the direct sum of thetangential and the orthogonal part, by 2.1; note that the projection operators arealgebraic. The tangential projection of the covariant derivative ∇ξη of smoothvector fields on M(M) × Ω2nd(M) which along H are tangential to H, is thusagain a smooth vector field, and the usual proof of Gauß’ formula involving thesix term expression of the Levi-Civita covariant derivative shows that PrT ∇ξηis the smooth Levi-Civita covariant derivative of G on H.So a curve σ(t) = (g(t), ω(t)) is a geodesic for the induced metric if and only ifthe covariant derivative of its tangent vector σt in M(M)×Ω2nd(M) is everywhereorthogonal to H; i. e. ∇∂tσt ∈N 4σ, or equivalently PrTσ (∇∂tσt) = 0.

6O. GIL-MEDRANO, P. W. MICHORIf we put X := g−1gt and W := ω−1ωt and use the Christoffel form from [7],(2.3 for α = 1/n), these conditions become, respectively:(1)JXt + 12 tr(X)JX = XtJ + 12 tr(X)XJXt + Wt = −12 tr(X)X −12 tr(W)W + 14(tr(X2) + tr(W 2))Id(2)3Xt + 32 tr(X)X −12 tr(X2)Id + JXtJ + 12 tr(X)JXJ++Wt + 12 tr(W)W −12 tr(W 2)Id −JWtJ −12 tr(W)JWJ = 03Wt + 32 tr(W)W −12 tr(W 2)Id + JWtJ + 12 tr(W)JWJ++Xt + 12 tr(X)X −12 tr(X2)Id −JXtJ −12 tr(X)JXJ = 02.3.The submanifold HJ of H. For a fixed almost complex structure Jon M let us consider HJ := {(g, ω) ∈H: g−1ω = J}, the space of almosthermitian structures with almost complex structure J.The tangent space isT(g,ω)HJ = N 3(g,ω), see 2.1Via the first projection the space HJ is diffeomorphic to the submanifold ofM(M) consisting of all g making J an isometry.

It is a totally geodesic sub-manifold in the Riemannian manifold (M(M), G). This follows from the generalresult:Lemma.

Let A: TM →TM be a vector bundle isomorphism covering the iden-tity. Then the space of all Riemannian metrics g on M such that g(AX, AY ) =g(X, Y ) for all X, Y ∈TxM is a geodesically closed submanifold of (M(M), G).This is also true for bilinear structures, or metrics with fixed signature.Proof.

The space of these g is the space of sections of the open subbundleS2+T ∗M ∩{g: A∗◦g ◦A = g} of the obvious vector subbundle, which is a smoothmanifold. A tangent vector h at such g is a tensor field with compact supportwith A∗◦h ◦A = h; for H = g−1h this is equivalent to A−1 ◦H ◦A = H. By[6], 3.2, the geodesic in M(M) starting at g in the direction h is the curveg(t) = g ea(t)Id+b(t)H0,where H0 is the traceless part of H and a(t) and b(t) are real valued functions.Then if g and h are A-invariant, so is the whole geodesic.□2.4.

The submanifold Hω. For a fixed non degenerate 2-form ω we considerHω := M(M) × {ω} ∩H, the space of almost hermitian structures with fixedω.

It is the space of sections of the pullback bundle ω∗(Herm, pr2, Λ2ndT ∗M) interms of 1.2, so a smooth manifold. Its tangent space is TgHω = N 1(g,ω).

Thespace Hω is a submanifold Mvol(ω) of the space of all metrics with fixed volumeequal to vol(ω), see 1.4.

GEODESICS ON SPACES OF ALMOST HERMITIAN STRUCTURES7Hω is a geodesically closed submanifold of Mvol(ω), see Blair [1], [2].Thegeodesics of Mvol(ω) have been determined by Ebin [3], see also [4]. The geodesicstarting at g in the direction h is given byg(t) = g etH0.Note that Mvol(ω) is not geodesically closed in M. These results are given forcompact M, but clearly they continue to hold for noncompact M in our setting.2.5.The submanifold Hg.

For a fixed metric g we may consider Hg :={g} × Ω2nd(M) ∩H, the space of almost hermitian structures with fixed g. It isthe space of all sections of the pullback bundle g∗(H, pr1, S2+T ∗M) in the notationof 1.2, so it is a smooth manifold. For the tangent space it easily follows thatTωHg = N 2(g,ω).

Since the situation is symmetric with respect to g or ω, it followsfrom 2.4 that Hg is a geodesically closed submanifold of the space Ω2nd(M)vol(g)of all almost symplectic structures with fixed volume equal to vol(g), and thegeodesics in Ω2nd(M)vol(g) are given byω(t) = ω etA0.3. The variational approach to the geodesic equation3.1.

It is not so easy to find an adapted chart for the subbundle Herm ⊂S2+T ∗M × Λ2ndT ∗M which would allow us to parameterize curves and their vari-ations in H. In order to achieve this parameterization we will use the followingscheme.We consider the bundle GL(TM, TM) of all isomorphisms of the tangent bun-dle. It is an open submanifold of the vector bundle L(TM, TM).

Any (fixed)almost hermitian structure (g0, ω0) ∈H induces a smooth mappingϕ = ϕ(g0,ω0): GL(TM, TM) →Herm ⊂S2+T ∗M × Λ2ndT ∗Mϕ(f) = ϕ(g0,ω0)(f) = (f ∗g0f, f ∗ω0f).The corresponding push forward mapping between the spaces of sections will bedenoted byΦ: G := C∞(GL(TM, TM)) →H = C∞(Herm) ⊂M × Ω2nd(M),Φ(f) = Φ(g0,ω0)(f) = ϕ ◦f = (f ∗g0f, f ∗ω0f).Remark. The mapping Φ: G = C∞(GL(TM, TM)) →H is not surjective ingeneral.

The mapping ϕ: GL(TM, TM) →Herm is the projection of a fiberbundle with typical fiber U(m). Since U(m) has nontrivial homotopy trying tolift a section s: M →Herm over ϕ will meet obstructions in general.

8O. GIL-MEDRANO, P. W. MICHOR3.2.Lemma.

For every curve (g(t), ω(t)) in H and also for every variation(g(t, s), ω(t, s)) of such a curve in H with (g(0), ω(0)) = (g0, ω0) there is a curvef(t) or variation f(t, s) in G = C∞(GL(TM, TM)) with (g, ω) = Φ(g0,ω0)(f).Proof. As noticed above ϕ: GL(TM, TM) →Herm is the projection of a smoothfiber bundle with compact fiber type U(m).

We choose a generalized connectionfor this bundle, see [11] or [9], section 9. Its parallel transportPt(c, t): GL(TM, TM)c(0) →GL(TM, TM)c(t)is globally defined for each curve c: R →Herm, and it is smooth in the choice ofthe curve, see loc.

cit.Then we just define f(t) := Pt((g(x,), ω(x,)), t) IdTxM and f(, t) will bea curve in G = C∞(GL(TM, TM)) with (g(t), ω(t)) = Φ(g0,ω0)(f(t)), and f(x, t)varies in t only for those x where also g(x, t) or ω(x, t) varies. So the compactsupport condition required of smooth curves of sections is automatically satisfied.For a variation of a curve we first define in turnf(x, t, 0) : = Pt((g(x,, 0), ω(x,, 0)), t) IdTxM,f(x, t, s) : = Pt((g(x, t,), ω(x, t,)), s)f(x, t, 0).□3.3.

Let (g(t), ω(t)) be a smooth curve in H = C∞(Herm), so it is smoothM × R →Herm and for each compact [a, b] ⊂R there is a compact set K ⊂Msuch that (g(x, t), ω(x, t)) is constant in t ∈[a, b] for each x ∈M \K, see ([11], 6.2,a slight mistake there). Then its energy with respect to the metric G of 1.4 isgiven by(1)Eba(g, ω) =Z baZMtr(g−1gtg−1gt) + tr(ω−1ωtω−1ωt)vol(g)dt.Since we cannot parameterize curves and their variations in H explicitly we willparameterize them with the help of Φ = Φ(g0,ω0): G →H, where (g0, ω0) =(g(0), ω(0)).

So let f(t) be a smooth curve in G = C∞(GL(TM, TM)). Then wehaveΦ(f) = Φ(g0,ω0)(f) = (f ∗g0f, f ∗ω0f),TfΦ(ft) = (f ∗t g0f + f ∗g0ft, f ∗t ω0f + f ∗ω0ft),so the energy of the curve Φ(f(t)) in H is given by(2)Eba(Φ(f)) ==Z baZMtr(f −1g−10 (f ∗)−1(f ∗t g0f + f ∗g0ft)f −1g−10 (f ∗)−1(f ∗t g0f + f ∗g0ft))+ tr(f −1ω−10 (f ∗)−1(f ∗t ω0f + f ∗ω0ft)f −1ω−10 (f ∗)−1(f ∗t ω0f + f ∗g0ft))det(f) vol(g0)dt.

GEODESICS ON SPACES OF ALMOST HERMITIAN STRUCTURES9Lemma. A curve f(t) in G is a critical point of the functional (2) if and onlyif Φ(f(t)) is a critical point of the functional (1), i. e. t 7→Φ(f(t)) is a geodesic.Proof.

An (infinitesimal) variation in G can (with the help of a connection forϕ: GL(TM, TM) →Herm) be written as a sum of two variations: the horizontalone corresponds exactly to a variation of Φ(f) in H, and along the vertical onethe functional is stationery anyhow.□3.4. Lemma.

In the setting of 3.3, for a variation f(t, s) ∈G with fixed end-points we have the first ‘variation formula’∂∂s0Eba(Φ(g0,ω0)(f(, s))) ==Z baZMtrE(g0, ω0, f; t)fsf −1det(f) vol(g) dt.whereE(g0, ω0, f; t) = −2fttf −1 −2g−10 (f ∗)−1f ∗ttg0 + 2ftf −1ftf −1−2 tr(ftf −1)ftf −1 + tr(ftf −1ftf −1)Id −2g−10 (f ∗)−1f ∗t g0ftf −1+ 2g−10 (f ∗)−1f ∗t (f ∗)−1f ∗t g0 + 2ftf −1g−10 (f ∗)−1f ∗t g0−2 tr(ftf −1)g−10 (f ∗)−1f ∗t g0 + tr(g−10 (f ∗)−1f ∗t g0ftf −1)Id−2fttf −1 −2ω−10 (f ∗)−1f ∗ttω0 + 2ftf −1ftf −1−2 tr(ftf −1)ftf −1 + tr(ftf −1ftf −1)Id−2ω−10 (f ∗)−1f ∗t ω0ftf −1 + 2ω−10 (f ∗)−1f ∗t (f ∗)−1f ∗t ω0+ 2ftf −1ω−10 (f ∗)−1f ∗t ω0 −2 tr(ftf −1)ω−10 (f ∗)−1f ∗t ω0+ tr(ω−10 (f ∗)−1f ∗t ω0ftf −1)IdProof. This is a long but straightforward computation.We may interchange∂∂s|0 with the first integral since this is finite dimensional analysis, and we mayinterchange it with the second one, sinceRM is a continuous linear functional onthe space of all smooth densities with compact support on M, by the chain rule.Then we use that tr∗is linear and continuous, d(vol)(g)h = 12 tr(g−1h) vol(g), andthat d(()−1)∗(g)h = −g−1hg−1 and partial integration; there are no boundaryterms since we assumed the variation to have fixed endpoints.□3.5.Lemma.

For curve f(t) in G the curve Φ(g0,ω0)(f(t)) is a geodesic in(H, G) if and only if f(t) satisfies the following equation:E(g0, ω0, f, t) = 0.Proof. This follows from 3.4 since the integral in (3) describes a nondegenerateinner product on G, given byGf(h, k) =ZMtr(hf −1kf −1) det(f) vol(g0).□

10O. GIL-MEDRANO, P. W. MICHOR3.6.Comparison with section 2.

Let Φ(g0,ω0)(f(t)) = (f ∗g0f, f ∗ω0f) =:(g(t), ω(t)). Then the expressions used in section 2 becomeX = g−1gt = f −1g−10 (f ∗)−1f ∗t g0f + f −1ft,W = ω−1ωt = f −1ω−10 (f ∗)−1f ∗t ω0f + f −1ft,J = g−1ω = f −1g−10 ω0f.If we compute Xt, Wt and insert this into the second equation of 2.2.

(1), we getexactly f −1E(g0, ω0, f, t)f = 0. So we get the same geodesic equation as in 2.24.

Some properties of the geodesicsWe are not able to give the explicit solution of the geodesic equation on H.But we can give some explicit formulas of the time evolution of some functionsof the structures.4.1. Proposition.

Let (g(t), ω(t)) be the geodesic of H starting at (g0, ω0) in thedirection (h, α), let H = g−10 h and A = ω−10 α, and let (X, W) = (g−1gt, ω−1ωt)as in 2.2. Then we havetr(X2) + tr(W 2) = (det(g−10 g))−1/2(tr(H2) + tr(A2)).Proof.

Since (g, ω) is in H its tangent vector (gt, ωt) satisfiesJXJ −X = JWJ −W(1)tr(X) = tr(W)(2)so that 2.2. (2) becomes3Xt + JXtJ + Wt −JWtJ + 2 tr(X)X −12(tr(X2) + tr(W 2))Id = 03Wt + JWtJ + Xt −JXtJ + 2 tr(W)W −12(tr(W 2) + tr(X2))Id = 0We multiply now the first equation by X, the second equation by W and addthem to obtain2(tr(XtX) + tr(WtW)) + 12(tr(X2) + tr(W 2)) tr(X) = 0.From that it is easy to see that the derivative of (tr(X2)+tr(W 2))(det(g−10 g))1/2is zero, where we also use ((det(g−10 g))1/2)t = 12(det(g−10 g))1/2 tr(X).□

GEODESICS ON SPACES OF ALMOST HERMITIAN STRUCTURES114.2. Proposition.

Let (g(t), ω(t)) be the geodesic of H starting at (g0, ω0) in thedirection (h, α), let H = g−10 h and A = ω−10 α, and let (X, W) = (g−1gt, ω−1ωt).We put p(t) := 12(det(g−10 g))1/2. Then we havep(t) = n32(tr(H2) + tr(A2))t2 + 12 tr(H)t + 1,(1)tr(X) = tr(W) = 2p′(t)p(t)(2)= 4n(tr(H2) + tr(A2))t + 8 tr(H)n(tr(H2) + tr(A2))t2 + 8 tr(H)t + 32,X + W =1p(t)14(tr(H2) + tr(A2))tId + H + A(3)Proof.

We take the trace in the second expression of 2.2. (1) and use (2) from theproof of 4.1 to obtain2 tr(X)′ = −tr(X)2 + n4 (tr(X2) + tr(W 2)).Inserting 4.1 we get2 tr(X)′ = −tr(X)2 + n4Cp(t),where C = tr(H2) + tr(A2).

From the proof of 4.1 we have in turnp′(t) = 12p(t) tr(X)p′′(t) = 14p(t) tr(X)2 + 12p(t) tr(X)′ = nC16 .For the initial conditions p(0) = 1 and p′(0) = 12 tr(H) this givesp(t) = nC32 t2 + 12 tr(H)t + 1and consequently assertions (1) and (2).Now we take the tracefree part of the second expression in 2.2. (1)(X + W)′0 = −14 tr(X + W)(X + W)0,and by an argument similar to that used in the proof of [7], 2.5 we getX + W = a(t)Id + b(t)(H0 + A0), where a(t) = 2 tr(X)n= 4np′(t)p(t) ,and it just remains to find b(t) when (H0 + A0) ̸= 0.We have (X + W)0 = b(t)(H0 + A0), thus (X + W)′0 = b′(t)(H0 + A0).

But wealso know that (X + W)′0 = −n4 a(t)b(t)(H0 + A0) and so we getb′b = −na4 = −p′pwith p(0) = 1 and b(0) = 1, so b = 1/p and we get assertion (3).□

12O. GIL-MEDRANO, P. W. MICHORReferences1.

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Kriegl, Andreas; Michor, Peter W., A convenient setting for real analytic mappings, ActaMathematica 165 (1990), 105–159.9. Kol´aˇr, Ivan; Slov´ak, Jan; Michor, Peter W., Natural operations in differential geometry, toappear, Springer-Verlag, Heidelberg-Berlin, 1993.10.

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Gil-Medrano: Departamento de Geometr´ıa y Topolog´ıa, Facultad de Mate-m´aticas, Universidad de Valencia, 46100 Burjassot, Valencia, SpainP. W. Michor: Institut f¨ur Mathematik, Universit¨at Wien, Strudlhofgasse 4, A-1090 Wien, Austria; ‘Erwin Schr¨odinger International Institute of MathematicalPhysics’, Pasteurgasse 6/7, A-1090 Wien, Austria.E-mail address: michor@pap.univie.ac.at

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