GENERALIZED PLANAR CURVES AND QUATERNIONIC GEOMETRY JAROSLAV HRDINA AND JAN SLOVAK Abstract. Motivated by the analogies between the projective and the almost quaternionic geometries, we first study the generalized planar curves and mappings. We follow, recover, and extend the classical approach, see e.g. [fO, 11]. Then we exploit the impact of the general results in the almost quaternionic geometry. In particular we show, that the natural class of H-planar curves coincides with the class of all geodesies of the so called Weyl connections and preserving this class turns out to be the necessary and sufficient condition on diffeomorphisms to become morphisms of almost quaternionic geometries. Various concepts generalizing geodesies of amne connections have been studied for almost quaternionic and similar geometries. Let us point out the generalized geodesies defined via generalizations of normal coordinates, cf. [2] and [3], or more recent [4, 12]. Another class of curves was studied in [11] for the hypercomplex structures with additional linear connections. The latter authors called a curve c quaternionic planar if the parallel transport of each of its tangent vectors c(t0) along c was quaternionic colinear with the tangent field c to the curve. Yet another natural class of curves is given by the set of all unparameterized geodesies of the so called Weyl connections, i.e. the connections compatible with the almost quaternionic structure with normalized minimal torsion. The latter connections have remarkably similar properties for all parabolic geometries, cf. [3], and so their name has been borrowed from the conformal case. In the setting of almost quaternionic structures there were studied first in [8] and they are also called Oproiu connections, see [1]. The first author showed in [6] that actualy the concept of quaternionic planar curves was well defined for the almost quaternionic geometries and their Weyl connections. Moreover, it did not depend on Date: November 14, 2005. The essential part of writing of this paper has been done with support of........... during the stay of the second author at the University of Auckland. The authors announce the support of grants GACR Nr. 201/05/H005 and 201/05/2117, respectively. 1 2 jaroslav hrdina and jan slovák the choice of a particular Weyl connection and it turned out that the quaternionic planar curves were just all unparameterized geodesies of all Weyl connections. The aim of this paper is to find further analogies of Mikes's classical results in the realm of the almost quaternionic geometry. On the way we simplify, recover, and extend results on generalized planar mappings, explain results from [6], and finally we show that morphisms of almost quaternionic geometries are just those diffeomorphisms which leave invariant the class of all unparameterized geodesies of Weyl connections. 1. Motivation and background on quaternionic geometry There are many equivalent definitions of almost quaternionic geometry to be found in the literature. Let us start with the following one: Definition 1.1. Let M be a smooth manifold of dimension An. An almost hypercomplex structure on M is a triple (I, J, K) of smooth affinors in Y(T*M (g> TM) satisfying I2 = J2 = —E, K = I o J = -J o I where E = idxM- An almost quaternionic geometry is a rank four subbundle Q C T*M (g) TM locally generated by the identity E and a hypercomplex structure. An almost complex geometry on a 2m-dimensional manifold M is given by the choice of the affinor J satisfying J2 = —E. Let us observe, that such a J is uniquely determined within the rank two subbundle {E, J) C TM, up to its sign. (Indeed, if J = aE+b,J, then the condition J2 = —E implies a = 0 and b = ±1.) Thus we may view the almost quaternionic geometry as a straightforward generalization of this case. Here, a similar simple computation reveals that the rank three subbundle {I, J, K) is invariant of the choice of the generators and this is the definition we may find in [1]. More explicitly, different choices will always satisfy I = al + bJ + cK with a2 + b2 + d2 = 1, and similarly for J and K. Let us also remark that the 4-dimensional almost quaternionic geometry coincides with 4-dimensional conformal Riemannian geometries. 1.2. The frame bundles. Equivalently, we can define an almost quaternionic structure Q on M as a reduction of the linear frame bundle P1M to an appropriate structure group, i.e. as a G-structure with the structure group of all automorphisms preserving the subbundle Q. We may view such frames as linear mappings TXM —> EI™ which carry over generalized planar curves and quaternionic geometry 3 the multiplications by i,j,k g EI onto some of the possible choices for /, J, K. Thus, a further reduction to a fixed hypercomplex structure leads to the structure group GL(n, EI) of all quaternionic linear mappings on EI™. Additionaly, we have to allow morphisms which do not leave the affinors /, J, K invariant but change them within the subbundle Q. As well known, the resulting group is where Sp(l) are the unit quaternions in GL(1,M), see e.g. [9]. We shall write Q0 C PXM for this principal G0-bundle defining our structure. The simplest example of such a structure is well understood as the homogeneous space Since P is a parabolic subgroup in the semisimple Lie group G, the almost quaternionic geometry is an instance of the so called parabolic geometries. All these geometries enjoy a rich and quite uniform theory similar to the classical development of the conformal Riemannian and projective geometries, but we shall not need much of this here. We refer the reader to [3] and the references therein. 1.3. Weyl connections. The classical prolongation procedure for G-structures starts with finding a minimal available torsion for a connection belonging to the structure on the given manifold M. Unlike the projective and conformal Riemannian structures where torsion free connections always exist, the torsion has to be allowed for the almost quaternionic structures in general in dimensions bigger than four. The standard normalization comes from the general theory of parabolic geometries and we shall not need this in the sequel. The details may be found for example in [5], [2], another and more classical point of view can be found in [9]. The only essential point for us is that all connections compatible with the given geometry sharing the unique normalized torsion are parametrized by smooth one-forms on M. In analogy to the conformal Riemannian geometry we call them Weyl connections for the given almost quaternionic geometry on M. Go = GL{n,M) xZ2Sp{l) FnU = G/P where Gq C P are the subgroups in G = SL(n + 1, EI) 4 jaroslav hrdina and jan slovák The almost quaternionic geometries with Weyl connections without torsion are called quaternionic geometries. From the point of view of prolongations of G-structures, the class of all Weyl connections defines a reduction Q of the semiholonomic second order frame bundle over the manifold M to the structure group P and the Weyl connections V are in bijective correspondence with Go~equivariant sections a : Qq —> Q of the natural projection. As mentioned above, the difference of two Weyl connections is a one-form and, also in full analogy to the conformal geometry, there are neat formulae for the change of the covariant derivatives of two such connections V and V in terms of their difference Y = V — V g Q1(M). 1.4. Adjoint tractors. In order to understand the latter formulae, we introduce the so called adjoint tractors. They are sections of the vector bundle A = G0 xGog. The Lie algebra g = sl(n + 1, HI) carries the G0~invariant grading 0 = 0-1 ©00 ©01 where So = { (o a) ; a G B'fa'11)' Re(fl + Tr A) = 0 Moreover, TM = Q0 xGo q_x, T*M = Q0 xGo qx, and we obtain on the level of vector bundles A = A-1®Ao®A1 = TM®A0® T*M. The key feature of A is that all further Go~irrvariant objects on g are carried over to the adjoint tractors, too. In particular, the Lie bracket on G induces an algebraic bracket { , } on A. Now we may write down easily the transformation formula. Let V and V be two Weyl connections, V — V = T g T(Ai). Then for all tangent vector fields X, Y g T(A-i), (1) VXY = VXY + {{X,T},Y}, see [3] or [2, 5] for the proof. Notice that the internal bracket results in an endomorphism on TM, while the external bracket is just the evaluation of this endomorphism on Y (all this is read of the brackets in the Lie algebra easily). generalized planar curves and quaternionic geometry 5 2. Generalized planar curves and mappings Various geometric structures on manifolds are denned as smooth distributions in the vector bundle T*M ®TM of all endomorphisms of the tangent bundle. We have seen the two examples of almost complex and almost quaternionic structures above. Let us extract some formal properties from these examples. Definition 2.1. Let A be a smooth C-dimensional vector subbundle in T*M ®TM, such that the identity affinor E = i&tm restricted to TXM belongs to AXM C T*M ® TXM at each point x G M. We say that M is equipped by an A-structure. For any tangent vector X G TXM we shall write A(X) for the vector subspace A(X) = {F(X); F G AXM} c TXM and we call A(X) the A-hull of the vector X. Similarly, the A-hull of a vector field will be the subbundle in TM obtained pointwise. Notice that the dimension of such a subbundle in TM may vary pointwise. We say that the ^-structure has generic rank I if the dimension of A is £, and for each x G M, the subset of vectors (X, Y) G TXM x TXM, such that the A-hulls A(X) and A(Y) generate a vector subspace A(X) © A(Y) of dimension 2£, is open and dense. Let us point out some examples: • The (E')-structure is of generic rank one on all manifolds of dimensions at least 2. • Any almost complex structure or almost product structure {E, J) is of generic rank two on all manifolds of dimensions at least 4. • Any almost quaternionic structure is of generic rank four on all manifolds of dimensions at least 8. Definition 2.2. Let M be a smooth manifold with a given A-structure and a linear connections V. A smooth curve c : R —> M is told to be A-planar if Vtc eA(c). Clearly, A planarity means that the parallel transport of any tangent vector to c has to stay within the A-hull A(c) of the tangent vector field c along the curve. Moreover, this concept does not depend on the parametrization of the curve c. Definition 2.3. Let M be a manifold with a linear connection V and an A-structure, while N be another manifold with a linear connection 6 jaroslav hrdina and jan slovák V and a B-structure. A diffeomorphism f : M —> N is called (A,B)-planar if each A-planar curve c on M is mapped onto the B-planar curve F o c on N. Example 2.4. The 1-dimensional A structures are all trivial since the distribution A = (E) must be given just as the linear hull of the identity affinor E, by the definition. Obviously, the (E)-planar curves on a manifold M with a linear connection V are exactly the unparametrized geodesies. Moreover, two connections V and V without torsion are protectively equivalent (i.e. they share the same unparametrized geodesies) if and only if their difference satisfies ViF-Vi-F = a(X)Y + a(Y)X for some one-form a on M. The latter condition can be rewritten as The latter condition on projective structures may be also rephrased in the terms of morphisms: A diffeomorphism / : M —> M is geode-tical (an automorphism of the projective structure) if / o c is an unparametrized geodesic for each geodesic c and this happens if and only if the symmetrization of the difference /*V —V is a section of T*M&(E). We are going to generalize the above example in the rest of this section. In the case A = (E), the ((E), S)-planar mappings are called simply B-planar. They map each geodesic curve on (M, V) onto a 5-planar curve on (N, V, B). Each I dimensional A structure A C T*M TM determines the distribution in S2T*M TM, given at any point x E M by A^M = {a1QF1 + --- + aeQFe; a, g T*M, Fl g AXM}. Theorem 2.5. Let M be a manifold with a linear connection V, let N be a manifold of the same dimension with a linear connection V and an A-structure of generic rank £, and suppose dimM > 11. Then a diffeomorphism f : M —> N is A-planar if and only if where Sym denotes the symmetrization of the difference of the two connections. Proof. Let us first observe that the entire claim of the theorem is of local character. Thus, identifying the objects on N with their pullbacks on M, we may assume that M = N and / = id^- Next, let us observe that the A-planarity of / : M —> N does not at all depend on the possible torsions of the connection. Indeed, we always test expressions od the type VcC for a curve c and thus deforming V into V' = V + T by adding some torsion will not effect the results. (2) V - V g T (T* M 0 {E}) c T{S2T*M ® TM). (3) Sym(rV-V)Gr(^) generalized planar curves and quaternionic geometry 7 Thus, without any loss of generality, we may assume that the connections V and V share the same torsion, and then we may ommit the symmetrization from equation (3). Finally, we may fix some (local) basis E = F0, Fl, i = 1,... £ — 1, of A, i.e. A = (Fq, ..., Fe-i). Then the condition in equation (3) says e-i (4) V = V + J^al0Fl i=0 for some suitable one-forms at on M. Of course, the existence of such forms does not depend on our choice of the basis of A. The quite simplified statement we now have to prove is: Claim 1. Let M be a manifold of dimension at least 21, V and V two connections on M with the same torsion, and consider an A-structure of generic rank I on M. Then each geodesic curve with respect to V is A-planar with respect to V if and only if there are one-forms at satisfying equation (4)- Assume first we have such forms al5 and let c be a geodesic for V. Then equation (4) implies V^c G A(c) so that c is A-planar, by definition. The other implication is the more difficult one. Assume each (un-parametrized) geodesic c is A-planar. This implies that the symmetric difference tensor P = V - V G T(S2T*M ® TM) satisfies P{c,c) = VcCG 1R are smooth and their restrictions to the individual rays (half-lines) generated by vectors in V are linear. Let us fix a local smooth basis e% G TM, the dual basis el, and consider the induced dual bases ej and e1 on the multivectors and 8 jaroslav hrdina and jan slovak exterior forms. Let us consider the smooth mapping X : AeTM \ {0} - AeT*M, xiE^1) = £ ^2e'- 1^ ai Now, for all non-zero tensors S = ^aje1, the evaluation is the constant function 1, while x(k ' = Next, we define for each X G V r(X) = X(X A Fx(X) A • • • A F^X)) and we may compute the unique coefficients at from (5): ao(X) = {P(X, X) A A F2(X) A • • • A t(X)) «i(X) = {XAP(X,X) AF2(X) A-- - AFn(I),r(I)) a^-i(X) = (XAFi(X) AF2(X) A--- A X), r(X)). In particular, this proves the first part our Claim 2. Let us now consider a fixed vector X £ V. The defining formula (5) for ctj implies at(kX) = kat(X), for each real number k ^ 0. Passing to zero with positive k shows that a does have the limit 0 in the origin and so we may extend the definition of ctj's (and validity of formula (5) ) to the entire cone V U {0} by setting ctj(0) = 0 for all i. Finally, along the ray {tX;t > 0} C V the derivative a'(tX) = j-ta(tX) satisfies a'ikX) = a'(X) for all X G V, and thus a'itX) is constant along the ray. This proves the rest of Claim 2. Now, in order to complete the proof of Theorem 2.5, we have to prove the following assertion. Claim 3. // a smooth symmetric tensor P is determined over the above defined subspace VU{0} by (5), then the coefficients al are smooth one-forms on M and the tensor P is given by 1 e~X P(X,Y) = -^2(ai(Y)Fi(X) + ai(X)Fi(Y)). i=0 The entire tensor P is obtained through polarization from its evaluations P{X,X), X G TM, (6) P(X, Y) = \ (P(X + Y,X + Y)- P(X, X) - P(Y, Y)), and again, the entire tensor is determined by its values on arbitrarily small non-empty open subset of X and Y in each fiber. generalized planar curves and quaternionic geometry 9 The summands on the right hand side have values in the following subspaces: P(X + Y, X + Y) G (X + Y, + Y),..., F^X + Y)) C (X, Fx (X),... Fe-!(X), Y, F1 (Y),..., (Y)), P(X,X) G (X^X),...^^)), PiYX) e (Y,F1(Y),...Fe-1(Y)). Since we have assumed that A has generic rank £, the subspace W G TM xMTM of vectors (X, F) such that all the values {X, FX(X),..., F,_x(X), Y, -Fi(F),..., F,_x(Y)} are linearly independent is open and dense. Clearly W C VxMV. Moreover, if (X, F)GW than F0(X + Y),F^X + Y) are independent, i.e. X + Y G V. Inserting (5) into (6), we obtain e-i P(X, Y) = J^(X, Y)Fi(X) + ei(X, Y)Ft(Y)). i=0 For all (X, Y) G W, the coefficients d,(X, F) = \{a,{X + Y) — a,(X)) at Fi(X), and e^X,!") = ^(X + Y) - a,(Y)) at Ft(Y) in the latter expression are uniquely determined. The symmetry of P implies di(X,Y) = e^Y, X). If (X,Y) G W then also (sX,tF) G W for all non-zero reals s, t and the linearity of P in the individual arguments yields for all real parameters s, t stdi(X,Y) = sdi(sX,tY). Thus the functions al satisfy a^sX + tY) - Oi(sX) = t(ai(X + Y)- at(X)). Since ctj(tX) = tal(X), in the limit s —> 0 this means ai(Y) = ai(X + Y)-ai(X). Thus ctj are additive over the open and dense set (X, Y) G W. Choosing a basis of TXM such that each couple of basis elements is in W, this shows that at are restrictions of linear forms, as required. □ Theorem 2.6. Let M be a manifold with a linear connection V and an A-structure, N be a manifold of the same dimension with a linear connection V, and suppose that B has generic rank £. Then a diffeo-morphism f : M —> X is (A, B)-planar if and only if f is B-planar and A{X) C {f*{B)){X) for all X G TM. 10 jaroslav hrdina and jan slovák Proof. As in the proof of Theorem 2.5, we may restrict ourselves to some open submanifolds, fix generators Ft for B, assume that / = id^ and both connections V and V share the same torsion, and prove the equivalent local assertion to our theorem: Claim. Each A-planar curve c with respect to V is B-planar with respect to V, if and only if the symmetric difference tensor P = V — V is of the form (5) with smooth one-forms al} i = 0, ...,£ — 1 and A(X) C B(X) for each X G TM. Obviously, the condition in this statement is sufficient. So let us deal with its neccessity. Since every (A, S)-planar mapping is also 5-planar, Theorem 2.5 (or the equivalent Claim 1 in its proof) says that e P(X,X) = ^al(X)Fl(X) 3=0 for uniquely given smooth one-forms al. Now, consider a fixed Fei and suppose F(X) ^ B(X). Since we assume that all {E, F)-planar curves c in M are 5-planar, we may proceed exactly as in the beginning of the proof of Theorem 2.5 to deduce that e P(X, X) = ^ ai(X)Fi(X) + P(X)F(X) 3=0 on a neighborhood of X, with some unique functions at and [3. The comparison of the latter two unique expressions for P(X, X) shows that (3(X) vanishes. But since F(X) ^ X, there definitly are curves which are {E, F)-planar and tangent to X, but not (E')-planar. Thus, the assumption in the theorem would lead to (3(X) 7^ 0. Consequently, our choice F(X) ^ B(X) cannot be achieved and we have proved A(X) C B(X) for all X G TM. □ 3. Results on quaternionic geometries The main result of this section is: Theorem 3.1. Let f : M —> M' be a diffeomorphism between two almost quaternionic manifolds of dimension at least eight. Then f is a morphism of the geometries if and only if it preserves the class of unparametrized geodesies of all Weyl connections on M and M'. This theorem will follow easily from the results of Section 2 and its proof requires only a few quite simple formal steps. Let Q be the GENERALIZED PLANAR CURVES AND QUATERNION!C GEOMETRY 11 subbundle in T*M ® TM denning the almost quaternionic geometry on M as in 1.1. Lemma 3.2. A curve c : R —> M is Q-planar with respect to at least one Weyl connection V on M if and only if c is Q-planar with respect to all Weyl connections on M. Proof. For a Weyl connection V and a curve c : R —> M, the defining equation for Q-planarity reads Vcc G Q(c). If we choose some hypercomplex structure within Q, we may rephraze this condition as: Vcc = c-q for a quaternion q. Now the formula (1) for the deformation of the Weyl connections implies Indeed, this is the consequence of the computation of the Lie bracket in g of the corresponding elements c G T G Q±: Thus we see that if there is such a quaternion q for one Weyl connection, Definition 3.3. A curve c : R —> M is called W-planar if it is Q-planar with respect to each Weyl connection V on M. Theorem 3.4. Let M be a manifold with an almost quaternionic geometry. Then a curve c : R —> M is W-planar if and only if c a geodesic of some Weyl connection, up to parametrization. Proof. Let us remark that c is a geodesic for V if and only if Vcc = 0. Thus, the statement follows immediately from the computation in the proof of Lemma 3.2. Indeed, if c is H-planar, then choose any Weyl connection V and pick up T so that Vcc vanishes. □ 3.5. Proof of Theorem 3.1. Every morphism of almost quaternionic geometries preserves the class of Weyl connections and thus also the class of their geodesies. We have to prove the opposit implication. This means, we have two manifolds with almost quaternionic structures (M,Q), (N,Qr) and a diffeomorphism / : M —> N which is (Q, Q')—planar. Then Theorem 2.6 implies that Q(X) = (f*Q')(X) for each X G TM (since they both have the same dimension). But this is equivalent to the statement Q C f*Q', i.e. we have proved that / preserves the defining subbundles and thus is a morphism of the almost quaternionic structures. This still has to be checked! Probably nearly the same computation as showing the relations of hypercomplex structures within q, but we have to show that the coefficients do not depend on X!! Vcc = Vcc + {{c, T}, c} = 2c • T(c). then it exists also for all of them. □ 12 jaroslav hrdina and jan slovák 3.6. Final remarks. All curves in the four-dimensional quaternionic geometries are H-planar by the definition. Thus this concept starts to be interesting in higher dimensions only, and all of them are covered by Theorem 3.1. The class of the unparametrized geodesies of Weyl connections is well defined for all parabolic geometries. Our result for the quaternionic geometries suggests the question, whether a similar statement holds for other geometries as well. ?????Some more???? References [1] Alekseevsky, D.V.; Marchiafava, S.; Pontecorvo, M., Compatible Complex Structures on Almost Quaternionic Manifolds, Trans. Amer. Math. Soc 351 (1999), no. 3, 997-1014. [2] Bailey, T.N.; Eastwood, M.G., Complex paraconformal manifolds - their differential geometry and twistor theory, Forum Math., 3 (1991), 61-103. [3] Cap, A.; Slovák, J., Weyl Structures For Parabolic geometries, Math. Scand. 93 (2003), 53-90. [4] Cap, A.; Slovák, J.; Zádník, V.; On distinguished curves in parabolic geometries, Transform. Groups, Vol. 9, No. 2, ()2004), 143-166. [5] Cover, A.R.; Slovák, J., Invariant local twistor calculus for quaternionic structures and related geometries, J. Geom. Physics, 32 (1999), no. 1, 14-56. 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