T,(V,,)=eU v. )KF and m I + )K + F K (1.2) T,(V,,) is the BUNDLE OF ORDER 2. (Received Febraury 21, 1991 and in revised form March 29, 1991)

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1 Internat. J. Math. & Math. Sci. VOL. 16 NO. (1993) PROLONGATIONS OF F-STRUCTURE TO. THE TANGENT BUNDLE OF ORDER 2 LOVEJOY S. DAS 201 Department of Mathematics Kent State University, Tuscarawas Campus New Philadelphia, Ohio (Received Febraury 21, 1991 and in revised form March 29, 1991) ABSTRACT. presented. A study of prolongations of F-structure to the tangent bundle of order 2 has been KEY WORDS AND PHRASES. Prolongations, tangent bundle, integrable, lift, F-structure AMS SUBJECT CLASSIFICATION CODE. 53C15 1. INTRODUCTION. Let F be a nonzero tensor field of type (1,1) and of class c such that [1] on an n-dimensional manifold V F K +(-)K+IF--O and F TM +(-)W+IF #0 for <W <K (1.1) where K is a fixed positive integer greater than 2. Such a structure on V, is called an F-structure of rank r and degree K. If the rank of F is constant and r r(f), then v, is called an F-structure manifold of degree K( _> 3). The case when K is odd has been considered in this paper. Let the operators on v, be defined as follows [1]: where I denotes the identity operator on V,. From the operators defined by (1.2) we have )KF and m I + )K + F K (1.2) + m I and 12 l; and m m (1.3) For F satisfying (1.1), there exist complementary distributions L and M corresponding to the projection operators and m respectively. If rank (F) constant on V,, then dim L r and dim M (n- r). We have the following results [l Fi IF F and Fm= mf 0 (1.4a) FK-l -I and FC-m 0 (1.4b) 2. PROLONGATIONS OF F-STRUCTURE IN THE TANGENT BUNDLE OF ORDER 2. Let V, be an n-dimensional differentiable manifold of class c and T,(V,,)=eU v T,(V,,) is the t, tangent bundle over the manifold v,. Let us denote T;(V,), the set of all tensor fields of class c and of the type (r,s) in V, and T(V,) be the tangent bundle over V,.

2 202 L.S. DAS Let us introduce an equivalence relation in the set of all differentiable mappings F:RV, where R is the real line. Let r >_ be a fixed integer. If two mappings F: R-.V, and G: R--V, satisfy the conditions Fh(O) Gh(O), dfh(o) dga(o) dr (o) dt dt dt the mapping F and G being represented respectively by Xh= Fh(t) and X h =Gh(t), (t E R) with respect to local coordinates x in a coordinate neighborhood {U,X} containing the point P F(0)=G(0), then we say that the mapping F is equivalent to G. Eah equivalence class determined by the equivalence relation is called an r-jet of V. and denoted by J(F). The set of all r-jets of V. is called the tangent bundle of order r and denoted by T.(Vn). The tangent bundle T2(V.) of order 2 has the natural bundle structure over Vn, its bundle projection being defined Iy 2(Jp2(F))= P. If we introduce a mapping such that P F(0), then T(V.) has a bundle structure over T(Vn) with projection Let us denote T(Vn) the second order tangent bundle over v and let F u be the second lift of F in T(Vn). The second lift F u which belong to T(T(Vn) has component of the form [3] F 0 0 FII: ".F + (/),.F.F F (2.1) with respect to the induced coordinates in T2(Vn), F being local components of F in V.. Now we obtain the following results on the second liit of F satisfying (1.1). For any F,G T(Vn), the following holds [3]: therefore we have (GIIFII)X II GII(FXII), :GII(Fx) II :(G(FX)) II (GF)fIX II for every X T(Vn) GUF II If P(s) denote a polynomial of variable s, then we have (GF) I (2.2) (P(F)) II p(fii), where F T](Vn) (2.3) We have the following theorem: THEOREM 2.1. The second lift F u defines a F-structure in T(V.) iff F defines a F-structure in V,. PROOF. Let F satisfy (1.1) then F defines F-structure in V. satisfying which in view of equation (2.3) yields F K

3 PROLONGATIONS OF F-STRUCTURE TO THE TANGENT BUNDLE 203 (FH) K + )K + 1FH 0. Therefore F H defines a F-structure in T2(Vn). The converse can be proved in a similar manner. THEOREM 2.2. The second lift F is integrable in T2(V,,), iff F is integrable in Vn. PROOF. Let us denote N H and N, the Nijenhuis tensors of F H and F respectively. Then we have [2] We know that F-structure is integrable in V,, iff N11(X,Y) (N(X,Y)) 11 (2.5) which in view of (2.5) is equivalent to N(X,V)=O, NII(X,Y =0. (2.6) Thus F H is integrable, iff F is integrable in V,,. THEOREM 2.3. The second lift F H of F is partially integrable in T2(V,,), iff F is integrable in V and PROOF. We know that for F to be partially integrable in V,,the following holds [2]: which, in view of equation (2.5), takes the form N(X, tv) o N(mX, my) O, NII(IIIXII, III, yii) 0 and (2.7) NII(mlIXII, mliy II) O. where ill, m II axe operators in T(V,,) which define the distribution L H and M II respectively. equation (2.7) gives the condition for F to be partially integrable. The converse follows in a similar manner. REFERENCES 1. KIM, J.B., Notes on l-manifold, Tensor (N.S.), Vol. 29 (1975), YANO, K. & ISHIHARA, S., On integrability of a structure y satisfying Is+ l J. Math. Oxford, Vol. 25 (1964), , Quart. 3. YANO, K. & ISHIHARA, S., York, Tangent and Cotangent Bundles, Marcel Dekkar, Inc., New 4. DOMBROWSKI, P., (1980), On the geometry of the tangent bundle, J. Reine Angewandte Math HELGASON, S., Differential Geometry, Lie Groups and Symmetric Spac.es, Academic Press, New York, CALABI, E., Metric Reimann Surfaces, Annals of Math Studies, No. 30, Princeton University Press, Princeton, 1953, BEJANCU, A. & YANO, K., (1981), CR-submanifolds of a complex spe form, J. Diff. Geom. 16 Thus

4 204 L.S. DAS 8. DAS, L.S. & UPADHYAY, M.D., F-structure manifold, Kyung Pook Math. J. 18, Korea (1978), DAS, L.S., Complete lift of F-structure manifold, Kyung Pook Math. J. 20, Korea (1980), DAS, L.S., On differentiable manifold with F(K, _(_)K+I) structure of rank r, Revista Mathematica, Univ. Nac. Tucuman, Argentina, Rev. Ser. A27, No 1-2 (1978),

5 Journal of Applied Mathematics and Decision Sciences Special Issue on Decision Support for Intermodal Transport Call for Papers Intermodal transport refers to the movement of goods in a single loading unit which uses successive various modes of transport (road, rail, water) without handling the goods during mode transfers. Intermodal transport has become an important policy issue, mainly because it is considered to be one of the means to lower the congestion caused by single-mode road transport and to be more environmentally friendly than the single-mode road transport. Both considerations have been followed by an increase in attention toward intermodal freight transportation research. Various intermodal freight transport decision problems areindemandofmathematicalmodelsofsupportingthem. As the intermodal transport system is more complex than a single-mode system, this fact offers interesting and challenging opportunities to modelers in applied mathematics. This special issue aims to fill in some gaps in the research agenda of decision-making in intermodal transport. The mathematical models may be of the optimization type or of the evaluation type to gain an insight in intermodal operations. The mathematical models aim to support decisions on the strategic, tactical, and operational levels. The decision-makers belong to the various players in the intermodal transport world, namely, drayage operators, terminal operators, network operators, or intermodal operators. Topics of relevance to this type of decision-making both in timehorizonasintermsofoperatorsare: Intermodal terminal design Infrastructure network configuration Location of terminals Cooperation between drayage companies Allocation of shippers/receivers to a terminal Pricing strategies Capacity levels of equipment and labour Operational routines and lay-out structure Redistribution of load units, railcars, barges, and so forth Scheduling of trips or jobs Allocation of capacity to jobs Loading orders Selection of routing and service Before submission authors should carefully read over the journal s Author Guidelines, which are located at Prospective authors should submit an electronic copy of their complete manuscript through the journal Manuscript Tracking System at according to the following timetable: Manuscript Due June 1, 2009 First Round of Reviews September 1, 2009 Publication Date December 1, 2009 Lead Guest Editor Gerrit K. Janssens, Transportation Research Institute (IMOB), Hasselt University, Agoralaan, Building D, 3590 Diepenbeek (Hasselt), Belgium; Gerrit.Janssens@uhasselt.be Guest Editor Cathy Macharis, Department of Mathematics, Operational Research, Statistics and Information for Systems (MOSI), Transport and Logistics Research Group, Management School, Vrije Universiteit Brussel, Pleinlaan 2, 1050 Brussel, Belgium; Cathy.Macharis@vub.ac.be Hindawi Publishing Corporation