Robotics and automation

1. Journal of Science and Technology ISSN: 2456-5660 Volume 2, Issue 01 (Jan -Feb 2017) www.jst.org.in DOI: https://doi.org/10.46243/jst.2017.v2.i01.pp66- 75 SOME LIFT AND STRUCTUR FROM A MANIFOLD TO ITS TANGENT BUNDLE Dr.B.P. Yadav Department of Mathematics, Allahabad Degree College, Allahabad To Cite this Article Dr.B.P. Yadav,“SOME LIFT AND STRUCTUR FROM A MANIFOLD TO ITS TANGENT BUNDLE” Journal of Science and Technology, Vol. 02, Issue 01, - Jan-Feb 2017, pp66-75 Article Info Received: 29-01-2017 Revised: 07-02-2017 Accepted: 17-02-2017 Published: 27-02-2017 Abstract: In this paper, Lift of Tensor field, almost complex structures, F-structure and Para compact structures are studied. 1. Lift of tensor fields: ( ) T V is its tangent bundle .The projection being   induced from in Let V be a n-dimensional differentiable manifold and * for a differentiable function  in V. The function : ( ) T V  → V denoted by ( ) T V is denoted by   =   * V (1.1) and is called the vertical lift of function. Any 1-form  given in differentiable manifold V is, in ( ) T V, which we be denoted by. If we are given a vector field X V X in ( ) T Vby natural way, regarded as a function in in V, then we define a vector field ( ) ( = ) V   V ( ) X X (1.2)  being an arbitrary1 vector field X. − form V X in V. The vector Field thus defined is called the vertical lift of the We define the vertical lift of (1.1) type d andd   by (1.3) by and ( ) ( ) (1.3) ( ) ( )  V V V V respectively andbeing arbitrary function in V  in V by   =    = V d d d d − of an arbitrary 1 form V and the vertical lift Published by: Longman Publishers Page | 66 www.jst.org.in

2. Journal of Science and Technology ISSN: 2456-5660 Volume 2, Issue 01 (Jan -Feb 2017) www.jst.org.in DOI: https://doi.org/10.46243/jst.2017.v2.i01.pp66- 75 ( ) dx    = ) ( V V V (1.4) −  1( ) U h , where U is a co-ordinates neighborhood with local coordinates ( dx   ) X In each open set  is given by in V and  = −  1( ) U −  of 1 form in U. It is easily verified that the vertical lift 1 form − V defied by (1.4) in each is a global in tangent bundle. When there is given function in V, we put ( )   = −    H C H    =    ( ) ( ) T V, where T V. (1.5) The horizontal lift of the function  for a vector field Xgiven in V. we define a vector field in tangent bundle by In and all the functions defined in ( )  H  =  −   H H C H H  being an arbitrary function in V and call X X X  (1.6) lift of vector field X, where  denotes the affine connection. the horizontal  in V , we define − Given 1 ( ) form T V by  H in ( )  = H H 0 X − X being an arbitrary vector field in V, the1 ( ) form T V is  H (1.7) called the Horizontal lift thus defined in ( ) H = H H F X FX  for any X V = X H V 0 F X S and (1.8) and We define Vertical lift ( ) T V for a tensor field S of type (1.2) given in V as H S Horizontal lift in ( ) ( ) = = V C C V H H H H , ( ( , )) S X Y , ( ( , )) S X Y S X Y S X Y (1.9) and X and Y being arbitrary vector fields in V.We shall now give local representation of the lifts. Let ( ( )) U X h hx be a coordinate neighborhood of differential manifold V , where ( ) coordinates defined in U. Let ( be the system of Cartesian coordinates in each tangent space   , where P is an arbitrary point in U, there is an open set U  of U  is a system of local ( ) PT V hy ) of =  1( ) U ( ) T U ) y h x V at P with respect to of x T U, we can introduce local coordinates ( from ( ) = 1( ) h h ,  ( ) ,where p U ,then in the open set , =  1( ) hx hx . Let there be a given function ( ) which are called coordinate induced in  in U, then  V H its vertical lift and its horizontal lift represented as Published by: Longman Publishers Page | 67 www.jst.org.in

3. Journal of Science and Technology ISSN: 2456-5660 Volume 2, Issue 01 (Jan -Feb 2017) www.jst.org.in DOI: https://doi.org/10.46243/jst.2017.v2.i01.pp66- 75 ( )   = and ( ) , x y where x  =  =  −    H C hx 1( ) U V hx ( ) (1.10) In with respect to induced coordinates    = y h h  inU, then its vertical lift H V H X X    X If a vector field X has components and its horizontal lift    and have 0 x V : X h respectively component of the form (1.11)     −   h X H : X x h  with respect to induced coordinates (  and its horizontal lifts )   =  U 1( ) U h h , x y − If 1 form  In v has components , hen its  V H vertical lifts have respectively components of the form ( ) ( ) =    V 1( ) U     : ,0 H h  : ,    (1.12) ( ) , x y and in with respect to induced coordinates  .If a tensor field Fof type (1,1)has components Fh F have respectively components of the form h h U V F , then its vertical lift and its H horizontal lift         H 0 F         and 0 F  0 0 H : F V : F   − +  h  h h  F F h  (1.13) with respect to the induced coordinates ( ) =  1( ) U h h , x y in . Thus the horizontal lift of the identity tensor T V. On taking account of the definition of lift or their local ( ) field of V is the identity tensor field in representation (1.10),(1.11),(.2) and(1.13), we get formulae (1.14) ( (  ) ( ) ( ) ;( ) H V H V  =    =    =   =  V V H H V V H , X X X X ) ;( ) ;( ) V H H =    =      =    V V H H H H For any function, any vector field X form  given in V X  =  =  V V H H V 0 ( ) X X (1.15) ; X  = (X )  − ((d )(X)   H V C X  = H H 0 Published by: Longman Publishers Page | 68 www.jst.org.in

4. Journal of Science and Technology ISSN: 2456-5660 Volume 2, Issue 01 (Jan -Feb 2017) www.jst.org.in DOI: https://doi.org/10.46243/jst.2017.v2.i01.pp66- 75 Fr any function  and any vector field X given in V;  =  = ( ( ))  V V V H V ( ) 0 ( ) X X X (1.16) ;  = ( ( ))   = H V V H H ( ) ( ) 0 X X X ; For any vector field X and any 1-form  given in V,    V    =  −  = −  V H V V ,Y , ( ) ( X) X X Y Y Y X    −    C H , ( ) X Y Y X (1.17)  = − H H H [X , ] [ , ] X Y ( , ) R X Y Y For any vector field X and Y given in V, where  denotes the affine connections in V defined by     =  + , Y X X Y  For any two vector field X and Y given in V = = V V H V V 0; ( ) F X F X FX (1.18) = = V H H H H H (FX) F ( ) (FX) X F X , For any vector field X and any tensor field F of type (1,1) given in V. Le there be given any two tensor field F and K of type (1,1) in V , then we get = H H H (K ) (FK) F (1.19)  Thus, if there are given a tensor field F of type (1, 1) and a polynomial (t); from (1.19) of a variable t, then we have  = ( ) F  H H (1.20) ( )) F For example + = + 2 2 H H ( ) ( ) F I F I (1.21) + = + 2 3 H H H (F ) ( ) F F F Published by: Longman Publishers Page | 69 www.jst.org.in

5. Journal of Science and Technology ISSN: 2456-5660 Volume 2, Issue 01 (Jan -Feb 2017) www.jst.org.in DOI: https://doi.org/10.46243/jst.2017.v2.i01.pp66- 75 Where I denotes the identity tensor field of type (1, 1) in the corresponding manifold V of T (V). Let there be a given a tensor field of type (1, 1) in V, then Nijenhuis tensor N is defined by = + − − 2 ( , ) N X Y [ , ] [ , ] X Y [ , ] , ] FX FY F F FX Y Fx FY (1.22) X and Y being arbitrary vector field in V. On taking horizontal lift of both sides and taking account of1.9),(1.170 and (1.18), we have = ] ( + − 2 H H H H H H H H H H H H H H N ( , ) [ , ) [ , ] [ , ] X Y F X F Y F X Y F F X Y (1.23) ~ H H ( , ) N X Y H F The right hand side of (1,23) is nothing but the Nijenhuis tensor of F, thus we have of the horizontal lift ~ = NH N (1.24) ~ N denotes the Nijenhuis tensor of H F . Where V F is of the rank r and its horizontal lift Let F be the tensor field of type (1,1) in V, then the vertical lift H F is of rank 2r iff F is of rank r.  = 2 Let there be a given projection tensor m ,which is determined by m. On taking account (1.21) we get, V m m such that , then there exist a distribution D in V = 2 H H m ) m ( which means mHis also projection tensor in T(V), thus there exist in T(V)a distribution corresponding to mH and which we call the horizontal lift of distribution D. The horizontal lift H D H D X and XH, X being an arbitrary vector field belonging to the distribution D V by all vector field of type because we have = = H V V H H H (1.25) m (m ) ,m (m ) X X X X The distribution D is integrated iff I mX mY = (1.26) ( , ) 0 = − For any vector field X and Y in V, where I we obtain I m , taking the horizontal lift of the both sides in(1.26), Published by: Longman Publishers Page | 70 www.jst.org.in

6. Journal of Science and Technology ISSN: 2456-5660 Volume 2, Issue 01 (Jan -Feb 2017) www.jst.org.in DOI: https://doi.org/10.46243/jst.2017.v2.i01.pp66- 75 ( , ) I m X m Y = H H H H H 0 H mis Theorem 1.1: Let therebe a given distribution D with projection tenor in V, the horizontal lift integrable in T(V) if and only if D is integrable in V. Proof: Let there be a given a differentiable transformation  → :V V : ( ) T V  → ( ) T V and denoted simply . , we define 1-form   → :V V  V The differential of transformation for a 1-form of  = ( ( ))   ( ) X X : ( ) T V  → ( ) T V X being an arbitrary vector field in V, if the differential of the transformation  is → H : ( ( )) T T V ( )) T V denoted by Then we have the following formulae  =   =  V C V H H H ( ) ,( ) X X X X  =    =   V C V H H H ( ) ;( ) For any vector field X and 1-form  given in V for a tensor field F of type (1, 1) given in V, we define a F  by tensor field  =  X ( ) ( ( F ) F X then we get  =   =  V C V H H H ( ) ,( ) F F F F   → :V F V The tensor field is that induced from the given F by the transformation . 2. Almost Complex Structure: Let there be a given field F of type (1, 1) in V. On taking account of (1.21) we have + = 2 H ( ) 0 F I = − , thus we have 2 ( ) I F if and only of Published by: Longman Publishers Page | 71 www.jst.org.in

7. Journal of Science and Technology ISSN: 2456-5660 Volume 2, Issue 01 (Jan -Feb 2017) www.jst.org.in DOI: https://doi.org/10.46243/jst.2017.v2.i01.pp66- 75 Theorem2: The horizontal liftFH of a tensor field of type F of type (1, 1) given in differentiable manifold V is an almost complex structure in T (V) if and only if so F in V. + = 3 0 F F Proof: A tensor field of type (1, 1) is called f—structure of rankr, when F satisfies F F = − and the rank of is a constant reverywhere, r being necessarily even. On taking account of (1.24), we have the theorem, or 3 Theorem 2.2: An almost complex structure F in an almost complex space V, a transformation :V V  → preserves the structure F if and only if its differential function FH. : ( ) T V  → ( ) T V preserves 3. F-Structure: Let there be given n dimensional differentiable manifold V of classC on V and a tensor of type F of type (1,1) of rank r(1 ) r n such that   + = 3 0 F F (3.1) , then F is called F-structure. If we put = − = + 2 2 2 , I F m F I (3.2) , then we have + = = − I F  = = = 2 4 2 , I m I I F F I = m F  + = = = 2 2 ;Im m 0 m m mI (3.3) = − = , 0 FI IF Fm mF Q 1P and These equations shows that there exist two complementary distribution in V corresponding to ; m Q 1P is r-dimensional and the projection tensor I and m respectively, when the rankr of F in V, ( ) n r − dimensional, wheredim.m is ; m = n . We have the following integrability conditions: Q (a)A necessary and sufficient condition for the distribution to be integrble is ; m = (3.4) N(mX,mY) 0 X Y , where N denotes the Nijenhuis tensor of the F-structure. 0 1 , For any 1P to be integrable is (b)A necessary and sufficient condition for the distribution Published by: Longman Publishers Page | 72 www.jst.org.in

8. Journal of Science and Technology ISSN: 2456-5660 Volume 2, Issue 01 (Jan -Feb 2017) www.jst.org.in DOI: https://doi.org/10.46243/jst.2017.v2.i01.pp66- 75 ( ,Y) 0 N X = (3.5)  For any vector field X,Y V 1Pis integrable , then we have the relation = 2 F ,X' being an arbitrary vector field tangent I If the distribution structure F on each integral manifold of D such as , the F operators on V and almost FX = ' IX' 1P, thus rank r of F must be even. When the distribution 1Pis integrable and to the integral manifold of 1P. Hence we say that F- induced almost complex F is complex analytic on each integral manifold of structure is partially integrable. (a) A necessary and sufficient condition for the distribution for F-structure to be partially integrable that X Y . 0 1 , N IX IY = ( , ) 0 (3.6) for any Let us suppose that there exists, in each coordinates neighbourhoods of V, local coordinates with respect to which F-structure has numerical components −        0 Im 0 0 0 0 0 : Im    F 0  = where Im denotes the unit m m structure F is integrable. 2 r m matrix and is the rank of F. If this is the case, we call F- (b)A necessary and sufficient condition for F-structure to be integrable is that ( , ) N X Y = 0 (3.7) For any vector field X and Y in V. Let F be a tensor field of type (1,1) in V, then we have, on taking account of (1.21), that the equation + = is equal to ( ) 0 F F have + = 3 H H 3 ,where the rank of FH is 2r and r is the rank of F, thus we 0 F F H F of an element F of F is of rank 2r inT(V). Theorem 3.1: Let there be given a differentiable manifold, the horizontal lift T(V) is an F-structure iff so in F in V, When F is of the rank r in V and H H H m are two I Proof: Let there be an F-structure of rank rin V, then the horizontal lift and H m H P Q complementary projection in T(V),thus there exists in T(V) two complementary distribution and 1 Published by: Longman Publishers Page | 73 www.jst.org.in

9. Journal of Science and Technology ISSN: 2456-5660 Volume 2, Issue 01 (Jan -Feb 2017) www.jst.org.in DOI: https://doi.org/10.46243/jst.2017.v2.i01.pp66- 75 H m H H P Q P H H m respectively .The distribution I and determined by and are respectively lifts of and 1 1 ~ H m 1P and Qm denoted by N and (1.9),(1.18) and (1.24), the conditions (a),(b),(c) and (d)are equivalent to the following conditions: Q Nthe Nijenhuis tensor of F and H F respectively, then by means of of = H H H H H ( , ) 0 N m X m Y ( )' a = H H H H (X , ) 0 m N Y (b)' = H H H H H (I X ,I ) 0 N Y (c)' = H H H (X , ) 0 N Y (d)' for any vector field X and Y in V, therefore we have, H F of an F-structure F in V satisfies one of the integrabilty conditions Theorem 3.2: The Horizontal lift ( )' a ,(b)',(c)'and(d)' in T(V) iff the given F-structure F satisfies the corresponding integrability condition in V. 4. Almost Para contact structure and F-structure: Let there be given n-dimensional differentiable manifold V, a tensor field of type (1,1), a vector field  and a co-vector field  satisfying = −    2 F =  =   = ; F I 0 ,( ) 0 ,( ) 1 FX For any vector field X in v, then is necessarily odd and we call a structure defined by the set (F, , ) of such tensor field F,and an almost para-contact structure, on taking account of (4.1), we have F 0 F + = and that F of rank ( 1) everywhere in V. If there is given an almost para-contact structure (F, , )   n − 3 V=   in V o type (1, 2) is = + (X( (T))  − Y( (X))  [X,Y]  −  (4.2) ( , ) (X,Y) S X Y N For any X and Y in V, where N denotes the Nijenhuis tensor of J. The almost para-cotact structure (F, , )   is normal if 0 S = .The set is a framed F-structure F in V. If there is given an F-structure F of rank ( 1) in an n-dimensional differentiable manifold V, then there exists an almost Para-contact structure (F, , ) structure (F, , ) (4.1) n −   in V. when the framed F-structure (F, )   is said to be normal. On taking account of (1.14),(1.16),(1.18) and (1.9), we have from  is normal, the given almost Para-contact = −    +    = −    −    2 H H H V V H H V ( ) ( ) F I I (4.3) Published by: Longman Publishers Page | 74 www.jst.org.in

10. Journal of Science and Technology ISSN: 2456-5660 Volume 2, Issue 01 (Jan -Feb 2017) www.jst.org.in DOI: https://doi.org/10.46243/jst.2017.v2.i01.pp66- 75 ~  =  =  = V V H H V H 0; 0; ( ) 0 F F F X ~  =   =   =   == H H V H H V H H ( ) 0; ( ) 1; ( ) 1; ( ) 0 F X ~ Xin T(V). Now, if we define tensor field of type (1,1)in T(V) by For any vector field ~ J = +    −    H V V H H F (4.4) 2 ~ Jdenoted by (4.) is ~ J = − as a direct consequence of (4, 3) ,therefore the tensor field I Then we obtain ~ Jis expressed locally by an almost complex structure in T(V). The almost complex structure      − +   −  h h ~ J  =     h h  h   i h h With respect to induced coordinates( . ) x y , we have Theorem4.1: The almost complex structure defined by (4.1) is complex analytic if the almost para- contact structure (F, , )   is normal in V. References [1] Yano, K. and S. Kobayashi: Prolongation of tensor fields and connections to tangent bundles,I general theory J. of mathematical .Japan 1 (1996) 19-210 [2] Yano, K. and S. Kobayashi: Prolongation of tensor fields and connections to tangent bundles, II affine automorphism J. of mathematical. Japan 18 (1996) 236-246 .[3] Ishihara S. and K.Yano, on integrability 0 f f + = , Quart. J. Math Oxford (2) 15-(1964)217-222. 3 of structure f satisfying [4] Sato I., Almost analytic vector field in almost complex manifold, Tohoku Math.J.17 (1965), 185-199. + = 3 0 f f [5] Sato I., on a structure defined by tensor field F of type (1,1) satisfying 14(1963) 99-109 ,Tensor, N.S. Published by: Longman Publishers Page | 75 www.jst.org.in