Cartan Spaces and Natural Foliations on the Cotangent Bundle
aa r X i v : . [ m a t h . D S ] A ug August 13, 2020 1:3 WSPC/INSTRUCTION FILE Attarchi
International Journal of Geometric Methods in Modern Physicsc (cid:13)
World Scientific Publishing Company
Cartan Spaces and Natural Foliations on the Cotangent Bundle
H. Attarchi
Department of Mathematics and Computer ScienceAmirkabir University of Technology, Tehran, [email protected]
M. M. Rezaii
Department of Mathematics and Computer ScienceAmirkabir University of Technology, Tehran, [email protected]
Received (Day Month Year)Revised (Day Month Year)In this paper, the natural foliations in cotangent bundle T ∗ M of Cartan space ( M, K ) isstudied. It is shown that geometry of these foliations are closely related to the geometryof the Cartan space (
M, K ) itself. This approach is used to obtain new characterizationsof Cartan spaces with negative constant curvature.
Keywords : Cartan Spaces; foliation; Cartan space of negative constant curvature.
1. Introduction
Lagrange space has been certified as an excellent model for some important prob-lems in Relativity, Gauge Theory and Electromagnetism [1,2]. The geometry ofLagrange spaces gives a model for both the gravitational and electromagnetic field.P. Finsler in his Ph.D. thesis introduced the concept of general metric function,which can be studied by means of variational calculus. Later, L. Berwald, J.L.Synge and E. Cartan precisely gave the correct definition of a Finsler space [3,4].These structures play a fundamental role in study of the geometry of tangent bun-dle
T M . The geometry of cotangent bundle T ∗ M and tangent bundle T M whichfollows the same outlines are related by Legendre transformation. From this duality,the geometry of a Hamilton space can be obtained from that of certain Lagrangespace and vice versa. As a particular case, we can associate to a given Finsler spaceits dual, which is a Cartan space [5,6]. Using this duality several important re-sults in the Cartan spaces can be obtained: the canonical nonlinear connection, thecanonical metrical connection etc. Therefore, the theory of Cartan spaces has thesame symmetry and beauty like Finsler geometry. Moreover, it gives a geometricalframework for the Hamiltonian theory of Mechanics or Physical fields. With respectto the importance of these spaces in Physical areas and the quick growth of Finsler ugust 13, 2020 1:3 WSPC/INSTRUCTION FILE Attarchi H. Attarchi and M. M. Rezaii geometry in recent decades, present work is formed to develop some properties ofFinsler geometry to the Cartan spaces. In [7] and [8] the natural foliations on tan-gent bundle of a Finsler manifold have been studied. Here, some theorems of theseobjects are reconsidered on cotangent bundle of a Cartan space for some naturalfoliations such as Liouville-Hamilton vector field and its complement distributionin
T T ∗ M .Let ( M, K ) be a Cartan space then to achieve these aims, the present work isorganized in the following way. In Section 2, some definitions and results on Cartanspaces which is needed in following sections is provided. In particular, canonicalnonlinear connections, Sasaki lift of the metric on cotangent bundle and coefficientsof Levi-Civita connection are presented, for more details see [4]. In Section 3, tosimplify some equations and proofs in following sections a new frame is set on
T T ∗ M such that T T ∗ M is decomposed to two foliations Liouville-Hamilton vectorfield and its complement distribution in T T ∗ M . The Levi-Civita connection on aCartan space is computed in the new basis and relation of curvature tensor fields oflevel hypersurfaces K = const. and T ∗ M is calculated. In Section 4, the ideas of [7]are developed to cotangent bundle of a Cartan space and six natural foliations ofcotangent bundle are introduced. They are studied from different viewpoints suchas being totally geodesic, bundle like and etc. As a main result of present paper,the condition of being a Cartan space with negative constant curvature is found.Finally in Section 5, -indicatrix bundles in cotangent bundle of the Cartan space( M, K ) which is denoted by I ∗ M (1) is studied. Similar to Finsler geometry [8], it isshown that it naturally has contact structure. In addition, it is shown that I ∗ M (1)cannot have Sasakian structure with lifted Sasaki metric G .
2. Preliminaries and notations
Let M be a real n -dimensional differentiable manifold and let ( T ∗ M, π ∗ , M ) be itscotangent bundle. If ( x i ), ( i = 1 , ..., n ), is a local coordinate system on a domain U of a chart on M , the induced system of coordinates on π ∗− ( U ) are ( x i , p i ). Thecoordinates p i are called momentum variables . The Liouville-Hamilton vector field , Liouville 1-form and canonical symplectic structure on T ∗ M are denoted by C ∗ , ω and θ , respectively, and their local expressions are as follows: C ∗ = p i ∂ i , ω = p i dx i , θ = dp i ∧ dx i (1)where ∂ i := ∂∂P i . Let { ., . } be the Poisson bracket on T ∗ M , defined by: { f, g } = ∂f∂p i ∂g∂x i − ∂g∂p i ∂f∂x i , ∀ f, g ∈ C ∞ ( T ∗ M )A Cartan space is a pair ( M, K ( x, p )) such that following axioms hold:(1) K is a real positive function on T ∗ M , differentiable on T ∗ M \{ } and continuouson the null section of the projection π ∗ .(2) K is positively 1-homogeneous with respect to the momenta p i .ugust 13, 2020 1:3 WSPC/INSTRUCTION FILE Attarchi Cartan Spaces and Natural Foliations on the Cotangent Bundle (3) The Hessian of K , with elements g ij ( x, y ) = ∂ i ∂ j K is positive-defined.The nonlinear connection coefficient N ij of the Cartan space ( M, K ) is given by: N ij = 14 { g ij , K } − (cid:18) g ik ∂ K ∂p k ∂x j + g jk ∂ K ∂p k ∂x i (cid:19) where ( g ij ) is the inverse matrix of ( g ij ). The adapted basis of T T ∗ M and T ∗ T ∗ M with respect to the natural nonlinear connection coefficient N ij are expressed asfollows: T T ∗ M = < δδx i , ∂ i > , δδx i = ∂∂x i + N ij ∂ j T ∗ T ∗ M = < dx i , δp i > , δp i = dp i − N ij dx j (2)In addition, with respect to these bases the Sasakian lift of the metric tensor g ij on T ∗ M is shown by G and given by: G := g ij dx i ⊗ dx j + g ij δp i ⊗ δp j (3)The almost complex structure compatible with metric G is shown by J and haslocal expression as follows: J := g ij δδx j ⊗ δp i − g ij ∂ j ⊗ dx i Then by direct calculations and using (2) it is obtained that:[ δδx i , δδx j ] = R ijk ∂ k , [ ∂ j , δδx i ] = N jik ∂ k (4)where R ijk = δN jk δx i − δN ik δx j and N jik = ∂ j N ik . Now, let ∇ be the Levi-Civita connec-tion on Riemannian manifold ( T ∗ M, G ). Then we can prove the following.
Theorem 1.
Let ( M, K ) be a Cartan space. Then the Levi-Civita connection ∇ on ( T ∗ M, G ) is locally expressed as follows: ∇ δδxi δδx j = Γ kij δδx k + ( R ijk + g ijk ) ∂ k ∇ δδxi ∂ j = ∇ ∂ j δδx i − N jih ∂ h = − ( g jki + R ish g hj g sk ) δδx k + ( δg jk δx i + ∂ k ( N is g sj ) − ∂ j ( N is g sk )) g kh ∂ h ∇ ∂ i ∂ j = − ( δg ij δx k + N iks g sj + N jks g si ) g kh δδx h + g ijk ∂ k where Γ kij = g kh { δg ih δx j + δg jh δx i − δg ij δx h } g ijk = g is g jt g stk = g is g jt g kh g sth = g is g jt g kh ∂ h ( g st ) Proof.
By using (4), definition of Levi-Civita connection (cid:26) G ( ∇ X Y, Z ) = XG ( Y, Z ) +
Y G ( X, Z ) − ZG ( X, Y ) − G ([ X, Z ] , Y ) − G ([ Y, Z ] , X ) + G ([ X, Y ] , Z )and a straightforward calculation the proof is complete.ugust 13, 2020 1:3 WSPC/INSTRUCTION FILE Attarchi H. Attarchi and M. M. Rezaii
3. Adapted frame on indicatrix bundle of a Cartan space
Supposed that (
M, K ) is a n -dimensional Cartan space. In natural way, the verticaland horizontal distributions of T T ∗ M are given by: V T ∗ M = < ∂ , ..., ∂ n > , HT ∗ M = < δδx , ..., δδx n > Consider the Riemannian manifold ( T ∗ M, G ), then, orthogonal distribution toLiouville-Hamilton vector field C ∗ in vertical distribution V T ∗ M is denoted by V ′ T ∗ M . By definition of V ′ T ∗ M , it is easy to see that V ′ T ∗ M is a foliation in T T ∗ M . Therefore, there is the local chart ( U, ϕ ) on T ∗ M such that: T T ∗ M | U = < ¯ ∂ , ..., ¯ ∂ n − , v , ..., v n +1 > where V ′ T ∗ M | U = < ¯ ∂ , ..., ¯ ∂ n − > It is obvious that ¯ ∂ a for all a = 1 , ..., n − V T ∗ M and theycan be written as a linear combination of the natural basis { ∂ , ..., ∂ n } of V T ∗ M as follows: ¯ ∂ a = E ai ∂ i ∀ a = 1 , ..., n − E ai is the ( n − × n matrix of maximum rank. The first property of thismatrix is E ai g ij p j = 0. Moreover, on U the vertical distribution V T ∗ M is locallyspanned by: { C ∗ , ¯ ∂ , ..., ¯ ∂ n − } (1)Then the almost complex structure J acts on the basis (1) as follows: J ( ¯ ∂ a ) = E aj g ji δδx i , ξ = J ( C ∗ )We put ¯ E ia = E bj g ba g ji where g ab := G ( ¯ ∂ a , ¯ ∂ b ) = E ai g ij E bj and ( g ab ) = ( g ab ) − .Then, it is obtained that: J ( ¯ ∂ a ) = g ab ¯ E ib δδx i Now, if we let ¯ δ ¯ δx a = ¯ E ia δδx i , then we obtain: J ( ¯ ∂ a ) = g ab ¯ δ ¯ δx b , J ( ¯ δ ¯ δx a ) = g ab ¯ ∂ b and we can set a new local vector fields of T T ∗ M as follows: T T ∗ M = < ξ, ¯ δ ¯ δx a , C ∗ , ¯ ∂ a > (2)ugust 13, 2020 1:3 WSPC/INSTRUCTION FILE Attarchi Cartan Spaces and Natural Foliations on the Cotangent Bundle and the Sasakian metric G on T ∗ M can be shown in these new local vector fieldsas follows: G := K g ab ) 0 00 0 K
00 0 0 ( g ab ) (3)The Lie brackets of the vector fields in (2) are presented as follows: [ ¯ δ ¯ δx a , ¯ δ ¯ δx b ] = ( ¯ δ ¯ E ib ¯ δx a − ¯ δ ¯ E ia ¯ δx b ) δδx i + ¯ E ia ¯ E jb R ijs ∂ s , [ ¯ δ ¯ δx a , ¯ ∂ b ] = ( ¯ δE bi ¯ δx a − ¯ E ka E bj N jki ) ∂ i − ¯ ∂ b ( ¯ E ia ) δδx i , [ ¯ ∂ a , ¯ ∂ b ] = ( ∂ a E bi − ∂ b E ai ) ∂ i , [ ¯ δ ¯ δx a , ξ ] = ( ¯ E ja N jk g ki + p j ¯ δg ji ¯ δx a − ξ ( ¯ E ia )) δδx i + ¯ E ia p h g hj R ijs ∂ s , [ ¯ ∂ a , ξ ] = g ab ¯ δ ¯ δx b + ( E aj p k g kh N jhi − ξ ( E ai )) ∂ i , [ ¯ δ ¯ δx a , C ∗ ] = − C ∗ ( ¯ E ia ) δδx i , [ ¯ ∂ a , C ∗ ] = ¯ ∂ a − C ∗ ( E ai ) ∂ i , [ ξ, ξ ] = [ C ∗ , C ∗ ] = [ ξ, C ∗ ] + ξ = 0 . Now, the Levi-Civita connection ∇ on the Riemannian manifold ( T ∗ M, G ) for thebasis (2) is given by: ∇ ¯ δ ¯ δxa ¯ δ ¯ δx b = (Γ cab + ¯ δ ¯ E ib ¯ δx a E ci ) ¯ δ ¯ δx c + ( R abc − g abc ) ¯ ∂ c + K ( p i g ij ( ¯ δE cj ¯ δx a g cb + ¯ δE cj ¯ δx b g ca ) − ¯ E ia ¯ E jb ξ ( g ij )) ξ ∇ ¯ ∂ a ¯ ∂ b = ( ¯ ∂ a ( E bi ) ¯ E id + g abd ) ¯ ∂ d − K g ab C ∗ − E ai E bj ¯ E kc ( δg ij δx k + N ikh g hj + N jkh g hi ) g cd ¯ δ ¯ δx d ∇ ¯ δ ¯ δxa ¯ ∂ b = ( g bac − R bac ) g cd ¯ δ ¯ δx d − K R ba ξ + ¯ δE bi ¯ δx a ¯ E id ¯ ∂ d + E bi E cj ¯ E ka ( δg ij δx k + N jkh g hi − N ikh g hj ) g cd ¯ ∂ d ∇ ¯ ∂ b ¯ δ ¯ δx a = ( g bac − R bac ) g cd ¯ δ ¯ δx d − K ( R ba + 2 δ ba ) ξ + E bi E cj ¯ E ka ( δg ij δx k + N jkh g hi + N ikh g hj ) g cd ¯ ∂ d + ¯ ∂ b ( ¯ E ka ) E dk ¯ δ ¯ δx d (4) ∇ ¯ δ ¯ δxa ξ = ( ¯ E ia ¯ E jc ξ ( g ij ) + p j ¯ δg ji ¯ δx c ¯ E ka g ik + ¯ E ia ¯ E hc N ih − p j g ji ¯ δE bi ¯ δx a g bc ) g cd ¯ δ ¯ δx d + R ad ¯ ∂ d ∇ ξ ¯ δ ¯ δx a = ( ¯ E ia ¯ E jc ξ ( g ij ) + p j ¯ δg ji ¯ δx c ¯ E ka g ik + ¯ E ia ¯ E hc N ih + p j g ji ¯ δE bi ¯ δx a g bc ) g cd ¯ δ ¯ δx d + ξ ( ¯ E ia ) E di ¯ δ ¯ δx d − R ad ¯ ∂ d ∇ ¯ ∂ a ξ = ( g ad + R bc g ba g cd ) ¯ δ ¯ δx d ∇ ξ ¯ ∂ a = R bc g ba g cd ¯ δ ¯ δx d + ¯ E id ( ξ ( E ai ) − p k g kh E as N shi ) ¯ ∂ d (5) ∇ ¯ δ ¯ δxa C ∗ = ∇ C ∗ ¯ δ ¯ δx a − C ∗ ( ¯ E ia ) E di ¯ δ ¯ δx d = 0 ∇ ¯ ∂ a C ∗ − ¯ ∂ a = ∇ C ∗ ¯ ∂ a − C ∗ ( E ai ) ¯ E id ¯ ∂ d = 0 ∇ ξ C ∗ = ∇ C ∗ ξ − ξ = ∇ ξ ξ = ∇ C ∗ C ∗ − C ∗ = 0 (6)ugust 13, 2020 1:3 WSPC/INSTRUCTION FILE Attarchi H. Attarchi and M. M. Rezaii where R abc = ¯ E ia ¯ E jb ¯ E kc R ijk = g cd R dab , R ba = R ac g cb = ¯ E ia ¯ E jc R ij g cb g abc = ¯ E ia ¯ E jb ¯ E kc g ijk = g cd g dab = g ad g be g dec Γ cab = ¯ E ia ¯ E jb E ck Γ kij , N cab = ¯ E ia ¯ E jb E ck N kij The c- indicatrix bundle of T ∗ M denoted by I ∗ M ( c ) is defined as follows: I ∗ M ( c ) = { ( x, p ) ∈ T ∗ M | K ( x, p ) = c > } . Where for each c ∈ R + normal vector field to I ∗ M ( c ) is given by: grad ( K ) = p i c ∂ i = 1 c C ∗ (7)Therefore, T ( I ∗ M ) = < ξ, ¯ δ ¯ δx a , ¯ ∂ a > (8)In following, the Levi-Civita connection and metric on c -indicatrix bundles aredenoted by ¯ ∇ and ¯ G , respectively, which ¯ G is the restriction of metric (3). Inorder to compute the components of Levi-Civita connection ¯ ∇ on indicatrix bundle I ∗ M ( c ) for the basis (8) the Gauss Formula [9]: ∇ X Y = ¯ ∇ X Y + H ( X, Y )where H is the second fundamental form of I ∗ M ( c ) is needed. It is obvious that allcomponents in (4)–(6) except ∇ ¯ ∂ a ¯ ∂ b are tangent to I ∗ M ( c ). Therefore, ¯ ∇ is equalto ∇ for the other components of (4)–(6). Moreover, the curvature tensor R of ∇ defined by R ( X, Y ) Z = ∇ X ∇ Y Z − ∇ Y ∇ X Z − ∇ [ X,Y ] Z is related to the curvaturetensor ¯ R of ¯ ∇ in following equations: R ( ¯ δ ¯ δx a , ¯ δ ¯ δx b ) ¯ ∂ c = ¯ R ( ¯ δ ¯ δx a , ¯ δ ¯ δx b ) ¯ ∂ c + K R abe g ec C ∗ R ( ¯ δ ¯ δx a , ¯ ∂ b ) ¯ δ ¯ δx c = ¯ R ( ¯ δ ¯ δx a , ¯ ∂ b ) ¯ δ ¯ δx c + K ( R acd − g acd ) g db C ∗ R ( ¯ ∂ a , ¯ ∂ b ) ¯ ∂ c = ¯ R ( ¯ ∂ a , ¯ ∂ b ) ¯ ∂ c + K ( g ac ¯ ∂ b − g bc ¯ ∂ a ) R ( ¯ δ ¯ δx a , ¯ ∂ b ) ¯ ∂ c = ¯ R ( ¯ δ ¯ δx a , ¯ ∂ b ) ¯ ∂ c − K ¯ E ka E bi E cj ( δg ij δx k + N ikh g hj + N jkh g hi ) C ∗ R ( ¯ δ ¯ δx a , ¯ ∂ b ) ξ = ¯ R ( ¯ δ ¯ δx a , ¯ ∂ b ) ξ + K R ad g db C ∗ R ( ¯ ∂ a , ξ ) ¯ δ ¯ δx b = ¯ R ( ¯ ∂ a , ξ ) ¯ δ ¯ δx b + K R bd g da C ∗ R ( ¯ δ ¯ δx a , ξ ) ¯ ∂ b = ¯ R ( ¯ δ ¯ δx a , ξ ) ¯ ∂ b + K R ad g db C ∗ for the other combinations of ¯ δ ¯ δx a , ¯ ∂ a and ξ , tensor fields R and ¯ R coincide witheach other.ugust 13, 2020 1:3 WSPC/INSTRUCTION FILE Attarchi Cartan Spaces and Natural Foliations on the Cotangent Bundle
4. Foliations on T ∗ M Let (
M, K ) be a Cartan space defined in Section 2. The purpose of this sectionis to prove some results of [7] on six natural foliations in the cotangent bundle ofa Cartan space. It is interesting to see, that a study of these foliations providesimportant information on the geometry of the Cartan spaces themselves. These sixfoliations are presented as follows:(1) C ∗ : Liouville-Hamilton vector field.(2) ξ : defined by ξ := JC ∗ .(3) C ∗ ⊕ ξ .(4) V T ∗ M : defined by V T ∗ M = < ∂ , ..., ∂ n > .(5) V ′ T ∗ M : which is perpendicular to C ∗ in V T ∗ M with respect to the metric G defined in (3).(6) V ⊥ T ∗ M : which is perpendicular to C ∗ in T T ∗ M with respect to the metric G defined in (3). Theorem 2. C ∗ , ξ and C ∗ ⊕ ξ are three totally geodesic foliations on ( T ∗ M, G ) . Proof.
According to (6) ∇ ξ C ∗ = ∇ C ∗ ξ − ξ = ∇ ξ ξ = ∇ C ∗ C ∗ − C ∗ = 0which this shows they are totally geodesic. Theorem 3.
The lifted metric G defined in (3) is bundle-like for the verticalfoliation V T ∗ M if and only if ( g ij ) is a Riemannian metric on M . Proof.
With respect to bundle-like condition (see [10]), G is bundle-like for V T ∗ M if and only if G ( ∇ δδxi δδx j + ∇ δδxj δδx i , ∂ k ) = 0 ∀ i, j, k = 1 , ..., n Then, by Theorem 1, it is deduced that G is bundle-like for V T ∗ M if and only if g ijk = 0. This completes the proof.By using the basis (2) and (4)–(6) the following theorem is obtained: Theorem 4.
The lifted metric G defined in (3) is bundle-like for the foliation V ′ T ∗ M if and only if ( g ij ) is a Riemannian metric on M . Proof.
From (4)-(6), it is obtained: G ( ∇ ξ ¯ δ ¯ δx a + ∇ ¯ δ ¯ δxa ξ, ¯ ∂ c ) = G ( ∇ C ∗ ¯ δ ¯ δx a + ∇ ¯ δ ¯ δxa C ∗ , ¯ ∂ c )= G ( ∇ ξ C ∗ + ∇ C ∗ ξ, ¯ ∂ c ) = G (2 ∇ C ∗ C ∗ , ¯ ∂ c ) = G (2 ∇ ξ ξ, ¯ ∂ c ) = 0ugust 13, 2020 1:3 WSPC/INSTRUCTION FILE Attarchi H. Attarchi and M. M. Rezaii and G ( ∇ ¯ δ ¯ δxa ¯ δ ¯ δx b + ∇ ¯ δ ¯ δxb ¯ δ ¯ δx a , ¯ ∂ c ) = − g abd g dc Therefore, the lifted metric G defined in (3) is bundle-like for the foliation V ′ T ∗ M if and only if g abd = 0 and it is equivalent to g ijk = 0. This completes the proof. Theorem 5.
The lifted metric G defined in (3) is not bundle-like for the foliation C ∗ , or C ∗ is not a Killing vector field of metric G . Proof.
From (4), it is obtain that: L C ∗ G ( ¯ ∂ a , ¯ ∂ b ) = G ( ∇ ¯ ∂ a C ∗ , ¯ ∂ b ) + G ( ¯ ∂ a , ∇ ¯ ∂ b C ∗ )= − G ( C ∗ , ∇ ¯ ∂ a ¯ ∂ b ) − G ( ∇ ¯ ∂ b ¯ ∂ a , C ∗ ) = 2 g ab Therefore, C ∗ is not a Killing vector field and G is not bundle-like for C ∗ . Theorem 6.
The vertical distribution
V T ∗ M is totally geodesic if and only if thefollowing holds: δg ij δx k + N iks g sj + N jks g si = 0 Proof.
It is a straight conclusion of Theorem 1.
Theorem 7.
The foliations V ′ T ∗ M and V ⊥ T ∗ M are not totally geodesic in Rie-mannian manifold ( T ∗ M, G ) . Proof.
From (4), we obtain H ( ¯ ∂ a , ¯ ∂ b ) = − K g ab C ∗ which it cannot be vanish. Theorem 8.
The foliation V ′ T ∗ M is a totally umbilical foliation of T T ∗ M . Proof.
From (4), we obtain H ( ¯ ∂ a , ¯ ∂ b ) = − K g ab C ∗ that it shows V ′ T ∗ M is totally umbilical in T T ∗ M . Theorem 9.
The tangent bundles
T I ∗ M ( c ) for all c ∈ R + is just the foliationdetermined by the integrable distribution V ⊥ T ∗ M . Also, it can be shown that C ∗ and ξ are orthogonal and tangent to I ∗ M ( c ) , respectively. Proof.
In (7), it is shown that C ∗ is orthogonal to T I ∗ M ( c ). Therefore, foliationof tangent bundles of level hypersurfaces of K is equal to V ⊥ T ∗ M . On other hand,it is easy to see that ξ ( K ) = 0 and ξ is tangent to level hypersurfaces of K .ugust 13, 2020 1:3 WSPC/INSTRUCTION FILE Attarchi Cartan Spaces and Natural Foliations on the Cotangent Bundle From (4) the followings are obtained: R ijk g kh p h = 0 , R ijk + R jki + R kij = 0Therefore, R ij defined by p h g hk R ikj is a symmetric tensor for indices i and j . In [4],it is proved that Cartan space ( M, K ) is of constant curvature c if and only if thefollowing holds: R ij = cK h ij (1)where h ij is angular metric tensor field of a Cartan space defined by: h ij = g ij − K p i p j (2)Let ∧ ∗ = ( ∧ ∗ ij ) given by ∧ ∗ ij = R ij + h ij (3)be a symmetric bilinear tensor field on C ∞ ( M )-module Γ( HT ∗ M ), and call it angular curvature of M . Lemma 10.
For any X ∈ Γ( HT ∗ M ) , the following holds: ∧ ∗ ( ξ, X ) = 0 . Proof. ∧ ∗ ij p h g hi X j = R ij p h g hi X j + g ij p h g hi X j − K p i p j p h g hi X j = 0 + p j X j − p j X j = 0. Theorem 11.
Let ( M, K ) be a Cartan space and I ∗ M ( c ) be a c -indicatrix over M . Then, metric G is bundle-like on I ∗ M ( c ) for ξ if and only if ∧ ∗ = 0 on I ∗ M ( c ) . Proof. ξ is bundle-like on I ∗ M ( c ) if and only if the following holds: G ( ∇ X Y, ξ ) + G ( ∇ Y X, ξ ) = 0where X = X i δδx i + ¯ X j ∂ j , Y = Y i δδx i + ¯ Y j ∂ j , X i p i = Y i p i = 0 and ¯ X j p i g ij =¯ Y j p i g ij = 0. By help of Theorem 1, it can be obtain that: G ( ∇ δδxi ξ, δδx j ) = G ( ∇ ∂ i ξ, ∂ j ) = 0and G ( ∇ ∂ i ξ, δδx j ) = G ( ∇ δδxj ξ, ∂ i ) + δ ij = δ ij + 12 R js g si Therefore, we obtain that:0 = G ( ∇ X Y, ξ ) + G ( ∇ Y X, ξ ) = − G ( ∇ X ξ, Y ) − G ( ∇ Y ξ, X )ugust 13, 2020 1:3 WSPC/INSTRUCTION FILE Attarchi H. Attarchi and M. M. Rezaii = − ¯ X j Y i ( δ ji + R is g sj ) − ¯ Y j X i ( δ ji + R is g sj ) ⇐⇒ ( ¯ X j Y i + ¯ Y j X i )( δ ji + R is g sj ) = 0 ⇐⇒ ( ¯ X j Y i + ¯ Y j X i )( δ ji + R is g sj − K p i p j ) = 0 ⇐⇒ g ij − K p i p j + R ij = 0 ⇐⇒ ∧ ∗ ij = h ij + R ij = 0. Theorem 12.
Let ( M, K ) be a Cartan space. Then, ξ is a Killing vector field on I ∗ M ( c ) if and only if ∧ ∗ = 0 on I ∗ M ( c ) . Proof.
Here, it is shown that ξ is a Killing vector field on I ∗ M ( c ) if and only ifmetric G is bundle-like on I ∗ M ( c ) for ξ . Then by Theorem 11 the proof is complete.It is obvious that if ξ is a Killing vector field then G is bundle like on I ∗ M ( c ) for ξ . The converse state is true if G be bundle like on I ∗ M ( c ) for ξ and the followingshold: G ( ∇ ξ ξ, ξ ) = G ( ∇ ξ ξ, X ) + G ( ∇ X ξ, ξ ) = 0for all X ∈ Γ( I ∗ M ( c )) where G ( X, ξ ) = 0. Since ∇ ξ ξ = 0, it is enough to showthat G ( ∇ ¯ δ ¯ δxa ξ, ξ ) = G ( ∇ ¯ ∂ a ξ, ξ ) = 0 which it is obvious from (5) and it makes proofcomplete. Theorem 13.
Let ( M, K ) be a Cartan space. Then, M is a Cartan space ofnegative constant curvature k if and only if ∧ ∗ be vanish on I ∗ M ( c ) , where c = − k . Proof.
Let M be a Cartan space of negative constant curvature k , then by (1) weobtain: R ij ( x, p ) = − h ij ( x, p ) ∀ ( x, p ) ∈ I ∗ M ( c ) (4)Thus, by (3) we have ∧ ∗ = 0 on I ∗ M ( c ). Conversely, suppose that ∧ ∗ = 0 on I ∗ M ( c ). To complete the proof it is enough to show that (4) is valid for ( x, p ) ∈ T ∗ M \ I ∗ M ( c ). For all ( x, p ) ∈ T ∗ M there exists a constant ¯ c which ( x, p ¯ c ) ∈ I ∗ M ( c ).By definition R ij and equations (4) and (2), it is easy to see that R ij and h ij arehomogeneous of degree two and zero, respectively. Thus: R ij ( x, p ¯ c ) = − h ij ( x, p ¯ c ) ⇐⇒ R ij ( x, p ) = − ¯ c h ij ( x, p ) ⇐⇒ R ij ( x, p ) = kK ( x, p ) h ij ( x, p ) ∀ ( x, p ) ∈ T ∗ M and this completes the proof.Combining Theorems 11, 12 and 13 the following is obtained: Theorem 14.
Let ( M, K ) be a Cartan space, and k < < c two constant where − c k = 1 . Then the followings are equivalent: ugust 13, 2020 1:3 WSPC/INSTRUCTION FILE Attarchi Cartan Spaces and Natural Foliations on the Cotangent Bundle (1) M is a Cartan space of constant curvature k.(2) G is bundle-like on I ∗ M ( c ) for ξ .(3) ξ is a Killing vector field on I ∗ M ( c ) .(4) Angular curvature ∧ ∗ is vanish on I ∗ M ( c ) .
5. The contact structure on indicatrix bundle of a Cartan space
In the first part of this section, it is shown that each I ∗ M ( c ) naturally has contactstructure. Then in the next part, it is proved that this contact structure cannot bea Sasakian one.The (1,1)-tensor field ϕ is set on the indicatrix bundle I ∗ M (1) as follows: ϕ := − J | D , ϕ ( ξ ) = 0 (1)where D = { X ∈ T T ∗ M | G ( X, ξ ) = G ( X, C ∗ ) = 0 } . The distribution D is called contact distribution in contact manifolds. Also, notation ¯ G is used to restrict metric G to the c-indicatrix bundle. It is obvious that the dual 1-form ξ with respect tothe metric G is lioville 1-form ω defined in (1) in Cartan space ( M, K ). Now, thefollowing theorem can be expressed:
Theorem 15.
Let the 4-tuple ( ϕ, ω, ξ, ¯ G ) be defined as above. Then 1-indicatrixbundle of a Cartan space with ( ϕ, ω, ξ, ¯ G ) is a contact manifold. Proof.
The compatibility of ϕ and the metric ¯ G is equivalent to compatibility of J and G . Also, The conditions ω ◦ ϕ = 0 , ϕ ( ξ ) = 0 , ϕ = − I + ξ ⊗ ω are easy to be proved by considering Eq. (1) and definitions given in above. Tocomplete the proof, the condition dω ( X, Y ) = ¯ G ( X, ϕY ) for the vector fields
X, Y ∈ Γ( T I ∗ M ) needs to be checked. By calculating dω the following can beobtained: dω = N ij dx j ∧ dx i + δp i ∧ dx i = δp i ∧ dx i Also, G ( δδx i , J δδx j ) = G ( ∂ i , J∂ j ) = 0and G ( δδx i , J∂ j ) = − G ( ∂ i , J δδx j ) = δ ji = ⇒ G ( ., J. ) = dx i ∧ δp i = ⇒ dω ( X, Y ) = − G ( X, JY )ugust 13, 2020 1:3 WSPC/INSTRUCTION FILE Attarchi H. Attarchi and M. M. Rezaii
Since ¯ G is the restriction of G to the indicatrix and dω ( ξ, . ) = 0 thus dω ( X, Y ) = ¯ G ( X, ϕY ) ∀ X, Y ∈ Γ D So, I ∗ M (1) with ( ϕ, ω, ξ, ¯ G ) is a contact manifold [11].Now, we show that this contact structure cannot be a Sasakian one. First, let( M, ϕ, η, ξ, g ) be a contact Riemannian manifold. In [12], it was proved that M isSasakain manifold if and only if( ˜ ∇ X ϕ ) Y = 0 ∀ X, Y ∈ Γ( T M )where ˜ ∇ X Y = ∇ X Y − η ( X ) ∇ Y ξ − η ( Y ) ∇ X ξ + ( dη + 12 ( L ξ g ))( X, Y ) ξ and ∇ is Levi-Civita connection on ( M, g ).Since the indicatrix bundle has the contact metric structure in Cartan spaces byTheorem 15, the following question cross our mind that ”
Can the indicatrix bundlein a Cartan space be a Sasakian manifold? ”. First, the following Lemma is provedin order to reduce the number of calculations.
Lemma 16. If ( M, ϕ, ω, ξ, g ) be a contact metric manifold with contact distribution D , then M is Sasakian manifold if and only if: ( ˜ ∇ X ϕ ) Y = 0 ∀ X, Y ∈ Γ( D ) Proof.
For all ¯ X ∈ Γ( T M ), they can be written in the form X + f ξ where X ∈ Γ D , f ∈ C ∞ ( M ) and ξ is Reeb vector field of the contact structure on M . Therefore:( ˜ ∇ ¯ X ϕ ) ¯ Y = ( ˜ ∇ X + fξ ϕ )( Y + gξ ) = ( ˜ ∇ X ϕ ) Y + ( ˜ ∇ fξ ϕ ) Y + ( ˜ ∇ X ϕ ) gξ +( ˜ ∇ fξ ϕ ) gξ = ( ˜ ∇ X ϕ ) Y + f ( ˜ ∇ ξ ϕY − ϕ ˜ ∇ ξ Y ) + ˜ ∇ X ϕ ( gξ ) − ϕ ( ˜ ∇ X gξ )+ f ( ˜ ∇ ξ ϕgξ − ϕ ˜ ∇ ξ gξ ) = ( ˜ ∇ X ϕ ) Y The lemma is proved using Theorem 3.2 in [12] and the last equation.Now, the following theorem can be expressed:
Theorem 17.
Let ( M, K ) be a Cartan space. Then, indicatrix bundle I ∗ M (1) withcontact structure ( I ∗ M (1) , ϕ, ω, ξ, ¯ G ) can never be a Sasakian manifold. Proof.
From lemma 16, I ∗ M (1) is a Sasakian manifold if and only if:( ˜ ∇ ¯ δ ¯ δxa ϕ ) ¯ δ ¯ δx b = ( ˜ ∇ ¯ δ ¯ δxa ϕ ) ¯ ∂ b = ( ˜ ∇ ¯ ∂ a ϕ ) ¯ δ ¯ δx b = ( ˜ ∇ ¯ ∂ a ϕ ) ¯ ∂ b = 0Using (4)–(6), one of the components in above equations is g ab = 0which demonstrates a contradiction and shows that the indicatrix bundle cannotbe a Sasakian manifold with contact structure ( ϕ, ω, ξ, ¯ G ).ugust 13, 2020 1:3 WSPC/INSTRUCTION FILE Attarchi Cartan Spaces and Natural Foliations on the Cotangent Bundle References [1] R. Miron and M. Anastasiei,
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