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The natural transformations between T-th order prolongation of tangent and cotangent bundles over Riemannian manifolds

The natural transformations between T-th order prolongation of tangent and cotangent bundles over... If (M, g) is a Riemannian manifold then there is the well-known base preserving vector bundle isomorphism T M T M given by v g(v, -) between the tangent T M and the cotangent T M bundles of M . In the present note first we generalize this isomorphism to the one J r T M J r T M between the r-th order prolongation J r T M of tangent T M and the r-th order prolongation J r T M of cotangent T M bundles of M . Further we describe all base preserving vector bundle maps DM (g) : J r T M J r T M depending on a Riemannian metric g in terms of natural (in g) tensor fields on M . 1. Introduction. All manifolds are smooth, Hausdorff, finite dimensional and without boundaries. Maps are assumed to be smooth, i.e. of class C . Let Mfm denote category of m-dimensional manifolds and their embeddings. From the general theory it is well known that the tangent T M and the cotangent T M bundles of M are not canonically isomorphic. However, if g is a Riemannian metric on a manifold M , there is the base preserving vector bundle isomorphism ig : T M T M given by ig (v) = g(v, -), v Tx M, x M . In the second section of the present note we give necessary definitions. 2010 Mathematics Subject Classification. 58A05, 58A20, 58A32. Key words and phrases. Riemannian manifold, higher order prolongation of a vector bundle, natural tensor, natural operator. In the third section first we generalize the isomorphism ig : T M T M depending on g to a base preserving vector bundle isomorphism J r ig : J r T M J r T M canonically depending on g between the r-th order prolongation J r T M of tangent T M and the r-th order prolongation J r T M of cotangent T M bundles of M . Next we construct another more advanced base preserving vector bundle isomorphism i : J r T M J r T M canonically g depending on g. In the fourth section we consider the problem of describing all Mfm natural operators D : Riem Hom(J r T, J r T ) transforming Riemannian metrics g on m-dimensional manifolds M into base preserving vector bundle maps DM (g) : J r T M J r T M . Our studies lead to the reduction of this problem to the one of describing all Mfm -natural operators t : Riem T S l T T S k T (for l, k = 1, . . . , r) sending Riemannian metrics g on M into tensor fields tM (g) of types T S l T T S k T . 2. Definitions. Now we give some necessary definitions. Definition 1. The r-th order prolongation of tangent bundle is a functor J r T : Mfm VB sending any m-manifold M into J r T M and any embedding : M1 M2 of two manifolds into J r T : J r T M1 J r T M2 given by J r T (jx X) = j(x) X, where X X (M1 ) and X = T X is the image of a vector field X by . Definition 2. The r-th order prolongation of cotangent bundle is a functor J r T : Mfm VB sending any m-manifold M into J r T M and any embedding : M1 M2 of two manifolds into J r T : J r T M1 J r T M2 given by J r T := J r (T ) . Definition 3. The dual bundle of the r-th order prolongation of tangent bundle is a functor (J r T ) : Mfm VB sending any m-manifold M into (J r T ) M := (J r T M ) and any embedding : M1 M2 of two manifolds into (J r T ) : (J r T ) M1 (J r T ) M2 given by (J r T ) := (J r T ) . Definition 4. The dual bundle of the r-th order prolongation of cotangent bundle is a functor (J r T ) : Mfm VB sending any m-manifold M into (J r T ) M := (J r T M ) and any embedding : M1 M2 of two manifolds into (J r T ) : (J r T ) M1 (J r T ) M2 given by (J r T ) := (J r T ) . The general concept of natural operators can be found in [4]. In particular, we have the following definitions. Definition 5. An Mfm -natural operator D : Riem Hom(J r T, J r T ) transforming Riemannian metrics g on m-dimensional manifolds M into base preserving vector bundle maps DM (g) : J r T M J r T M is a system D = {DM }M obj(Mfm ) of regular operators DM : Riem(M ) HomM (J r T M, J r T M ) satisfying the Mfm -invariance condition, where HomM (J r T M, J r T M ) is the set of all vector bundle maps J r T M J r T M covering the identity map idM of M . The Mfm -invariance condition of D is following: for any g1 Riem(M1 ) and g2 Riem(M2 ) if g1 and g2 are -related by an embedding : M1 M2 of m-manifolds (i.e. is (g1 , g2 )-isomorphism) then DM1(g1 ) and DM2(g2 ) are also -related (i.e. DM2 (g2 ) J r T = J r T DM1 (g1 )). Equivalently, the above Mfm -invariance means that for any g1 Riem(M1 ) and g2 Riem(M2 ) if the diagram T M1 T M 1 (1) g1 M1 T T T M 2 T M2 g2 M2 commutes for an embedding : M1 M2 (i.e. (T T ) g1 = g2 ) then the diagram J r T M1 DM1 (g1 ) J r T M1 J rT J r T J r T M2 DM2 (g2 ) J r T M2 commutes also. We say that operator DM is regular if it transforms smoothly parameterized families of Riemannian metrics into smoothly parameterized ones of vector bundle maps. Similarly, we can define the following concepts: - an Mfm -natural operator D : Riem Hom(J r T, J r T ), Hom(J r T, (J r T ) ), - an Mfm -natural operator D : Riem Hom(J r T, (J r T ) ), - an Mfm -natural operator D : Riem - an Mfm -natural operator D : Riem - an Mfm -natural operator D : Riem - an Mfm -natural operator D : Riem - an Mfm -natural operator D : Riem - an Mfm -natural operator D : Riem - an Mfm -natural operator D : Riem - an Mfm -natural operator D : Riem - an Mfm -natural operator D : Riem - an Mfm -natural operator D : Riem - an Mfm -natural operator D : Riem - an Mfm -natural operator D : Riem - an Mfm -natural operator D : Riem Now we have the following definition. Hom(J r T , J r T ), Hom(J r T , J r T ), Hom(J r T , (J r T ) ), Hom(J r T , (J r T ) ), Hom((J r T ) , J r T ), Hom((J r T ) , J r T ), Hom((J r T ) , (J r T ) ), Hom((J r T ) , (J r T ) ), Hom((J r T ) , J r T ), Hom((J r T ) , J r T ), Hom((J r T ) , (J r T ) ), Hom((J r T ) , (J r T ) ). (T S l T , T S k T ) Definition 6. An Mfm -natural operator A : Riem transforming Riemannian metrics g on m-dimensional manifolds M into base preserving vector bundle maps AM (g) : T M S l T M T M S k T M is a system A = {AM }M obj(Mfm ) of regular operators AM : Riem(M ) C (T M S l T M, T M S k T M ) satisfying the Mfm -invariance condition, where C (T M S l T M, T M S k T M ) is the set of all vector bundle maps T M S l T M T M S k T M covering the identity map idM of M . The Mfm -invariance condition of A is following : for any g1 Riem(M1 ) and g2 Riem(M2 ) if g1 and g2 are -related by an embedding : M1 M2 of m-manifolds (i.e. (T T )g1 = g2 ) then AM1 (g1 ) and AM2 (g2 ) are also -related (i.e. AM2 (g2 )(T S l T ) = (T S k T )AM1 (g1 )). Equivalently, the above Mfm -invariance means that for any g1 Riem(M1 ) and g2 Riem(M2 ) if the diagram (1) commutes for an embedding : M1 M2 then the diagram T M1 S k T M 1 AM1 (g1 ) T M1 S l T M 1 T Sk T T M 2 S k T M2 AM2 (g2 ) T M2 S l T M2 T SlT commutes also. The regularity means almost the same as in Definition 5. Similarly, we can define the following concepts: (T S l T, T S k T ), - an Mfm -natural operator A : Riem (T S l T, T S k T ), - an Mfm -natural operator A : Riem - an Mfm -natural operator A : Riem (T S l T , T S k T ), (T S l T, T S k T ), - an Mfm -natural operator A : Riem (T S l T, T S k T ), - an Mfm -natural operator A : Riem (T S l T , T S k T ), - an Mfm -natural operator A : Riem (T S l T, T S k T ), - an Mfm -natural operator A : Riem (T S l T, T S k T ), - an Mfm -natural operator A : Riem (T S l T , T S k T ), - an Mfm -natural operator A : Riem (T S l T , T S k T ), - an Mfm -natural operator A : Riem (T S l T, T S k T ), - an Mfm -natural operator A : Riem (T S l T , T S k T ), - an Mfm -natural operator A : Riem (T S l T , T S k T ), - an Mfm -natural operator A : Riem (T S l T, T S k T ), - an Mfm -natural operator A : Riem (T S l T , T S k T ). - an Mfm -natural operator A : Riem Next we have an important general definition of natural tensor. p T Definition 7. An Mfm -natural operator (natural tensor) t : Riem q T transforming Riemannian metrics g on m-dimensional manifolds M into tensor fields of type (p, q) on M is a system t = {tM }M obj(Mfm ) of regular operators tM : Riem(M ) T (p,q) (M ) satisfying the Mfm -invariance condition, where T (p,q) (M ) is the set of tensor fields of type (p, q) on M . The Mfm -invariance condition of t is following : for any g1 Riem(M1 ) and g2 Riem(M2 ) if g1 and g2 are -related by an embedding : M1 M2 of m-manifolds (i.e. (T T ) g1 = g2 ) then tM1 (g1 ) and tM2 (g2 ) are also -related (i.e. tM2 (g2 ) = ( p T q T ) tM1 (g1 )). Equivalently, the above Mfm -invariance means that for any g1 Riem(M1 ) and g2 Riem(M2 ) if the diagram (1) commutes for an embedding : M1 M2 , then the diagram p p T M1 T M T M2 T M2 tM1 (g1 ) tM2 (g2 ) M1 M2 commutes also. We say that operator tM is regular if it transforms smoothly parametrized families of Riemannian metrics into smoothly parametrized ones of tensor fields. Now we have a definition of a special kind of natural tensor. Definition 8. An Mfm -natural operator (natural tensor) t : Riem T l T T S k T transforming Riemannian metrics g on m-dimensional S manifolds M into tensor fields of type T S l T T S k T on M is a system t = {tM }M obj(Mfm ) of regular operators tM : Riem(M ) C (T M S l T M T M S k T M ) satisfying the Mfm -invariance condition, where C (T M S l T M T M S k T M ) is the set of all tensor fields of type T S l T T S k T on M . The Mfm -invariance condition of t is following: for any g1 Riem(M1 ) and g2 Riem(M2 ) if g1 and g2 are -related by an embedding : M1 M2 of m-manifolds (i.e. (T T ) g1 = g2 ), then tM1 (g1 ) and tM2 (g2 ) are also -related (i.e. tM2 (g2 ) = (T S l T T S k T )tM1 (g1 )). Equivalently, the above Mfm -invariance means that for any g1 Riem(M1 ) and g2 Riem(M2 ) if the diagram (1) commutes for an embedding : M1 M2 , then the diagram T M1 S l T M1 T M1 S k T M1 T M2 S l T M2 T M2 S k T M2 tM1 (g1 ) tM2 (g2 ) M1 M2 commutes also, where = T S l T T S k T . The regularity means almost the same as in Definition 7. Similarly, we can define the following concepts: - an Mfm -natural operator (natural tensor) t : Riem T SkT , - an Mfm -natural operator (natural tensor) t : Riem T SkT , - an Mfm -natural operator (natural tensor) t : Riem T SkT , - an Mfm -natural operator (natural tensor) t : Riem T SkT , - an Mfm -natural operator (natural tensor) t : Riem T Sk T , - an Mfm -natural operator (natural tensor) t : Riem T SkT , - an Mfm -natural operator (natural tensor) t : Riem T SkT , T SlT T SlT T SlT T SlT T SlT T SlT T SlT - an Mfm -natural operator (natural tensor) t : Riem T Sk T , - an Mfm -natural operator (natural tensor) t : Riem T SkT , - an Mfm -natural operator (natural tensor) t : Riem T Sk T , - an Mfm -natural operator (natural tensor) t : Riem T Sk T , - an Mfm -natural operator (natural tensor) t : Riem T SkT , - an Mfm -natural operator (natural tensor) t : Riem T Sk T , - an Mfm -natural operator (natural tensor) t : Riem T Sk T , - an Mfm -natural operator (natural tensor) t : Riem T Sk T . T SlT T SlT T SlT T SlT T SlT T SlT T SlT T SlT In the third section we present also explicit examples of Mfm -natural operators D : Riem Hom(J r T, J r T ). p T A full description of all polynomial natural tensors t : Riem q T transforming Riemannian metrics on m-manifolds into tensor fields of types (p, q) can be found in [1]. This description is following. Each covariant derivative of the curvature R(g) T (0,4) (M ) of a Riemannian metric g is a natural tensor and the metric g is also a natural tensor. Further all the p T q T can be obtained by a procedure: natural tensors t : Riem (a) every tensor multiplication of two natural tensors give a new natural tensor, (b) every contraction on one covariant and one contravariant entry of a natural tensor give a new natural tensor, (c) we can tensorize any natural tensor with a metric independent natural tensor, (d) we can permute any number of entries in the tensor product, (e) we can repeat these steps, (f) we can take linear combinations. Furthermore, if we take respective type natural tensors and apply respective symmetrization, then we can produce many natural tensors t : Riem T SlT T Sk T . 3. Constructions. Example 1. Let (M, g) be a Riemannian manifold. Then we have a base preserving vector bundle isomorphism ig : T M T M given by ig (v) = g(v, -), v Tx M, x M. Next we can obtain a base preserving vector bundle isomorphism J r ig : J r T M J r T M defined by a formula J r ig (jx X) = jx (ig X), where X X (M ). Similarly we receive also a base preserving vector bundle isomorphism (J r i ) : (J r T M ) (J r T M ) . g Because of the canonical character of the above constructions we get the following propositions. Proposition 1. The family A(r) : Riem (r) Hom(J r T, J r T ) of operators AM (g) = J r ig (r) AM : Riem(M ) HomM (J r T M, J r T M ), for all M obj(Mfm ) is an Mfm -natural operator. Proposition 2. The family A : Riem ators Hom((J r T ) , (J r T ) ) of operAM (g) = (J r i ) g AM : Riem(M ) HomM ((J r T M ) , (J r T M ) ), for all M obj(Mfm ) is an Mfm -natural operator. Now we are going to present another more advanced example of an Mfm natural operator D : Riem Hom(J r T, J r T ). Recall that if g is a Riemannian tensor field on a manifold M and x M , then there is g-normal coordinate system : (M, x) (Rm , 0) with centre x. If : (M, x) (Rm , 0) is another g-normal coordinate system with centre x, then there is A O(m) such that = A near x. Let r m k m = m k m (see [3]) and I : J 0 T Rm S R k=1 T0 R S T0 T Rm Rm S k T Rm = m S k Rm (see [7]) be I1 : J0 0 k=1 T0 the standard O(m)-invariant vector space isomorphisms. We have the following important proposition. Proposition 3. Let g be a Riemannian tensor field on a manifold M . Then there are (canonical in g) vector bundle isomorphisms Ig : J r T M T M S k T M, T M S k T M, Jg : J r T M (Ig ) : r (J T M ) TM S T M T M S k T M, k=0 (Jg ) : (J T M ) k=0 T M S T M T M S k T M. k=0 Proof. Let v = jx X Jx T M , where X X (M ), x M . Let : (M, x) m , 0) be a g-normal coordinate system with centre x. We define ( Ig (v) := Ig (v) k=0 T S k T I J r T (v). If : (M, x) is another g-normal coordinate system with centre x, then = A (near x) for some A O(m). The O(m)-invariance of I means that (Rm , 0) (2) I J rT A = k=0 T0 A S k T0 A I. Hence we deduce that r Ig (v) = T S k T I J r T (v) (T (A ) S k T (A ) ) I J r T (A )(v) ((T T A ) S k T ( A )) I (J r T A J r T )(v) = = k=0 ((T T A ) S k T ( A )) (I J r T A) J r T (v) =: L. Now using (2), we receive L= ((T T A )S k T (A )) k=0 T AS k T A I J r T (v) ((T T A )T A) (S k T ( A )S k T A) I J r T (v) ((T T A T A) S k T ( A A)) I J r T (v) (T S k T ) I J r T (v) = Ig (v). k=0 = = Therefore, the definition of Ig (v) is independent of the choice of . So, isomorphism Ig : J r T M r T M S k T M is well defined. k=0 Similarly, we put Jg (v) := Jg (v) k=0 T S k T I1 J r T (v). Using O(m)-invariance of I1 (i.e. I1 J r T A = ( r T0 A S k T0 A) k=0 I1 ) analogously as before, we show that Ig (v) = Ig (v). This proves that the definition of Jg (v) is independent of the choice of g-normal coordinate system with centre x and the isomorphism Jg : J r T M r T M k=0 S k T M is well defined. Finally we obtain (canonical in g) vector bundle isomorphisms r (Ig ) : (J r T M ) (Jg ) : (J r T M ) k=0 k=0 T M S k T M. k=0 Remark 1. W. Mikulski (in [7]) has recently constructed a (canonical in r k ) vector bundle isomorphism I : J r T M k=0 T M S T M for a classical linear connection on a manifold M . Now we have further important identifications. Example 2. Let (M, g) be a Riemannian manifold and ig : T M T M be a base preserving vector bundle isomorphism recalled in Example 1. Using the base preserving vector bundle isomorphisms Ig and Jg from Proposition 3, we receive the following vector bundle isomorphisms r i := Jg g i := Jg g k=0 ig S k T idM i S k T idM g Ig : J r T M J r T M, (Ig ) : (J r T M ) (J r T M ) . Because of canonical character of above constructions we obtain the following propositions. Proposition 4. The family B : Riem Hom(J r T, J r T ) of operators BM (g) = i g BM : Riem(M ) HomM (J r T M, J r T M ), for all M obj(Mfm ) is an Mfm -natural operator. Proposition 5. The family B : Riem ators Hom((J r T ) , (J r T ) ) of operBM (g) = i g BM : Riem(M ) HomM ((J r T M ) , (J r T M ) ), for all M obj(Mfm ) is an Mfm -natural operator. Example 3. Let (M, g) be a Riemannian manifold. In an article [5] J. Kurek and W. Mikulski constructed a base preserving vector bundle isomorphism i(r) : g k=1 S TM k=1 Sk T M given by i(r) (v1 g ··· vk ) = ig (v1 ) ··· ig (vk ). Now using this isomorphism, we get a base preserving vector bundle isomorphism r (r) Ig : T M Sk T M = SkT M = T M k=0 k=0 defined by a formula (r) Ig = ig (i(r) ) . g Similarly, we construct another base preserving vector bundle isomorphisms ~(r) Ig : ~(r) ~ Ig : ^(r) Ig : T M S k T M, T M S k T M, ~(r) Ig = idT M i(r) g ~(r) ~ Ig = i i(r) g g k=0 k=0 T M S k T M, ^(r) Ig = idT M i(r) g Thus we receive a base preserving vector bundle isomorphism Ig : J r T M (J r T M ) given by (r) = Ig Ig Ig . Ig Similarly, we construct another base preserving vector bundle isomorphisms ~ Ig : J r T M (J r T M ) , Ig : J r T M (J r T M ) , ~ = Jg Ig Ig , ~(r) Ig ^ I = I I (r) Jg , g g g ~ Ig : J r T M J T M (J T M ) , ~ Ig Jg ~(r) ~ Ig Jg . Using the base preserving vector bundle isomorphism J r ig : J r T M constructed in Example 1, we obtain also the following vector bundle isomorphisms = (J r i ) Ig : J r T M (J r T M ) , Jg g ~ ~ = (J r ig ) Ig : J r T M (J r T M ) , Jg Jg = (J r i ) Ig : J r T M (J r T M ) , g ~ ~ Jg = (J r ig ) Ig : J r T M (J r T M ) . Because of canonical character of the above constructions we get the following propositions. Proposition 6. The family C : Riem tors Hom(J r T, (J r T ) ) of opera CM (g) = Ig CM : Riem(M ) HomM (J r T M, (J r T M ) ), for all M obj(Mfm ) is an Mfm -natural operator. ~ Proposition 7. The family C : Riem Hom(J r T, (J r T ) ) of operators ~ ~ ~ C : Riem(M ) HomM (J r T M, (J r T M ) ), C (g) = Ig for all M obj(Mfm ) is an Mfm -natural operator. Proposition 8. The family C : Riem Hom(J r T , (J r T ) ) of operators CM (g) = Ig CM : Riem(M ) HomM (J r T M, (J r T M ) ), for all M obj(Mfm ) is an Mfm -natural operator. ~ Proposition 9. The family C : Riem Hom(J r T , (J r T ) ) of operators ~ ~ ~ C : Riem(M ) HomM (J r T M, (J r T M ) ), C (g) = Ig for all M obj(Mfm ) is an Mfm -natural operator. Proposition 10. The family D : Riem ators Hom(J r T, (J r T ) ) of oper DM (g) = Jg DM : Riem(M ) HomM (J r T M, (J r T M ) ), for all M obj(Mfm ) is an Mfm -natural operator. ~ Proposition 11. The family D : Riem Hom(J r T, (J r T ) ) of operators ~ ~ ~ D : Riem(M ) HomM (J r T M, (J r T M ) ), D (g) = Jg for all M obj(Mfm ) is an Mfm -natural operator. Proposition 12. The family D : Riem ators Hom(J r T , (J r T ) ) of oper DM (g) = Jg DM : Riem(M ) HomM (J r T M, (J r T M ) ), for all M obj(Mfm ) is an Mfm -natural operator. ~ Proposition 13. The family D : Riem Hom(J r T , (J r T ) ) of operators ~ ~ ~ D : Riem(M ) HomM (J r T M, (J r T M ) ), D (g) = Jg for all M obj(Mfm ) is an Mfm -natural operator. 4. The main results. Let g Riem(M ) be a Riemannian metric on an mmanifold M . By Proposition 3 and Examples 1, 2, 3 we have identifications J r T M = J r T M = (J r T M ) = (J r T M ) = r k=0 = k=0 = k=0 T M SkT M modulo the base preserving vector bundle isomorphisms canonically depending on g. Consequently, the problem of finding all Mfm -natural operators D : Riem Hom(J r T, J r T ) is reduced to the one of finding all systems (Al,k ) of (T S l T , T S k T ) transforming Mfm -natural operators Al,k : Riem Riemannian metrics g on m-manifolds M into base preserving vector bundle maps Al,k (g) : T M S l T M T M S k T M , where l, k = 1, . . . , r or M (equivalently) our problem is reduced to the one of finding all natural tenT S l T T S k T transforming Riemannian metrics sors tl,k : Riem g on m-manifolds M into tensor fields of types T S l T T S k T on M for l, k = 1, . . . , r. Thus we have proved the following theorem. Hom(J r T, J r T ) Theorem 1. The Mfm -natural operators D : Riem transforming Riemannian metrics g on m-manifolds M into base preserving vector bundle maps DM (g) : J r T M J r T M are in the bijection with systems (tl,k ) of Mfm -natural operators (natural tensors) tl,k : Riem T S l T T S k T transforming Riemannian metrics g on m-manifolds M into tensor fields of types T S l T T S k T on M for l, k = 1, . . . , r. Because of the isomorphism J r T M J r T M depending on g, we have = the following theorem. Theorem 2. The Mfm -natural operators D : Riem Hom(J r T, J r T ) transforming Riemannian metrics g on m-manifolds M into base preserving vector bundle maps DM (g) : J r T M J r T M are in the bijection with sys(T S l T , T S k T ) tems (Al,k ) of Mfm -natural operators Al,k : Riem transforming Riemannian metrics g on m-manifolds M into base preserving vector bundle maps Al,k (g) : T M S l T M T M S k T M for l, k = M 1, . . . , r or (equivalently) in the bijection with systems (tl,k ) of Mfm -natural T S l T T S k T transformoperators (natural tensors) tl,k : Riem ing Riemannian metrics g on m-manifolds M into tensor fields of types T S l T T S k T on M for l, k = 1, . . . , r. By the same reason, we have also the following corollary. Corollary 1. The Mfm -natural operators D : Riem Hom(J r T , J r T ) transforming Riemannian metrics g on m-manifolds M into base preserving vector bundle maps DM (g) : J r T M J r T M are in the bijection with (T S l T , T systems (Al,k ) of Mfm -natural operators Al,k : Riem k T ) transforming Riemannian metrics g on m-manifolds M into base S preserving vector bundle maps Al,k (g) : T M S l T M T M S k T for l, k = 1, . . . , r or (equivalently) in the bijection with systems (tl,k ) of T SlT T SkT Mfm -natural operators (natural tensors) tl,k : Riem transforming Riemannian metrics g on m-manifolds M into tensor fields of types T S l T T S k T on M for l, k = 1, . . . , r. By the same reason, we have another corollary. Corollary 2. The Mfm -natural operators D : Riem Hom(J r T , J r T ) transforming Riemannian metrics g on m-manifolds M into base preserving vector bundle maps DM (g) : J r T M J r T M are in the bijection with (T S l T , T systems (Al,k ) of Mfm -natural operators Al,k : Riem S k T ) transforming Riemannian metrics g on m-manifolds M into base preserving vector bundle maps Al,k (g) : T M S l T M T M S k T for l, k = 1, . . . , r or (equivalently) in the bijection with systems (tl,k ) of T S l T T S k T Mfm -natural operators (natural tensors) tl,k : Riem transforming Riemannian metrics g on m-manifolds M into tensor fields of types T S l T T S k T on M for l, k = 1, . . . , r. Because of the isomorphisms J r T M J r T M (J r T M ) depending = = on g, we have the following theorem. Hom(J r T, (J r T ) ) Theorem 3. The Mfm -natural operators D : Riem transforming Riemannian metrics g on m-manifolds M into base preserving vector bundle maps DM (g) : J r T M (J r T M ) are in the bijection with (T S l T , T systems (Al,k ) of Mfm -natural operators Al,k : Riem k T ) transforming Riemannian metrics g on m-manifolds M into base preS serving vector bundle maps Al,k (g) : T M S l T M T M S k T M for M l, k = 1, . . . , r or (equivalently) in the bijection with systems (tl,k ) of Mfm T SlT T SkT natural operators (natural tensors) tl,k : Riem transforming Riemannian metrics g on m-manifolds M into tensor fields of types T S l T T S k T on M for l, k = 1, . . . , r. By the same reason, we have the following theorem. Theorem 4. The Mfm -natural operators D : Riem Hom(J r T , (J r T ) ) transforming Riemannian metrics g on m-manifolds M into base preserving vector bundle maps DM (g) : J r T M (J r T M ) are in the bijection with (T S l T , T systems (Al,k ) of Mfm -natural operators Al,k : Riem S k T ) transforming Riemannian metrics g on m-manifolds M into base preserving vector bundle maps Al,k (g) : T M S l T M T M S k T M for M l, k = 1, . . . , r or (equivalently) in the bijection with systems (tl,k ) of Mfm T S l T T S k T transnatural operators (natural tensors) tl,k : Riem forming Riemannian metrics g on m-manifolds M into tensor fields of types T S l T T S k T on M for l, k = 1, . . . , r. By the same reason, we have also the following corollary. Corollary 3. The Mfm -natural operators D : Riem Hom((J r T ) , J r T ) transforming Riemannian metrics g on m-manifolds M into base preserving vector bundle maps DM (g) : (J r T M ) J r T M are in the bijection with (T S l T, T systems (Al,k ) of Mfm -natural operators Al,k : Riem k T ) transforming Riemannian metrics g on m-manifolds M into base S preserving vector bundle maps Al,k (g) : T M S l T M T M S k T M for M l, k = 1, . . . , r or (equivalently) in the bijection with systems (tl,k ) of Mfm T SlT T SkT natural operators (natural tensors) tl,k : Riem transforming Riemannian metrics g on m-manifolds M into tensor fields of types T S l T T S k T on M for l, k = 1, . . . , r. We have also the next similar corollary. Corollary 4. The Mfm -natural operators D : Riem Hom((J r T ) , J r T ) transforming Riemannian metrics g on m-manifolds M into base preserving vector bundle maps DM (g) : (J r T M ) J r T M are in the bijection with (T S l T, T systems (Al,k ) of Mfm -natural operators Al,k : Riem k T ) transforming Riemannian metrics g on m-manifolds M into base S preserving vector bundle maps Al,k (g) : T M S l T M T M S k T M for M l, k = 1, . . . , r or (equivalently) in the bijection with systems (tl,k ) of Mfm T SlT T SkT natural operators (natural tensors) tl,k : Riem transforming Riemannian metrics g on m-manifolds M into tensor fields of types T S l T T S k T on M for l, k = 1, . . . , r. We have also another corollary. Corollary 5. The Mfm -natural operators D : Riem Hom((J r T ), (J r T )) transforming Riemannian metrics g on m-manifolds M into base preserving vector bundle maps DM (g) : (J r T M ) (J r T M ) are in the bijection with (T S l T, T S k T ) systems (Al,k ) of Mfm -natural operators Al,k : Riem transforming Riemannian metrics g on m-manifolds M into base preserving vector bundle maps Al,k (g) : T M S l T M T M S k T M for l, k = M 1, . . . , r or (equivalently) in the bijection with systems (tl,k ) of Mfm -natural T S l T T S k T transformoperators (natural tensors) tl,k : Riem ing Riemannian metrics g on m-manifolds M into tensor fields of types T S l T T S k T on M for l, k = 1, . . . , r. Because of the isomorphisms J r T M J r T M (J r T M ) (J r T M ) = = = depending on g, we have the following theorem. Theorem 5. The Mfm -natural operators D : Riem Hom(J r T, (J r T ) ) transforming Riemannian metrics g on m-manifolds M into base preserving vector bundle maps DM (g) : J r T M (J r T M ) are in the bijection with systems (Al,k ) of Mfm -natural operators Al,k : Riem (T S l T , T S k T ) transforming Riemannian metrics g on m-manifolds M into base preserving vector bundle maps Al,k (g) : T M S l T M T M S k T M for l, k = 1, . . . , r M or (equivalently) in the bijection with systems (tl,k ) of Mfm -natural opT S l T T S k T transformerators (natural tensors) tl,k : Riem ing Riemannian metrics g on m-manifolds M into tensor fields of types T S l T T S k T on M for l, k = 1, . . . , r. By the same reason, we have the following theorem. Theorem 6. The Mfm -natural operators D : Riem Hom(J r T , (J r T )) transforming Riemannian metrics g on m-manifolds M into base preserving vector bundle maps DM (g) : J r T M (J r T M ) are in the bijection with (T S l T , T systems (Al,k ) of Mfm -natural operators Al,k : Riem k T ) transforming Riemannian metrics g on m-manifolds M into base preS serving vector bundle maps Al,k (g) : T M S l T M T M S k T M for M l, k = 1, . . . , r or (equivalently) in the bijection with systems (tl,k ) of Mfm T S l T T S k T transnatural operators (natural tensors) tl,k : Riem forming Riemannian metrics g on m-manifolds M into tensor fields of types T S l T T S k T on M for l, k = 1, . . . , r. By the same reason, we have also the following theorem. Theorem 7. The Mfm -natural operators D : Riem Hom((J r T ), (J r T )) transforming Riemannian metrics g on m-manifolds M into base preserving vector bundle maps DM (g) : (J r T M ) (J r T M ) are in the bijection with (T S l T, T S k T ) systems (Al,k ) of Mfm -natural operators Al,k : Riem transforming Riemannian metrics g on m-manifolds M into base preserving vector bundle maps Al,k (g) : T M S l T M T M S k T M for l, k = 1, . . . , r M or (equivalently) in the bijection with systems (tl,k ) of Mfm -natural opT S l T T S k T transformerators (natural tensors) tl,k : Riem ing Riemannian metrics g on m-manifolds M into tensor fields of types T S l T T S k T on M for l, k = 1, . . . , r. We have also the following corollary. Hom((J r T ), J r T ) Corollary 6. The Mfm -natural operators D : Riem transforming Riemannian metrics g on m-manifolds M into base preserving vector bundle maps DM (g) : (J r T M ) J r T M are in the bijection with (T S l T, T S k T ) systems (Al,k ) of Mfm -natural operators Al,k : Riem transforming Riemannian metrics g on m-manifolds M into base preserving vector bundle maps Al,k (g) : T M S l T M T M S k T M for l, k = 1, . . . , r M or (equivalently) in the bijection with systems (tl,k ) of Mfm -natural operT S l T T S k T transformators (natural tensors) tl,k : Riem ing Riemannian metrics g on m-manifolds M into tensor fields of types T S l T T S k T on M for l, k = 1, . . . , r. We have also the next corollary. Corollary 7. The Mfm -natural operators D : Riem Hom((J r T ), J r T ) transforming Riemannian metrics g on m-manifolds M into base preserving vector bundle maps DM (g) : (J r T M ) J r T M are in the bijection with (T S l T, T systems (Al,k ) of Mfm -natural operators Al,k : Riem k T ) transforming Riemannian metrics g on m-manifolds M into base S preserving vector bundle maps Al,k (g) : T M S l T M T M S k T M for M l, k = 1, . . . , r or (equivalently) in the bijection with systems (tl,k ) of Mfm T SlT T SkT natural operators (natural tensors) tl,k : Riem transforming Riemannian metrics g on m-manifolds M into tensor fields of types T S l T T S k T on M for l, k = 1, . . . , r. We have also the similar corollary. Corollary 8. The Mfm -natural operators D : Riem Hom((J r T ), (J r T )) transforming Riemannian metrics g on m-manifolds M into base preserving vector bundle maps DM (g) : (J r T M ) (J r T M ) are in the bijection with (T S l T, T S k T ) systems (Al,k ) of Mfm -natural operators Al,k : Riem transforming Riemannian metrics g on m-manifolds M into base preserving vector bundle maps Al,k (g) : T M S l T M T M S k T M for l, k = 1, . . . , r M or (equivalently) in the bijection with systems (tl,k ) of Mfm -natural operT S l T T S k T transformators (natural tensors) tl,k : Riem ing Riemannian metrics g on m-manifolds M into tensor fields of types T S l T T S k T on M for l, k = 1, . . . , r. Finally, we have the last corollary. Corollary 9. The Mfm -natural operators D : Riem Hom((J r T ) , (J r T ) ) transforming Riemannian metrics g on m-manifolds M into base preserving vector bundle maps DM (g) : (J r T M ) (J r T M ) are in the bijection with systems (Al,k ) of Mfm -natural operators Al,k : Riem (T S l T, T S k T ) transforming Riemannian metrics g on m-manifolds M into base preserving vector bundle maps Al,k (g) : T M S l T M T M S k T for l, k = 1, . . . , r or (equivalently) in the bijection with systems (tl,k ) of T S l T T S k T Mfm -natural operators (natural tensors) tl,k : Riem transforming Riemannian metrics g on m-manifolds M into tensor fields of types T S l T T S k T on M for l, k = 1, . . . , r. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Annales UMCS, Mathematica de Gruyter

The natural transformations between T-th order prolongation of tangent and cotangent bundles over Riemannian manifolds

Plaszczyk, Mariusz
Annales UMCS, Mathematica , Volume 69 (1) – Jun 1, 2015
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de Gruyter
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Abstract

If (M, g) is a Riemannian manifold then there is the well-known base preserving vector bundle isomorphism T M T M given by v g(v, -) between the tangent T M and the cotangent T M bundles of M . In the present note first we generalize this isomorphism to the one J r T M J r T M between the r-th order prolongation J r T M of tangent T M and the r-th order prolongation J r T M of cotangent T M bundles of M . Further we describe all base preserving vector bundle maps DM (g) : J r T M J r T M depending on a Riemannian metric g in terms of natural (in g) tensor fields on M . 1. Introduction. All manifolds are smooth, Hausdorff, finite dimensional and without boundaries. Maps are assumed to be smooth, i.e. of class C . Let Mfm denote category of m-dimensional manifolds and their embeddings. From the general theory it is well known that the tangent T M and the cotangent T M bundles of M are not canonically isomorphic. However, if g is a Riemannian metric on a manifold M , there is the base preserving vector bundle isomorphism ig : T M T M given by ig (v) = g(v, -), v Tx M, x M . In the second section of the present note we give necessary definitions. 2010 Mathematics Subject Classification. 58A05, 58A20, 58A32. Key words and phrases. Riemannian manifold, higher order prolongation of a vector bundle, natural tensor, natural operator. In the third section first we generalize the isomorphism ig : T M T M depending on g to a base preserving vector bundle isomorphism J r ig : J r T M J r T M canonically depending on g between the r-th order prolongation J r T M of tangent T M and the r-th order prolongation J r T M of cotangent T M bundles of M . Next we construct another more advanced base preserving vector bundle isomorphism i : J r T M J r T M canonically g depending on g. In the fourth section we consider the problem of describing all Mfm natural operators D : Riem Hom(J r T, J r T ) transforming Riemannian metrics g on m-dimensional manifolds M into base preserving vector bundle maps DM (g) : J r T M J r T M . Our studies lead to the reduction of this problem to the one of describing all Mfm -natural operators t : Riem T S l T T S k T (for l, k = 1, . . . , r) sending Riemannian metrics g on M into tensor fields tM (g) of types T S l T T S k T . 2. Definitions. Now we give some necessary definitions. Definition 1. The r-th order prolongation of tangent bundle is a functor J r T : Mfm VB sending any m-manifold M into J r T M and any embedding : M1 M2 of two manifolds into J r T : J r T M1 J r T M2 given by J r T (jx X) = j(x) X, where X X (M1 ) and X = T X is the image of a vector field X by . Definition 2. The r-th order prolongation of cotangent bundle is a functor J r T : Mfm VB sending any m-manifold M into J r T M and any embedding : M1 M2 of two manifolds into J r T : J r T M1 J r T M2 given by J r T := J r (T ) . Definition 3. The dual bundle of the r-th order prolongation of tangent bundle is a functor (J r T ) : Mfm VB sending any m-manifold M into (J r T ) M := (J r T M ) and any embedding : M1 M2 of two manifolds into (J r T ) : (J r T ) M1 (J r T ) M2 given by (J r T ) := (J r T ) . Definition 4. The dual bundle of the r-th order prolongation of cotangent bundle is a functor (J r T ) : Mfm VB sending any m-manifold M into (J r T ) M := (J r T M ) and any embedding : M1 M2 of two manifolds into (J r T ) : (J r T ) M1 (J r T ) M2 given by (J r T ) := (J r T ) . The general concept of natural operators can be found in [4]. In particular, we have the following definitions. Definition 5. An Mfm -natural operator D : Riem Hom(J r T, J r T ) transforming Riemannian metrics g on m-dimensional manifolds M into base preserving vector bundle maps DM (g) : J r T M J r T M is a system D = {DM }M obj(Mfm ) of regular operators DM : Riem(M ) HomM (J r T M, J r T M ) satisfying the Mfm -invariance condition, where HomM (J r T M, J r T M ) is the set of all vector bundle maps J r T M J r T M covering the identity map idM of M . The Mfm -invariance condition of D is following: for any g1 Riem(M1 ) and g2 Riem(M2 ) if g1 and g2 are -related by an embedding : M1 M2 of m-manifolds (i.e. is (g1 , g2 )-isomorphism) then DM1(g1 ) and DM2(g2 ) are also -related (i.e. DM2 (g2 ) J r T = J r T DM1 (g1 )). Equivalently, the above Mfm -invariance means that for any g1 Riem(M1 ) and g2 Riem(M2 ) if the diagram T M1 T M 1 (1) g1 M1 T T T M 2 T M2 g2 M2 commutes for an embedding : M1 M2 (i.e. (T T ) g1 = g2 ) then the diagram J r T M1 DM1 (g1 ) J r T M1 J rT J r T J r T M2 DM2 (g2 ) J r T M2 commutes also. We say that operator DM is regular if it transforms smoothly parameterized families of Riemannian metrics into smoothly parameterized ones of vector bundle maps. Similarly, we can define the following concepts: - an Mfm -natural operator D : Riem Hom(J r T, J r T ), Hom(J r T, (J r T ) ), - an Mfm -natural operator D : Riem Hom(J r T, (J r T ) ), - an Mfm -natural operator D : Riem - an Mfm -natural operator D : Riem - an Mfm -natural operator D : Riem - an Mfm -natural operator D : Riem - an Mfm -natural operator D : Riem - an Mfm -natural operator D : Riem - an Mfm -natural operator D : Riem - an Mfm -natural operator D : Riem - an Mfm -natural operator D : Riem - an Mfm -natural operator D : Riem - an Mfm -natural operator D : Riem - an Mfm -natural operator D : Riem - an Mfm -natural operator D : Riem Now we have the following definition. Hom(J r T , J r T ), Hom(J r T , J r T ), Hom(J r T , (J r T ) ), Hom(J r T , (J r T ) ), Hom((J r T ) , J r T ), Hom((J r T ) , J r T ), Hom((J r T ) , (J r T ) ), Hom((J r T ) , (J r T ) ), Hom((J r T ) , J r T ), Hom((J r T ) , J r T ), Hom((J r T ) , (J r T ) ), Hom((J r T ) , (J r T ) ). (T S l T , T S k T ) Definition 6. An Mfm -natural operator A : Riem transforming Riemannian metrics g on m-dimensional manifolds M into base preserving vector bundle maps AM (g) : T M S l T M T M S k T M is a system A = {AM }M obj(Mfm ) of regular operators AM : Riem(M ) C (T M S l T M, T M S k T M ) satisfying the Mfm -invariance condition, where C (T M S l T M, T M S k T M ) is the set of all vector bundle maps T M S l T M T M S k T M covering the identity map idM of M . The Mfm -invariance condition of A is following : for any g1 Riem(M1 ) and g2 Riem(M2 ) if g1 and g2 are -related by an embedding : M1 M2 of m-manifolds (i.e. (T T )g1 = g2 ) then AM1 (g1 ) and AM2 (g2 ) are also -related (i.e. AM2 (g2 )(T S l T ) = (T S k T )AM1 (g1 )). Equivalently, the above Mfm -invariance means that for any g1 Riem(M1 ) and g2 Riem(M2 ) if the diagram (1) commutes for an embedding : M1 M2 then the diagram T M1 S k T M 1 AM1 (g1 ) T M1 S l T M 1 T Sk T T M 2 S k T M2 AM2 (g2 ) T M2 S l T M2 T SlT commutes also. The regularity means almost the same as in Definition 5. Similarly, we can define the following concepts: (T S l T, T S k T ), - an Mfm -natural operator A : Riem (T S l T, T S k T ), - an Mfm -natural operator A : Riem - an Mfm -natural operator A : Riem (T S l T , T S k T ), (T S l T, T S k T ), - an Mfm -natural operator A : Riem (T S l T, T S k T ), - an Mfm -natural operator A : Riem (T S l T , T S k T ), - an Mfm -natural operator A : Riem (T S l T, T S k T ), - an Mfm -natural operator A : Riem (T S l T, T S k T ), - an Mfm -natural operator A : Riem (T S l T , T S k T ), - an Mfm -natural operator A : Riem (T S l T , T S k T ), - an Mfm -natural operator A : Riem (T S l T, T S k T ), - an Mfm -natural operator A : Riem (T S l T , T S k T ), - an Mfm -natural operator A : Riem (T S l T , T S k T ), - an Mfm -natural operator A : Riem (T S l T, T S k T ), - an Mfm -natural operator A : Riem (T S l T , T S k T ). - an Mfm -natural operator A : Riem Next we have an important general definition of natural tensor. p T Definition 7. An Mfm -natural operator (natural tensor) t : Riem q T transforming Riemannian metrics g on m-dimensional manifolds M into tensor fields of type (p, q) on M is a system t = {tM }M obj(Mfm ) of regular operators tM : Riem(M ) T (p,q) (M ) satisfying the Mfm -invariance condition, where T (p,q) (M ) is the set of tensor fields of type (p, q) on M . The Mfm -invariance condition of t is following : for any g1 Riem(M1 ) and g2 Riem(M2 ) if g1 and g2 are -related by an embedding : M1 M2 of m-manifolds (i.e. (T T ) g1 = g2 ) then tM1 (g1 ) and tM2 (g2 ) are also -related (i.e. tM2 (g2 ) = ( p T q T ) tM1 (g1 )). Equivalently, the above Mfm -invariance means that for any g1 Riem(M1 ) and g2 Riem(M2 ) if the diagram (1) commutes for an embedding : M1 M2 , then the diagram p p T M1 T M T M2 T M2 tM1 (g1 ) tM2 (g2 ) M1 M2 commutes also. We say that operator tM is regular if it transforms smoothly parametrized families of Riemannian metrics into smoothly parametrized ones of tensor fields. Now we have a definition of a special kind of natural tensor. Definition 8. An Mfm -natural operator (natural tensor) t : Riem T l T T S k T transforming Riemannian metrics g on m-dimensional S manifolds M into tensor fields of type T S l T T S k T on M is a system t = {tM }M obj(Mfm ) of regular operators tM : Riem(M ) C (T M S l T M T M S k T M ) satisfying the Mfm -invariance condition, where C (T M S l T M T M S k T M ) is the set of all tensor fields of type T S l T T S k T on M . The Mfm -invariance condition of t is following: for any g1 Riem(M1 ) and g2 Riem(M2 ) if g1 and g2 are -related by an embedding : M1 M2 of m-manifolds (i.e. (T T ) g1 = g2 ), then tM1 (g1 ) and tM2 (g2 ) are also -related (i.e. tM2 (g2 ) = (T S l T T S k T )tM1 (g1 )). Equivalently, the above Mfm -invariance means that for any g1 Riem(M1 ) and g2 Riem(M2 ) if the diagram (1) commutes for an embedding : M1 M2 , then the diagram T M1 S l T M1 T M1 S k T M1 T M2 S l T M2 T M2 S k T M2 tM1 (g1 ) tM2 (g2 ) M1 M2 commutes also, where = T S l T T S k T . The regularity means almost the same as in Definition 7. Similarly, we can define the following concepts: - an Mfm -natural operator (natural tensor) t : Riem T SkT , - an Mfm -natural operator (natural tensor) t : Riem T SkT , - an Mfm -natural operator (natural tensor) t : Riem T SkT , - an Mfm -natural operator (natural tensor) t : Riem T SkT , - an Mfm -natural operator (natural tensor) t : Riem T Sk T , - an Mfm -natural operator (natural tensor) t : Riem T SkT , - an Mfm -natural operator (natural tensor) t : Riem T SkT , T SlT T SlT T SlT T SlT T SlT T SlT T SlT - an Mfm -natural operator (natural tensor) t : Riem T Sk T , - an Mfm -natural operator (natural tensor) t : Riem T SkT , - an Mfm -natural operator (natural tensor) t : Riem T Sk T , - an Mfm -natural operator (natural tensor) t : Riem T Sk T , - an Mfm -natural operator (natural tensor) t : Riem T SkT , - an Mfm -natural operator (natural tensor) t : Riem T Sk T , - an Mfm -natural operator (natural tensor) t : Riem T Sk T , - an Mfm -natural operator (natural tensor) t : Riem T Sk T . T SlT T SlT T SlT T SlT T SlT T SlT T SlT T SlT In the third section we present also explicit examples of Mfm -natural operators D : Riem Hom(J r T, J r T ). p T A full description of all polynomial natural tensors t : Riem q T transforming Riemannian metrics on m-manifolds into tensor fields of types (p, q) can be found in [1]. This description is following. Each covariant derivative of the curvature R(g) T (0,4) (M ) of a Riemannian metric g is a natural tensor and the metric g is also a natural tensor. Further all the p T q T can be obtained by a procedure: natural tensors t : Riem (a) every tensor multiplication of two natural tensors give a new natural tensor, (b) every contraction on one covariant and one contravariant entry of a natural tensor give a new natural tensor, (c) we can tensorize any natural tensor with a metric independent natural tensor, (d) we can permute any number of entries in the tensor product, (e) we can repeat these steps, (f) we can take linear combinations. Furthermore, if we take respective type natural tensors and apply respective symmetrization, then we can produce many natural tensors t : Riem T SlT T Sk T . 3. Constructions. Example 1. Let (M, g) be a Riemannian manifold. Then we have a base preserving vector bundle isomorphism ig : T M T M given by ig (v) = g(v, -), v Tx M, x M. Next we can obtain a base preserving vector bundle isomorphism J r ig : J r T M J r T M defined by a formula J r ig (jx X) = jx (ig X), where X X (M ). Similarly we receive also a base preserving vector bundle isomorphism (J r i ) : (J r T M ) (J r T M ) . g Because of the canonical character of the above constructions we get the following propositions. Proposition 1. The family A(r) : Riem (r) Hom(J r T, J r T ) of operators AM (g) = J r ig (r) AM : Riem(M ) HomM (J r T M, J r T M ), for all M obj(Mfm ) is an Mfm -natural operator. Proposition 2. The family A : Riem ators Hom((J r T ) , (J r T ) ) of operAM (g) = (J r i ) g AM : Riem(M ) HomM ((J r T M ) , (J r T M ) ), for all M obj(Mfm ) is an Mfm -natural operator. Now we are going to present another more advanced example of an Mfm natural operator D : Riem Hom(J r T, J r T ). Recall that if g is a Riemannian tensor field on a manifold M and x M , then there is g-normal coordinate system : (M, x) (Rm , 0) with centre x. If : (M, x) (Rm , 0) is another g-normal coordinate system with centre x, then there is A O(m) such that = A near x. Let r m k m = m k m (see [3]) and I : J 0 T Rm S R k=1 T0 R S T0 T Rm Rm S k T Rm = m S k Rm (see [7]) be I1 : J0 0 k=1 T0 the standard O(m)-invariant vector space isomorphisms. We have the following important proposition. Proposition 3. Let g be a Riemannian tensor field on a manifold M . Then there are (canonical in g) vector bundle isomorphisms Ig : J r T M T M S k T M, T M S k T M, Jg : J r T M (Ig ) : r (J T M ) TM S T M T M S k T M, k=0 (Jg ) : (J T M ) k=0 T M S T M T M S k T M. k=0 Proof. Let v = jx X Jx T M , where X X (M ), x M . Let : (M, x) m , 0) be a g-normal coordinate system with centre x. We define ( Ig (v) := Ig (v) k=0 T S k T I J r T (v). If : (M, x) is another g-normal coordinate system with centre x, then = A (near x) for some A O(m). The O(m)-invariance of I means that (Rm , 0) (2) I J rT A = k=0 T0 A S k T0 A I. Hence we deduce that r Ig (v) = T S k T I J r T (v) (T (A ) S k T (A ) ) I J r T (A )(v) ((T T A ) S k T ( A )) I (J r T A J r T )(v) = = k=0 ((T T A ) S k T ( A )) (I J r T A) J r T (v) =: L. Now using (2), we receive L= ((T T A )S k T (A )) k=0 T AS k T A I J r T (v) ((T T A )T A) (S k T ( A )S k T A) I J r T (v) ((T T A T A) S k T ( A A)) I J r T (v) (T S k T ) I J r T (v) = Ig (v). k=0 = = Therefore, the definition of Ig (v) is independent of the choice of . So, isomorphism Ig : J r T M r T M S k T M is well defined. k=0 Similarly, we put Jg (v) := Jg (v) k=0 T S k T I1 J r T (v). Using O(m)-invariance of I1 (i.e. I1 J r T A = ( r T0 A S k T0 A) k=0 I1 ) analogously as before, we show that Ig (v) = Ig (v). This proves that the definition of Jg (v) is independent of the choice of g-normal coordinate system with centre x and the isomorphism Jg : J r T M r T M k=0 S k T M is well defined. Finally we obtain (canonical in g) vector bundle isomorphisms r (Ig ) : (J r T M ) (Jg ) : (J r T M ) k=0 k=0 T M S k T M. k=0 Remark 1. W. Mikulski (in [7]) has recently constructed a (canonical in r k ) vector bundle isomorphism I : J r T M k=0 T M S T M for a classical linear connection on a manifold M . Now we have further important identifications. Example 2. Let (M, g) be a Riemannian manifold and ig : T M T M be a base preserving vector bundle isomorphism recalled in Example 1. Using the base preserving vector bundle isomorphisms Ig and Jg from Proposition 3, we receive the following vector bundle isomorphisms r i := Jg g i := Jg g k=0 ig S k T idM i S k T idM g Ig : J r T M J r T M, (Ig ) : (J r T M ) (J r T M ) . Because of canonical character of above constructions we obtain the following propositions. Proposition 4. The family B : Riem Hom(J r T, J r T ) of operators BM (g) = i g BM : Riem(M ) HomM (J r T M, J r T M ), for all M obj(Mfm ) is an Mfm -natural operator. Proposition 5. The family B : Riem ators Hom((J r T ) , (J r T ) ) of operBM (g) = i g BM : Riem(M ) HomM ((J r T M ) , (J r T M ) ), for all M obj(Mfm ) is an Mfm -natural operator. Example 3. Let (M, g) be a Riemannian manifold. In an article [5] J. Kurek and W. Mikulski constructed a base preserving vector bundle isomorphism i(r) : g k=1 S TM k=1 Sk T M given by i(r) (v1 g ··· vk ) = ig (v1 ) ··· ig (vk ). Now using this isomorphism, we get a base preserving vector bundle isomorphism r (r) Ig : T M Sk T M = SkT M = T M k=0 k=0 defined by a formula (r) Ig = ig (i(r) ) . g Similarly, we construct another base preserving vector bundle isomorphisms ~(r) Ig : ~(r) ~ Ig : ^(r) Ig : T M S k T M, T M S k T M, ~(r) Ig = idT M i(r) g ~(r) ~ Ig = i i(r) g g k=0 k=0 T M S k T M, ^(r) Ig = idT M i(r) g Thus we receive a base preserving vector bundle isomorphism Ig : J r T M (J r T M ) given by (r) = Ig Ig Ig . Ig Similarly, we construct another base preserving vector bundle isomorphisms ~ Ig : J r T M (J r T M ) , Ig : J r T M (J r T M ) , ~ = Jg Ig Ig , ~(r) Ig ^ I = I I (r) Jg , g g g ~ Ig : J r T M J T M (J T M ) , ~ Ig Jg ~(r) ~ Ig Jg . Using the base preserving vector bundle isomorphism J r ig : J r T M constructed in Example 1, we obtain also the following vector bundle isomorphisms = (J r i ) Ig : J r T M (J r T M ) , Jg g ~ ~ = (J r ig ) Ig : J r T M (J r T M ) , Jg Jg = (J r i ) Ig : J r T M (J r T M ) , g ~ ~ Jg = (J r ig ) Ig : J r T M (J r T M ) . Because of canonical character of the above constructions we get the following propositions. Proposition 6. The family C : Riem tors Hom(J r T, (J r T ) ) of opera CM (g) = Ig CM : Riem(M ) HomM (J r T M, (J r T M ) ), for all M obj(Mfm ) is an Mfm -natural operator. ~ Proposition 7. The family C : Riem Hom(J r T, (J r T ) ) of operators ~ ~ ~ C : Riem(M ) HomM (J r T M, (J r T M ) ), C (g) = Ig for all M obj(Mfm ) is an Mfm -natural operator. Proposition 8. The family C : Riem Hom(J r T , (J r T ) ) of operators CM (g) = Ig CM : Riem(M ) HomM (J r T M, (J r T M ) ), for all M obj(Mfm ) is an Mfm -natural operator. ~ Proposition 9. The family C : Riem Hom(J r T , (J r T ) ) of operators ~ ~ ~ C : Riem(M ) HomM (J r T M, (J r T M ) ), C (g) = Ig for all M obj(Mfm ) is an Mfm -natural operator. Proposition 10. The family D : Riem ators Hom(J r T, (J r T ) ) of oper DM (g) = Jg DM : Riem(M ) HomM (J r T M, (J r T M ) ), for all M obj(Mfm ) is an Mfm -natural operator. ~ Proposition 11. The family D : Riem Hom(J r T, (J r T ) ) of operators ~ ~ ~ D : Riem(M ) HomM (J r T M, (J r T M ) ), D (g) = Jg for all M obj(Mfm ) is an Mfm -natural operator. Proposition 12. The family D : Riem ators Hom(J r T , (J r T ) ) of oper DM (g) = Jg DM : Riem(M ) HomM (J r T M, (J r T M ) ), for all M obj(Mfm ) is an Mfm -natural operator. ~ Proposition 13. The family D : Riem Hom(J r T , (J r T ) ) of operators ~ ~ ~ D : Riem(M ) HomM (J r T M, (J r T M ) ), D (g) = Jg for all M obj(Mfm ) is an Mfm -natural operator. 4. The main results. Let g Riem(M ) be a Riemannian metric on an mmanifold M . By Proposition 3 and Examples 1, 2, 3 we have identifications J r T M = J r T M = (J r T M ) = (J r T M ) = r k=0 = k=0 = k=0 T M SkT M modulo the base preserving vector bundle isomorphisms canonically depending on g. Consequently, the problem of finding all Mfm -natural operators D : Riem Hom(J r T, J r T ) is reduced to the one of finding all systems (Al,k ) of (T S l T , T S k T ) transforming Mfm -natural operators Al,k : Riem Riemannian metrics g on m-manifolds M into base preserving vector bundle maps Al,k (g) : T M S l T M T M S k T M , where l, k = 1, . . . , r or M (equivalently) our problem is reduced to the one of finding all natural tenT S l T T S k T transforming Riemannian metrics sors tl,k : Riem g on m-manifolds M into tensor fields of types T S l T T S k T on M for l, k = 1, . . . , r. Thus we have proved the following theorem. Hom(J r T, J r T ) Theorem 1. The Mfm -natural operators D : Riem transforming Riemannian metrics g on m-manifolds M into base preserving vector bundle maps DM (g) : J r T M J r T M are in the bijection with systems (tl,k ) of Mfm -natural operators (natural tensors) tl,k : Riem T S l T T S k T transforming Riemannian metrics g on m-manifolds M into tensor fields of types T S l T T S k T on M for l, k = 1, . . . , r. Because of the isomorphism J r T M J r T M depending on g, we have = the following theorem. Theorem 2. The Mfm -natural operators D : Riem Hom(J r T, J r T ) transforming Riemannian metrics g on m-manifolds M into base preserving vector bundle maps DM (g) : J r T M J r T M are in the bijection with sys(T S l T , T S k T ) tems (Al,k ) of Mfm -natural operators Al,k : Riem transforming Riemannian metrics g on m-manifolds M into base preserving vector bundle maps Al,k (g) : T M S l T M T M S k T M for l, k = M 1, . . . , r or (equivalently) in the bijection with systems (tl,k ) of Mfm -natural T S l T T S k T transformoperators (natural tensors) tl,k : Riem ing Riemannian metrics g on m-manifolds M into tensor fields of types T S l T T S k T on M for l, k = 1, . . . , r. By the same reason, we have also the following corollary. Corollary 1. The Mfm -natural operators D : Riem Hom(J r T , J r T ) transforming Riemannian metrics g on m-manifolds M into base preserving vector bundle maps DM (g) : J r T M J r T M are in the bijection with (T S l T , T systems (Al,k ) of Mfm -natural operators Al,k : Riem k T ) transforming Riemannian metrics g on m-manifolds M into base S preserving vector bundle maps Al,k (g) : T M S l T M T M S k T for l, k = 1, . . . , r or (equivalently) in the bijection with systems (tl,k ) of T SlT T SkT Mfm -natural operators (natural tensors) tl,k : Riem transforming Riemannian metrics g on m-manifolds M into tensor fields of types T S l T T S k T on M for l, k = 1, . . . , r. By the same reason, we have another corollary. Corollary 2. The Mfm -natural operators D : Riem Hom(J r T , J r T ) transforming Riemannian metrics g on m-manifolds M into base preserving vector bundle maps DM (g) : J r T M J r T M are in the bijection with (T S l T , T systems (Al,k ) of Mfm -natural operators Al,k : Riem S k T ) transforming Riemannian metrics g on m-manifolds M into base preserving vector bundle maps Al,k (g) : T M S l T M T M S k T for l, k = 1, . . . , r or (equivalently) in the bijection with systems (tl,k ) of T S l T T S k T Mfm -natural operators (natural tensors) tl,k : Riem transforming Riemannian metrics g on m-manifolds M into tensor fields of types T S l T T S k T on M for l, k = 1, . . . , r. Because of the isomorphisms J r T M J r T M (J r T M ) depending = = on g, we have the following theorem. Hom(J r T, (J r T ) ) Theorem 3. The Mfm -natural operators D : Riem transforming Riemannian metrics g on m-manifolds M into base preserving vector bundle maps DM (g) : J r T M (J r T M ) are in the bijection with (T S l T , T systems (Al,k ) of Mfm -natural operators Al,k : Riem k T ) transforming Riemannian metrics g on m-manifolds M into base preS serving vector bundle maps Al,k (g) : T M S l T M T M S k T M for M l, k = 1, . . . , r or (equivalently) in the bijection with systems (tl,k ) of Mfm T SlT T SkT natural operators (natural tensors) tl,k : Riem transforming Riemannian metrics g on m-manifolds M into tensor fields of types T S l T T S k T on M for l, k = 1, . . . , r. By the same reason, we have the following theorem. Theorem 4. The Mfm -natural operators D : Riem Hom(J r T , (J r T ) ) transforming Riemannian metrics g on m-manifolds M into base preserving vector bundle maps DM (g) : J r T M (J r T M ) are in the bijection with (T S l T , T systems (Al,k ) of Mfm -natural operators Al,k : Riem S k T ) transforming Riemannian metrics g on m-manifolds M into base preserving vector bundle maps Al,k (g) : T M S l T M T M S k T M for M l, k = 1, . . . , r or (equivalently) in the bijection with systems (tl,k ) of Mfm T S l T T S k T transnatural operators (natural tensors) tl,k : Riem forming Riemannian metrics g on m-manifolds M into tensor fields of types T S l T T S k T on M for l, k = 1, . . . , r. By the same reason, we have also the following corollary. Corollary 3. The Mfm -natural operators D : Riem Hom((J r T ) , J r T ) transforming Riemannian metrics g on m-manifolds M into base preserving vector bundle maps DM (g) : (J r T M ) J r T M are in the bijection with (T S l T, T systems (Al,k ) of Mfm -natural operators Al,k : Riem k T ) transforming Riemannian metrics g on m-manifolds M into base S preserving vector bundle maps Al,k (g) : T M S l T M T M S k T M for M l, k = 1, . . . , r or (equivalently) in the bijection with systems (tl,k ) of Mfm T SlT T SkT natural operators (natural tensors) tl,k : Riem transforming Riemannian metrics g on m-manifolds M into tensor fields of types T S l T T S k T on M for l, k = 1, . . . , r. We have also the next similar corollary. Corollary 4. The Mfm -natural operators D : Riem Hom((J r T ) , J r T ) transforming Riemannian metrics g on m-manifolds M into base preserving vector bundle maps DM (g) : (J r T M ) J r T M are in the bijection with (T S l T, T systems (Al,k ) of Mfm -natural operators Al,k : Riem k T ) transforming Riemannian metrics g on m-manifolds M into base S preserving vector bundle maps Al,k (g) : T M S l T M T M S k T M for M l, k = 1, . . . , r or (equivalently) in the bijection with systems (tl,k ) of Mfm T SlT T SkT natural operators (natural tensors) tl,k : Riem transforming Riemannian metrics g on m-manifolds M into tensor fields of types T S l T T S k T on M for l, k = 1, . . . , r. We have also another corollary. Corollary 5. The Mfm -natural operators D : Riem Hom((J r T ), (J r T )) transforming Riemannian metrics g on m-manifolds M into base preserving vector bundle maps DM (g) : (J r T M ) (J r T M ) are in the bijection with (T S l T, T S k T ) systems (Al,k ) of Mfm -natural operators Al,k : Riem transforming Riemannian metrics g on m-manifolds M into base preserving vector bundle maps Al,k (g) : T M S l T M T M S k T M for l, k = M 1, . . . , r or (equivalently) in the bijection with systems (tl,k ) of Mfm -natural T S l T T S k T transformoperators (natural tensors) tl,k : Riem ing Riemannian metrics g on m-manifolds M into tensor fields of types T S l T T S k T on M for l, k = 1, . . . , r. Because of the isomorphisms J r T M J r T M (J r T M ) (J r T M ) = = = depending on g, we have the following theorem. Theorem 5. The Mfm -natural operators D : Riem Hom(J r T, (J r T ) ) transforming Riemannian metrics g on m-manifolds M into base preserving vector bundle maps DM (g) : J r T M (J r T M ) are in the bijection with systems (Al,k ) of Mfm -natural operators Al,k : Riem (T S l T , T S k T ) transforming Riemannian metrics g on m-manifolds M into base preserving vector bundle maps Al,k (g) : T M S l T M T M S k T M for l, k = 1, . . . , r M or (equivalently) in the bijection with systems (tl,k ) of Mfm -natural opT S l T T S k T transformerators (natural tensors) tl,k : Riem ing Riemannian metrics g on m-manifolds M into tensor fields of types T S l T T S k T on M for l, k = 1, . . . , r. By the same reason, we have the following theorem. Theorem 6. The Mfm -natural operators D : Riem Hom(J r T , (J r T )) transforming Riemannian metrics g on m-manifolds M into base preserving vector bundle maps DM (g) : J r T M (J r T M ) are in the bijection with (T S l T , T systems (Al,k ) of Mfm -natural operators Al,k : Riem k T ) transforming Riemannian metrics g on m-manifolds M into base preS serving vector bundle maps Al,k (g) : T M S l T M T M S k T M for M l, k = 1, . . . , r or (equivalently) in the bijection with systems (tl,k ) of Mfm T S l T T S k T transnatural operators (natural tensors) tl,k : Riem forming Riemannian metrics g on m-manifolds M into tensor fields of types T S l T T S k T on M for l, k = 1, . . . , r. By the same reason, we have also the following theorem. Theorem 7. The Mfm -natural operators D : Riem Hom((J r T ), (J r T )) transforming Riemannian metrics g on m-manifolds M into base preserving vector bundle maps DM (g) : (J r T M ) (J r T M ) are in the bijection with (T S l T, T S k T ) systems (Al,k ) of Mfm -natural operators Al,k : Riem transforming Riemannian metrics g on m-manifolds M into base preserving vector bundle maps Al,k (g) : T M S l T M T M S k T M for l, k = 1, . . . , r M or (equivalently) in the bijection with systems (tl,k ) of Mfm -natural opT S l T T S k T transformerators (natural tensors) tl,k : Riem ing Riemannian metrics g on m-manifolds M into tensor fields of types T S l T T S k T on M for l, k = 1, . . . , r. We have also the following corollary. Hom((J r T ), J r T ) Corollary 6. The Mfm -natural operators D : Riem transforming Riemannian metrics g on m-manifolds M into base preserving vector bundle maps DM (g) : (J r T M ) J r T M are in the bijection with (T S l T, T S k T ) systems (Al,k ) of Mfm -natural operators Al,k : Riem transforming Riemannian metrics g on m-manifolds M into base preserving vector bundle maps Al,k (g) : T M S l T M T M S k T M for l, k = 1, . . . , r M or (equivalently) in the bijection with systems (tl,k ) of Mfm -natural operT S l T T S k T transformators (natural tensors) tl,k : Riem ing Riemannian metrics g on m-manifolds M into tensor fields of types T S l T T S k T on M for l, k = 1, . . . , r. We have also the next corollary. Corollary 7. The Mfm -natural operators D : Riem Hom((J r T ), J r T ) transforming Riemannian metrics g on m-manifolds M into base preserving vector bundle maps DM (g) : (J r T M ) J r T M are in the bijection with (T S l T, T systems (Al,k ) of Mfm -natural operators Al,k : Riem k T ) transforming Riemannian metrics g on m-manifolds M into base S preserving vector bundle maps Al,k (g) : T M S l T M T M S k T M for M l, k = 1, . . . , r or (equivalently) in the bijection with systems (tl,k ) of Mfm T SlT T SkT natural operators (natural tensors) tl,k : Riem transforming Riemannian metrics g on m-manifolds M into tensor fields of types T S l T T S k T on M for l, k = 1, . . . , r. We have also the similar corollary. Corollary 8. The Mfm -natural operators D : Riem Hom((J r T ), (J r T )) transforming Riemannian metrics g on m-manifolds M into base preserving vector bundle maps DM (g) : (J r T M ) (J r T M ) are in the bijection with (T S l T, T S k T ) systems (Al,k ) of Mfm -natural operators Al,k : Riem transforming Riemannian metrics g on m-manifolds M into base preserving vector bundle maps Al,k (g) : T M S l T M T M S k T M for l, k = 1, . . . , r M or (equivalently) in the bijection with systems (tl,k ) of Mfm -natural operT S l T T S k T transformators (natural tensors) tl,k : Riem ing Riemannian metrics g on m-manifolds M into tensor fields of types T S l T T S k T on M for l, k = 1, . . . , r. Finally, we have the last corollary. Corollary 9. The Mfm -natural operators D : Riem Hom((J r T ) , (J r T ) ) transforming Riemannian metrics g on m-manifolds M into base preserving vector bundle maps DM (g) : (J r T M ) (J r T M ) are in the bijection with systems (Al,k ) of Mfm -natural operators Al,k : Riem (T S l T, T S k T ) transforming Riemannian metrics g on m-manifolds M into base preserving vector bundle maps Al,k (g) : T M S l T M T M S k T for l, k = 1, . . . , r or (equivalently) in the bijection with systems (tl,k ) of T S l T T S k T Mfm -natural operators (natural tensors) tl,k : Riem transforming Riemannian metrics g on m-manifolds M into tensor fields of types T S l T T S k T on M for l, k = 1, . . . , r.

Journal

Annales UMCS, Mathematica – de Gruyter

Published: Jun 1, 2015

References

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