A formalism for the study of Natural Tensors Fields of type (0,2) on Manifolds and Fibrations
aa r X i v : . [ m a t h . DG ] D ec A new formalism for the study of Natural Tensor Fields oftype (0,2) on Manifolds and Fibrations.
Guillermo HenryDepartamento de Matem´atica, FCEyN,Universidad de Buenos Aires and CONICET, Argentina.
Abstract.
In order to study tensor fields of type (0,2) on manifolds and fibrationswe introduce a new formalism that we called s-space . With the help of these objects wegeneralized the concept of natural tensor without making use of the theory of naturaloperators and differential invariants.
Keywords:
Natural tensor fields · Fibrations · General connections · Riemannianmanifolds
Mathematics Subject Clasification (2000): · · · In [9], Kowalski and Sekizawa defined and characterized the natural tensor f ields of type(0 ,
2) on the tangent bundle
T M of a manifold M . More precisely, let ˜ g be a metricon T M which cames from a second order natural transformation of a metric g on M .Then there are natural F − metrics ξ , ξ and ξ (i.e. a bundle morphism of the form ξ : T M ⊕ T M ⊕ T M −→ M × IR linear in the second and in the third argument) derivedfrom g , such that ˜ g = ξ s,g + ξ h,g + ξ v,g with ξ and ξ symmetric , where ξ s,g , ξ h,g and ξ v,g are the classical Sasaki, horizontal and vertical lift of ξ , ξ and ξ respectively. AlsoKowalski and Sekizawa [10] study the natural tensor f ields on the linear frame bundles ofa manifold endowed with a linear connection.In [2], Calvo and Keilhauer showed that given a Riemannian manifold ( M, g ) any (0 , T M admits a global matrix representation. Using this one to one relationship,they defined and characterized what they called natural tensor . In the symmetric casethis concept coincide with the one of Kowalski and Sekizawa. Keilhauer [7] defined andcharacterized the tensor fields of type (0 ,
2) on the linear frame bundle of a Riemannianmanifold endowed with a linear connection. The natural tensors on the tangent andcotangent bundle of a semi Riemannian manifold was characterized by Araujo and Keilhauerin [1]. The idea of all these works ([1],[2] and [7]) is to lifted to a suitable fiber bundle atensor field on the tangent bundle, cotangent bundle and linear frame bundle respectively,1o that to look at them as a global matricial maps. The principal difference with the works[9] and [10] is that they do not make use of the theory of differential invariant developed byKrupka [11], (see also [8] and [12]).The aim of this work is generalized the notion of natural tensor fields in the sense of[1],[2] and [7] to manifolds and fibrations. With this purpose we introduce the concept of s-space . In Section 2, we define and give some examples of s-spaces . We also see generalproperties of s-spaces , for example that there exist a one to one relationship between thetensor fields of type (0 ,
2) and some types of matricial maps. This relationship allows usto study the tensor fields in the sense of [2]. We characterize the s-spaces which its groupacts without fixed point. We study some general statement of morphisms of s-spaces andtensor fields on manifolds in Section 3. In Section 4, we define connections on s-spaces (thatagree with the well known notion of connection when the s-space is also a principal fiberbundle). We give a condition that a s-space endowed with a connection has to satisfies tohas a parallelizable space manifold. Also, help by a connection we show an useful way oflift metrics on the manifold to the space manifold of the s-space . The concept of s-space gives several notions of naturality. The λ − natural and λ − natural tensors with respectto a fibration are define in section 5. We also give examples and we see that these notionsextend that one of [1],[2] and [7]. In Section 7 we define the notion of atlas of s-spaces andwe use them to generalized the λ − naturality . In Section 8, we consider some s − spaces over a Lie group and characterized the natural tensors fields on it. Finally, we study thebundle metrics on a principal fiber bundle endowed with a linear connection. Definition 1
Let M be a manifold of dimension n . A collection λ = ( N, ψ, O, R, { e i } ) iscalled a s-space over M if:a) N be a manifold.b) ψ : N −→ M is a submersion.c) O is a Lie group and R is a right action of the group O over N which is transitive ineach fibers. The action also satisfies that ψ ◦ R a = ψ for all a ∈ O .d) e i : N −→ T M , with ≤ i ≤ n , are differential functions such that { e ( z ) , . . . , e n ( z ) } is a base of M ψ ( z ) for all z ∈ N . If ψ ( z ) = p , then { e ( z ) , . . . , e n ( z ) } and { e ( z.a ) , . . . , e n ( z.a ) } are bases of M p . Thereforethere exists an invertible matrix L ( z, a ) such that { e i ( z.a ) } = { e i ( z ) } .L ( z, a ) , (i.e. e i ( z.a ) = P nj =1 e l ( z ) L li ( z, a ) for 1 ≤ i ≤ n ). If the matrix L only depends of the parameter of the Liegroup O , we have a differentiable map L : O −→ GL ( n ) such that { e i } ◦ R a = { e i } .L ( a )2hat we called the base change morphism of the s-space λ . It easy to see that L is a groupmorphism. In this case we said that λ have a rigid base change . From now on, we willconsider only this class of s-spaces.In the sequel, unless otherwise stated, dim M = n , dim O = k and we will denote theLie algebra of O by o . Also, we assume that all tensor are of type (0 , Example 2
Let LM be the frame bundle of a manifold M . LM induce a s-space λ =( LM, π, GL ( n ) , ( · ) , { π i } ) over M , where π is the projection of the bundle, ( · ) is thenatural action of the general linear group over LM and π i ( p, u ) = u i . The base changemorphism is L ( a ) = a for all a ∈ GL ( n ) . This example shows that every manifolds admitsat least one s-space. For simplicity of notation, let us denote this s-space by LM too. Ifwe consider a Riemannian metric on M or an orientation, then the bundle of orthonormalframes and the bundle of orientated bases induced similar s-spaces over M . Example 3
Let α = ( P, π, G, · ) be a principal fiber bundle over M , and ω be a connectionon α . Let λ = ( N, ψ, O, R, { e i } ) wherea) N = { ( p, u, w ) : p ∈ P, u is a base of M π ( p ) and w is a base of g } b) ψ ( p, u, w ) = p .c) O = GL ( n ) × GL ( k ) and R ( a,b ) ( p, u, w ) = ( p, u.a, w.b ) d) For ≤ i ≤ n and ≤ j ≤ k , e i ( p, u, w ) is the horizontal lift with respect to ω of u i at p and e n + j ( p, u, w ) is the only vertical vector on P p such that ω ( p )( e n + j ( p, u, w )) = w j . λ is a s-space over P and it’s base change morphism is given by L ( a, b ) = (cid:18) a b (cid:19) . Example 4
This example can be found in [7]. Let M be a manifold and ∇ be a linearconnection on it. Let K : T T M −→ T M be the connection function induced by ∇ ( i.e. K is the unique function that satisfies: for v ∈ M p , K | T M v : T M v −→ M p is a surjective linearmap and for any vector field Y on M such that Y ( p ) = v , we have that K ( Y ∗ p ( w )) = ∇ w Y ).For ≤ i, j ≤ n , consider the 1-forms θ i and ω ij defined by π ∗ ( p,u ) ( b ) = n X i =1 θ i ( p, u )( b ) u i and K (( π j ) ∗ ( p,u ) ( b )) = n X i =1 ω ij ( p, u )( b ) u i . Let λ = ( LM × GL ( n ) , ψ, GL ( n ) , R, { H i , V ij } ) where ψ ( p, u, b ) = ( p, u.b ) , the action is R a ( p, u, b ) = ( p, u.a, a − b ) and { H i , V ij } is dual to { θ i , ω ij } . λ is a s-space over the frame bundle of M with base change morphism L ( a ) ≡ Id n × n . The importance of the s-spaces for the study of the tensors on manifolds is given by thefollowing proposition: 3 roposition 5
Let λ = ( N, ψ, O, R, { e i } ) be a s-space over M and L be the base changemorphism of λ . There is a one to one correspondence between tensor fields of type (0 , on M and differentiable maps λ T : N −→ IR n × n that satisfy the invariance property λ T ◦ R a = ( L ( a )) t . λ T.L ( a ) Proof.
Let T be a tensor on M . Consider the matrix function λ T : N −→ IR n × n definedby [ λ T ( z )] ij = T ( ψ ( z ))( e i ( z ) , e j ( z )). For a ∈ O , we have that the ( i, j ) entry of the matrix λ T ( z.a ) is [ λ T ( z.a )] ij = T ( ψ ( z.a ))( e i ( z.a ) , e j ( z.a )) = T ( ψ ( z ))( n X r =1 e r ( z ) L ( a ) ri , n X s =1 e s ( z ) L ( a ) sj )= P nr,s =1 L ( a ) ri . λ T ( z ) rs .L ( a ) sj , hence λ T satisfies the invariance property. Let F : N −→ IR n × n be a differentiable function that satisfies the invariance property, we are going to showthat there exists a unique tensor T on M such that λ T = F . If X is a vector field on M , thenit induce a map λ X = ( x , . . . , x n ) : N −→ IR n where X ( ψ ( z )) = n X i =1 x i ( z ) e i ( z ). It is easyto check that λ X ◦ R a = λ X. [ L ( a ) t ] − . Then, we define T ( p )( X, Y ) = λ X ( z ) .F ( z ) . ( λ Y ( z )) t where ψ ( z ) = p . Consider z and ¯ z such that ψ ( z ) = ψ (¯ z ) = p . Since O acts transitively onthe fibers of N , there exists a ∈ O that satisfies ¯ z = z.a . Therefore, λ X (¯ z ) .F (¯ z ) . ( λ Y (¯ z )) t = λ X ( z ) . ( L ( a ) t ) − .L ( a ) t . λ F ( z ) .L ( a ) . ( L ( a )) − ( λ Y ( z )) t = λ X ( z ) .F ( z ) . ( λ Y ( z )) t , what it provethat T it is well defined. Given X and Y vector fields on M , T ( X, Y ) : M −→ IR is adifferentiable function because T ( X, Y ) ◦ ψ is differentiable and ψ is a submersion. Since T is F ( M )-bilinear, we conclude that T is a tensor of type (0,2) on M . Finally, it is clearthat λ T = F . Theorem 6
Let λ = ( N, ψ, O, R, { e i } ) be a s-space over M , such that O acts without fixedpoint (i.e. if z.a = a then a = e ), then ( N, ψ, O, R ) its a principal fiber bundle over M . Let us denote by z ∼ z ′ the equivalence relation induced by the action of the group O on the manifold N . To prove the previous Theorem we will need the following next twolemmas. Lema 7
Let λ = ( N, ψ, O, R, { e i } ) be a s-space over M . Then N/O has differentiablemanifold structure and π : N −→ N/O is a submersion.Proof.
Consider the map ρ : N × N −→ M × M defined by ρ ( z, z ′ ) = ( ψ ( z ) , ψ ( z ′ )). ρ is a submersion since ψ it is. Let the set ¯∆ = { ( z, z ′ ) : z ∼ z ′ } and ∆ be the diagonalsubmanifold of N × N . Since z ∼ z ′ if and only if ψ ( z ) = ψ ( z ′ ), we have that ¯∆ = ρ − (∆).Therefore ¯∆ is a closed submanifold of N × N . It is well know (see for example [3]) that if agroup O acts on a manifold N , N/O has a structure of differentiable manifold such that thecanonical projection π is a submersion if and only if ¯∆ is a closed submanifold of N × N .In this case, the differentiable structure of N/O is unique.4 ema 8
Under the hypotheses of the previous lemma:i)
N/O is diffeomorphic to M .ii) ker π ∗ = ker ψ ∗ . Proof.
Let f : N/O −→ M defined by f ([ z ]) = ψ ( z ). By definition f ◦ π = ψ , then f is differentiable and ker π ∗ ⊆ ker ψ ∗ . In the other hand, let g : M −→ N/O , defined by g ( p ) = π ( z ) where z ∈ N satisfies that ψ ( z ) = p . Since O acts transitively on the fibersof N , g is well defined. As π = g ◦ ψ we have that g is a differentiable function and thatker ψ ∗ ⊆ ker π ∗ . An easy verification shows that g ◦ f = Id N/O and f ◦ g = Id M . Remark 9 If λ = ( N, ψ, O, R, { e i } ) is a s-space over M , then ( N, ψ, O, R ) is a principalfiber bundle over N/O .Proof of Theorem 6.
It remains to prove that (
N, ψ, O, R ) satisfies the local triviality prop-erty, (i.e. all p ∈ M has an open neighbour U on M , and a diffeomorphism τ : ψ − ( U ) −→ U × O such that τ = ( ψ, φ ), where φ ( z.a ) = φ ( z ) .a for all a ∈ O ). Let p ∈ M , take [ z ] ∈ N/O such that f ([ z ]) = p . As ( N, ψ, O, R ) is a principal fiber bundle over
N/O , there exist anopen neighbour V of [ z ] and a diffeomorphism ¯ τ = ( π ( z ) , ¯ φ ( z )) such that satisfy the localtriviality property. U = f ( V ) is an open neighbour of p on M , since f is a diffeomorphism,and it satisfies that ψ − ( U ) = π − ( V ) . Finally, if we define τ : ψ − ( U ) −→ U × O by τ ( z ) = ( ψ ( z ) , ¯ φ ( z )), U and τ satisfy the local triviality property on p . Remark 10
Note that there exist s-spaces that are not principal fiber bundles. For example,let λ = ( IR n × (IR n − { } ) , pr , GL ( n ) , R, { e i } ) over R n , where pr ( p, q ) = p , R a ( p, q ) =( p, q.a ) and e i ( p, q ) = ∂∂u i | p is the base of IR np induced by the canonical coordinate system of IR n . If we say that a s-space λ = ( N, ψ, O, R, { e i } ) over M is a principal fiber bundle, wewant to say that ( N, ψ, O, R ) is a principal fiber bundle over M .We denote by S z = { a ∈ O : z.a = z } the stabilizer’s group of the action R at z . It iswell know that, if for a point z ∈ N the orbit z.O is locally closed (i.e. if w ∈ z.O , thereexist an open neighbour V of w on N , such that V ∩ z.O is a closed set of V ), then z.O isa submanifold of N and f z ([ a ]) = z.a is a diffeomorphism between O/S z and z.O , see [3]. Proposition 11
Let λ = ( N, ψ, O, R, { e i } ) be a s-space over M , theni) There exists s ∈ IN such that dim S z = s for all z ∈ N .ii) dim N = dim M + dim O − s . roof. Let z ∈ N and ψ ( z ) = p . That dim N = dim ker ψ ∗ z + dim M and dim ker ψ ∗ z =dim ψ − ( p ) follow from the fact that ψ is a submersion. Note that z.O = ψ − ( p ), since O acts transitively on the fibers. As ψ − ( p ) is locally closed, we have dim O/S z = dim ψ − ( p ).Therefore, dim N = dim M + dim O − dim S z for all z , so dim S z ≡ s is constant, whichcompletes the proof.Given a s-space λ over M , it will be very important to know the tensors on M thatsatisfy that λ T is a constant matrix. It is clear that not for every matrix A ∈ IR n × n thereexists a tensor T on M such that λ T = A . From proposition 5, we know that a necessaryand sufficient condition for this happens is that L ( a ) t .A.L ( a ) = A for all a ∈ O . In thatcase, we said that λ admits matrix representations of type A . In the last part of the Sectionwe show some conditions that a s-space has to satisfies to admits matrix representation ofcertain class of diagonal matrix. For ν = 0 , , · · · , n −
1, we denote by I ν the followingmatrix of IR n × n I ν = − ν − n − ν if ν ≥ I = Id n × n With O ν we denote the orthonormal group of index ν . If ν = 0 then O = O ( n ). Proposition 12
Let λ = ( N, ψ, O, R, { e i } ) be a s-space over M with base change morphism L . If ≤ ν ≤ n − , the following conditions are equivalent:i) Img ( L ) ⊆ O ν .ii) λ admits matrix representations of type I ν .iii) There is a semi-Riemannian metric on M of signature ν such that { e ( z ) , . . . , e n ( z ) } is an orthonormal base of M ψ ( z ) for all z ∈ N .iv) There exists a tensor T on M that satisfies λ T ( z ) = I ν for all z ∈ ψ − ( p ) and for a p ∈ M .Proof. i ) = ⇒ ii ) Consider the constant map F ≡ I ν . Since F satisfies the invariance prop-erty, it follows from the Proposition 5 the existence of a tensor that satisfies λ T = I ν . ii ) = ⇒ iii ) If λ T = I ν , then T is a semi-Riemannian metric of index ν and T ( ψ ( z ))( e i ( z ) , e j ( z )) =[ I ν ] ij . iii ) = ⇒ iv ) is immediately. iv ) = ⇒ i ) Let a ∈ O and z such that ψ ( z ) = p , then I ν = I ν ( z .a ) = L ( a ) t .I ν .L ( a ) for all a ∈ O . 6he next Proposition is a consequence of the fact that O ( m ) ∩ O ν = { D ∈ O ( m ) : D = (cid:18) A B (cid:19) con A ∈ O ( ν ) y B ∈ O ( m − ν ) } . Proposition 13
Let λ = ( N, ψ, O, R, { e i } ) be a s-space over M with base change morphism L and ≤ ν ≤ n − . λ admits matrix representation of type I and I ν if and only if thereexist differentiable functions L : O −→ O ( ν ) and L : O −→ O ( n − ν ) such that L ( a ) = (cid:18) L ( a ) 00 L ( a ) (cid:19) Proposition 14
Let λ = ( N, ψ, O, R, { e i } ) be a s-space over M with O connected. λ admits matrix representations of type I ν for all ≤ ν ≤ n − if and only if λ admits matrixrepresentation of type A , for all constant matrix A ∈ IR n × n .Proof. If λ admits matrix representations of type I , I , . . . , I ν , from the proposition abovewe have that L ( a ) = ± ν ± l ( a ) with l ( a ) ∈ O ( n − ν ). Since L is differentiableand L ( ab ) = L ( a ) .L ( b ), we see that L ( a ) = (cid:18) Id ν × ν f ( a ) (cid:19) . If ν = n , then L ≡ I n × n andthe proposition follows. Definition 15
Let λ = ( N, ψ, O, R, { e i } ) and λ ′ = ( N ′ , ψ ′ , O ′ , R ′ , { e ′ i } ) be s-spaces over M .We call a pair ( f, τ ) a morphism of s-spaces between λ and λ ′ ifa) f : N −→ N ′ be differentiable.b) τ : O −→ O ′ is a morphism of Lie groups.c) ψ ′ ◦ f = ψ .d) f ( z.a ) = f ( z ) .τ ( a ) for all z ∈ N and a ∈ O . Note that if λ and λ ′ are principal fiber bundles, ( f, τ ) is a principal bundle morphismbetween them. Example 16
Let λ = ( N, ψ, O, R, { e i } ) be a s-space over M and LM the s-space in-duced by the frame bundle of M . Consider the pair (Γ , L ) : λ −→ LM , where Γ( z ) =( ψ ( z ) , e ( z ) , . . . , e n ( z )) and L is the base change morphism of λ , then (Γ , L ) is a morphismof s-spaces. emark 17 Let λ and λ ′ be s-spaces over M and ( f, τ ) : λ −→ λ ′ be a morphism betweenthem. If λ ′ is a principal fiber bundle and τ is injective, then λ is a principal fiber bundle. Remark 18
It is easy too see that if τ is surjective then f is also surjective. If O ′ actswithout fixed point, then we have that τ is surjective if and only if f is surjective; theinjectivity of τ implies that of f ; and if τ is bijective then so is f . If O and O ′ act withoutfixed point, then f is injective if and only if τ is it. Let ( f, τ ) : λ −→ λ ′ be a morphism of s-spaces. As ψ ′ ( f ( z )) = ψ ( z ) we have that { e ′ i ( f ( z )) } and { e i ( z ) } are bases of M ψ ( z ) . Therefore, there exists C ( z ) ∈ GL ( n ) thatsatisfies { e ′ i ( f ( z )) } = { e i ( z ) } .C ( z ). We called to the function C : N −→ GL ( n ) the linkingmap of ( f, τ ). For example the linking map of the morphism given in Example 16 is C ( z ) = Id n × n . Let λ be a s-space over M with base change morphism L and a ∈ O .Consider ( f, τ ) : λ −→ λ defined by f ( z ) = R a and τ ( b ) = Ad ( a − )( b ), then C ( z ) = L ( a ).The linking map of a morphism ( f, τ ) satisfies that C ( z.a ) = ( L ( a )) − .C ( z ) .L ′ ( τ ( a )),where L and L ′ are the base change morphism of λ and λ ′ respectively, and the relationshipbetween two linking maps is given by C ( g,γ ) ( z ) = C ( f,τ ) ( z ) .L ′ ( a ( z )), where a : N −→ O is adifferentiable function.Let λ = ( N, ψ, O, R, { e i } ) be a s-space over M and consider F : N −→ IR n × n . We saythat F comes from a tensor if there exists a tensor T on M such that λ T = F . In this case,we say that F is the matrix representation (or the induced matrix function by) of T withrespect to λ . Proposition 19
Let λ = ( N, ψ, O, R, { e i } ) and λ ′ = ( N ′ , ψ ′ , O ′ , R ′ , { e ′ i } ) are s-spaces over M with base change morphism L and L ′ respectively, and let ( f, τ ) : λ −→ λ ′ be a morphism.If λ ′ T is the matrix representation of T with respect to λ ′ , then λ ′ T ◦ f comes from a tensorif and only if ( L ( a )) t . ( λ ′ T ◦ f )( z ) .L ( a ) = ( L ′ ( τ ( a ))) t . ( λ ′ T ◦ f )( z ) .L ′ ( τ ( a )) for all z ∈ N and a ∈ O .Proof. If λ ′ T ◦ f comes from a tensor, then it satisfies ( λ ′ T ◦ f )( z.a ) = ( L ( a )) t . λ ′ ( T ◦ f )( z ) .L ( a ). So by definition, we have that λ ′ T ( f ( z.a )) = L ′ ( τ ( a ))) t . λ ′ T ( f ( z )) .L ′ ( τ ( a )).The other implication follows by a verification of the invariance property. Remark 20
Let T be a tensor on M . From the above Proposition it follows that until the k th iteration of T by ( f, τ ) comes from a tensor on M if and only if L t . ( C t ) j . λ T.C j .L =( L ′ ◦ τ ) t . ( C t ) j . λ T.C j . ( L ′ ◦ τ ) for all ≤ j ≤ k . Corollary 21
The following sentences are equivalent: ) For all tensor T on M , λ ′ ( T ◦ f ) comes from a tensor on M .ii) L ′ ◦ τ = ± L . Proposition 22
Let ( f, τ ) : λ −→ λ ′ be a morphism of s-spaces and let T be a tensor on M then ( λ ′ T ◦ f )( z ) = ( C ( z )) t . λ T ( z ) .C ( z ) where C is the linking map of ( f, τ ) .Proof. [( λ ′ T ◦ f )( z )] ij = T ( ψ ′ (( f ( z ))))( e ′ i ( f ( z )) , e ′ j ( f ( z ))) == T ( ψ ( z ))( m X r =1 ( C ( z )) ri e r ( z ) , m X s =1 ( C ( z )) sj e s ( z )) = m X r,s =1 ( C ( z )) ri [ λ T ( z )] rs . ( C ( z )) sj Definition 23
Let ( f, τ ) : λ −→ λ ′ be a morphism of s-spaces and T be a tensor on M .We say that T is invariant by ( f, τ ) if λ ′ T ◦ f = λ T . Let us denote with I ( f,τ ) the subspaceof χ ( M ) given by the invariant tensors of ( f, τ ) . For example, let λ be a s-space over M , if ( f, τ ) : λ −→ LM is the morphism given in theExample 16, then I ( f,τ ) = χ ( M ). Given a s-space λ = ( N, ψ, O, R, { e i } ) and T = 0, thenthere exists a ∈ GL ( n ) and z ∈ N such that a t .T ( z ) .a = T ( z ). Therefore, if we considerthe s-space λ ′ = ( N, ψ, O, R, { e ′ i } ), where { e ′ i } = { e i } .a , we have that T is not an invarianttensor by the morphism ( Id N , Id O ). Proposition 24
Let ( f, τ ) : λ −→ λ ′ be a morphism and T be a tensor on M . If thereexists k ∈ IN such that the k th iteration by ( f, τ ) of T is an invariant tensor, then T is aninvariant tensor.Proof. Let us denoted by λ T j and λ ′ T j the matrix representation of the j th iteration of T with respect to λ and λ ′ respectively. λ T k = λ ′ T k ◦ f = C t . λ T k .C , since the k th iteration is an invariant tensor. On the other hand, λ T k = ( λ ′ T k − ◦ f ) = C t λ T k − C = C t . ( λ ′ T k − ◦ f ) .C = ( C t ) . λ T k − .C = ( C t ) k − . λ T.C k − , hence λ T = C t . λ T.C .
Let T be a tensor on M and λ = ( N, ψ, O, R, { e i } ) be a s-space over M . For each z ∈ N ,consider the lie subgroup of GL ( n ) defined by G T ( z ) = { D ∈ GL ( n ) : D t . λ T ( z ) .D = λ T ( z ) } .We call it the group of invariance of T at z . For simplicity of notation we write G T ( z )instead of G λT ( z ) which is more convenient. In these terms, a tensor T is invariant by ( f, τ )if and only if C ( z ) ∈ G T ( z ) for all z ∈ N .If ψ ( z ) = ψ ( z ′ ) we have G T ( z ) ≃ G T ( z ′ ), because ϕ a : G T ( z ′ ) −→ G T ( z ) defined by ϕ a ( D ) = L ( a ) .D.L ( a − ) = Ad ( L ( a ))( D ) for a ∈ O such that z ′ = z.a , is a homomorphismof Lie groups. We called the subset F T = { ( z, g ) : z ∈ N and g ∈ G T ( z ) } of N × GL ( n ) the nvariance set of T . If there is a tensor T on M that admits a matrix representation ofthe form λ T = α.Id n × n , with α = 0, then F T = N × O ( n ). Let λ be the s-space of Example4. If T is the tensor on LM that satisfies λ T = (cid:18) Id m × m − Id m × m (cid:19) here m = n + n , then F T = LM × GL ( n ) × S m where S m denotes the symplectic group of IR m × m . In general F T does not has a manifold structure. The invariant tensor by a morphism ( f, τ ) : λ −→ λ ′ they are those that satisfy that ( z, C ( z )) ∈ F T for all z ∈ N . Remark 25
Let ( f, τ ) : λ −→ λ ′ be a morphism with linking map C . If T ∈ I ( f,τ ) and T is non degenerated, then det( C ( z )) = ± for all z ∈ N . Given λ = ( N, O, ψ, IR , { e i } ) a s-space over M , for z ∈ N let us denote by V z the verticalsubspace at z induced by the projection ψ (i.e. V z = ker ψ ∗ z ). Note that dim V z = k − s where s is the dimension of the stabilizer S z and k = dim O . As when we deal with fibrations(see [13]), we have a notion of connections for s-spaces. Definition 26
A connection on a s-space λ over M is (1 , tensor φ on N that satisfies:1) φ z : N z −→ V z is a linear map.2) φ = φ , φ is a projection to the vertical subspace.3) φ z.a (( R a ) ∗ z ( b )) = ( R a ) ∗ z ( φ ( b )) . Note that 3) has sense because ( R a ) ∗ z ( V z ) = V z.a .We called to H z = ker φ z the horizontal subspace at z . It is clear that N z = H z ⊕ V z .Since φ za (( R a ) ∗ z ( φ ( z )( b ))) = ( R a ) ∗ z ( φ ( z )( b )) = ( R a ) ∗ z (0) = 0, ( R a ) ∗ z ( H z ) = H z.a . As inthe case of connections in principal fiber bundles we have that: There is a connection φ on λ if and only if there exists a differentiable distribution on N ( z −→ H z ) such that N z = H z ⊕ V z and H z.a = ( R a ) ∗ z ( H z ). If we have a distribution with these properties, wedefine φ ( z )( b ) = b v where b = b h + b v .Let λ = ( N, ψ, O, R, { e i } ) be a s-space over M endowed with a connection φ , then wehave the concept of horizontal lift . Definition 27
Let v ∈ M p and z ∈ ψ − ( p ) . We called horizontal lift of v at z to the uniquevector v hz ∈ N z such that ψ ∗ z ( v hz ) = v and v hz ∈ H z . Given a vector field X on N , let H ( X ) a V ( X ) the vector fields that satisfy that H ( X )( z ) ∈ H z , V ( X )( z ) ∈ V z and X ( z ) = H ( X )( z ) + V ( X )( z ) for all z ∈ N . We called H ( X ) and V ( X ) the horizontal and the vertical projections of X . Is easy to see that H ( X )and V ( X ) are smooth vector fields if X is a smooth vector field.10 roposition 28 Let X be a vector field on M . Then there exists a unique vector field X h on N such that X h ( z ) ∈ H z and ψ ∗ z ( X h ( z )) = X ( ψ ( z )) for all z ∈ N .Proof. Let p ∈ M and z ∈ N such that ψ ( z ) = p . As ψ is a submersion, thereexist ( U, x ) and (
V, y ) centered at p and z respectively that satisfy ψ ( U ) ⊆ V and y ◦ ψ ◦ x − ( a , . . . , a n , a n +1 , . . . , a m ) = ( a , . . . , a n ). If X ( p ) = P ni =1 ρ i ( p ) ∂∂y i | p for p ∈ U ,let the vector field on V defined by ˜ X U ( z ) = P ni =1 ( ρ i ◦ ψ )( z ) ∂∂x i | z , then we have that ψ ∗ ( ˜ X ) = X ◦ ψ . For this reason, we can take an open covering { U i } i ∈ I of N such thatfor each U i we have a field ˜ X i ∈ χ ( U i ) that satisfies the previous property. Let { ζ i } i ∈ I bea unit partition subordinate to the covering { U i } i ∈ I . Consider the vector field ˜ X ∈ χ ( N )given for ˜ X = P i ∈ I ζ i . ˜ X i . ˜ X satisfies that ψ ∗ z ( ˜ X ( z )) = X ( ψ ( z )) for all z ∈ N . Finally, H ( ˜ X ) is the vector fields that we looked for. The uniqueness follows from the fact that ψ ∗ z | H z : H z −→ M ψ ( z ) is an isomorphism. Remark 29
The horizontal distribution z −→ H z is trivial since { e hi ( z ) = ( e i ( z )) hz } ni =1 isa base of H z for all z ∈ N and { e hi } ni =1 are smooth vector fields. For all z ∈ N we have defined the function σ z : O −→ N given by σ z ( a ) = z.a .If X ∈ o , let V ( X )( z ) = ( σ z ) ∗ e ( X ) ∈ V z , where e is the unit element of O . If thegroup O acts effectively and X = 0 is easy to see that V is not the null vector field.If O acts without fixed point, then V ( X )( z ) = 0 for all z ∈ N and X = 0. Any-way if { X , · · · X k } is a base of o , then { V ( X )( z ) , · · · , V ( X k )( z ) } spanned V z . It isnot difficult to see that ker( σ z ) ∗ e = T e S z . Consider the 1-forms θ i on N defined by ψ ∗ z ( b ) = P ni =1 θ i ( z )( b ) e i ( z ). { θ ( z ) , · · · , θ n ( z ) } are lineally independent and they are abase of the null space of the vertical subspace. Straightforward calculations show that the1-forms θ i satisfy that L ( a ) . θ ( z.a )(( R a ) ∗ z ( b ))... θ n ( z.a )(( R a ) ∗ z ( b )) = θ ( z )( b )... θ n ( z )( b ) for all z ∈ N and a ∈ O . Proposition 30
Let λ be a s-space over M such that exists a subspace ˜ V of o that satisfies dim ˜ V = k − s ( s = dim S z ) and ˜ V ∩ T e S z = { } for all z ∈ N . If λ admits a connection,then the tangent bundle of N is trivial.Proof. Let { X , . . . , X k − s } be a base of ˜ V , then the vertical vector fields V i ( z ) = ( σ z ) ∗ e ( X i )with i = 1 , . . . , k − s are a base of V z for all z ∈ N . We have that { e h , . . . , e hn , V , . . . , V k − s } trivialized the tangent bundle of N . Remark 31
With the same hypothesis of the Proposition, we a natural dual frame of N .For i = 1 , . . . , k − s , let the 1-forms W i on N defined by φ z ( b ) = P k − si =1 W i ( z )( b ) V i ( z ) . Thenis easy to see that { θ ( z ) , · · · , θ n ( z ) , W ( z ) , · · · , W k − s ( z ) } is a base of N ∗ z for all z ∈ N andit is the dual base of { e h ( z ) , · · · , e hn ( z ) , V ( z ) , · · · , V k − s ( z ) } . emark 32 Let λ = ( N, ψ, O, R, { e i } ) be a s-space over M that is also a principal fiberbundle. Is well know that every principal fiber bundle admits a smooth distribution that istransversal to the vertical distribution and is invariant by the action of the group O , see [5],so there exists a connection on λ . On the other hand, the group O acts on N without fixedpoint and the hypothesis of the Proposition 30 are satisfied. Therefore, the tangent bundleof N is trivial. Remark 33
Let G be a metric on N such that the maps R a are isometries for all a ∈ O .If O is compact and N is a closed manifold, then N admits a metric with this property (see[5]). Let H z be the subspace of N z orthogonal to V z . Is easy to see that z → H z induces aconnection on λ . Remark 34
In the situation of Proposition 30, we can lift a metric G on M to a metric ˜ G on N in a very natural way. Given G a Riemannian metric on M let ˜ G = ψ ∗ ( G ) + k − s X i =1 W i ⊗ W i . ˜ G is a metric on N and ψ : ( N, ˜ G ) −→ ( M, G ) is a Riemannian submersion. To keep inmind the metric ˜ G can be very useful. For example, using the fundamental equations of aRiemannian submersion [16] we can relate the curvature tensors of both metrics. Sometimesif we chose appropriately the s-space over M , we can simplify considerably the calculationof the curvature tensor of ( M, G ) . This is the case when the base manifolds is the tangentbundle of a Riemnannian manifold. In [6], we use a s-space λ and the metric ˜ G to computethe curvature tensor of the tangent bundle endowed with certain class of λ natural metricswith respect to the bundle. Remark 35
Let λ be a s-space over M and let ∇ be a linear connection on M with con-nection function K . Consider K i : T N −→ T M defined by K iz ( b ) = K (cid:16) ( e i ) ∗ z ( b ) (cid:17) and let H z = { b ∈ N z : K iz ( b ) = 0 for i = 1 , . . . , n } . This smooth distribution is invariantby the group action but it is not necessary complementary to V z . If F z : N z −→ M ψ ( z ) × n times z }| { M ψ ( z ) × . . . × M ψ ( z ) is given by F z ( b ) = ( ψ ∗ z ( b ) , K z ( b ) , . . . , K nz ( b )) it is not difficult to seethat there are equivalent:i) F z is injective and ( M ψ ( z ) × × . . . × ∈ Img F z .ii) N z = H z ⊕ V z .So if λ satisfies i ) − ii ) we have that z → H z induces a connection on λ . If G is a metricon M let the (0,2) symmetric tensor on N given by ˜ G ( A, B ) = c ( z ) G ( ψ ∗ z ( A ) , ψ ∗ z ( B )) + n X i =1 l i ( z ) G ( K i ( A ) , K i ( B ))12 here c, l i are positive differentiable functions. If F is injective, the ˜ G is a Riemmannianmetric. If λ is the s-space LM and c = 1 and l i = 1 for i = 1 , . . . n , then ˜ G is the well knowSasaki-Mok metric (see [15] and [4]). In this section we will study certain class of tensors on a manifolds and fibrations. With atensor T on a fibration we want to mean that T is a tensor on the space manifold of thefibration. If α = ( P, π,
IF) is a fibration we will consider a particular class of s-spaces over P in order to take into account the structure of the fibration for the study of the tensorson it. Definition 36
Let α = ( P, π,
IF) be a fibration on M and λ = ( N, ψ, O, R, { e i } ) be a s-spaceover P . We say that λ is a trivial s-space over α if N = N ′ × IF . Example 37
The s-space λ = ( LM × GL ( n ) , ψ, GL ( n ) , R, { H i , V ij } ) given in the example4 is a trivial s-space over the frame bundle of M . Definition 38
Let α = ( P, π,
IF) be a fibration and λ = ( N × IF , ψ, O, R, { e i } ) be a trivial s-space over α . We say that a tensor T on P is λ -natural with respect to α if λ T ( z, w ) = λ T ( w ) (i.e. its matrix representation depends only of the parameter w of the fiber IF ). Remark 39
Let M be a manifold endowed with a linear connection ∇ and a Riemannianmetric g . If we consider the s-spaces λ = ( LM × GL ( n ) , ψ, GL ( n ) , R, { H i , V ij } ) (Example 4)and λ ′ = ( O ( M ) × GL ( n ) , ψ, O ( n ) , R, { H i , V ij } ) , where O ( M ) is the manifold of orthonormalbases of ( M, g ) and the action of the orthonormal group and the projection are similar to thatones of λ , then the concept of λ − natural and λ ′ − natural with respect to ( LM, π, GL ( n )) agree with that ones of natural tensor with respect to the connection ∇ and with respect tothe metric g given in [7]. Remark 40
There exist s-spaces such that the concept of λ − natural with respect to thefibration agree with the known cases of naturality. So, our definition also generalizes thenotion of natural tensor on the tangent and the cotangent bundle of a Riemannian (see [2]and Example 53) and semi-Riemannian manifold (see [1]). In view of the definition of λ − natural with respect to a fibration, it seems interesting toask what it means to be λ − natural with respect to a manifold. A manifold M can be viewas a trivial fibration α M = ( M × { a } , pr , { a } ) and there is a one to one correspondence13etween the s-spaces over λ and the trivial s-spaces over α . A s-space λ = ( N, ψ, O, R, { e i } )over M induced the λ ′ = ( N × { a } , ψ, O, R, { e i } ) over α . A tensor T on M induce a tensor T ′ on M ×{ a } . Then T ′ is λ ′ − natural with respect to a α if and only if λ ′ T ′ ( z, a ) = λ ′ T ′ ( a ),therefore T ′ is λ ′ − natural with respect to a α if and only if λ T is a constant map. Thissuggests the following definition: Definition 41
Let λ be a s-space over M and T a tensor on M . We say that T is λ − natural if λ T is a constant map. Example 42
Let ( M, g ) be a Riemannian manifold and let λ = ( O ( M ) , π, O ( n ) , · , { π i } ) the s-space over M induced by the orthonormal frame bundles of M . Since L ( a ) = a for all a ∈ O ( n ) , T is λ − natural if and only if λ T = k.Id n × n , that is T is an scalar multiple ofthe metric g . Example 43
Suppose that the map F of the Remark 35 is bijective. Let β = ( N, id N , { } ,, ( · ) , { ( e i ( z )) h , ( e j ( z )) v ( i ) z } ) be the s-space over the space manifold of λ , where { } is thetrivial group and ( · ) is the trivial action, ( e i ( z )) h is the horizontal lift of e i ( z ) at z and ( e j ( z )) v ( i ) z satisfies that K i (( e j ( z )) v ( i ) z ) = e j ( z ) . If G is a metric on M and ˜ G is the gener-alizes Sasaki-Mok metric on N then β ˜ G ( z ) = [ λ G ] 0 · · ·
00 [ λ G ] 0 00 0 . . . · · · · · · [ λ G ] so ˜ G is β natural if and only if G is λ -natural. Remark 44
Let α = ( P, π,
IF) be a fibration on M and λ a trivial s-space over α . λ is also as-space over P . If a tensor T on P is λ − natural then T is λ − natural with respect to α . Theconverse implication not necessarily holds. Let λ = ( O ( M ) × GL ( n ) , ψ, O ( n ) , R, { H i , V ij } ) over LM , there are more λ − natural tensors with respect to LM than constant maps, see[7]. Remark 45
Consider the s-space LM and let T be a LM − natural tensor on M . Let A ∈ R n × n such that LM T ≡ A . Since the base change morphism of LM is the identity of GL ( n ) , A = a t .A.a for all a ∈ GL ( n ) , hence T must be the null tensor. Therefore, for amanifold M the null tensor is the only one that is λ − natural for all the s-spaces over M . Remark 46 If T is λ − natural , we have that N × Im ( L ) ⊆ F T where F T = N × G with G a subgroup of GL ( n ) . Let λ = ( N, O, ψ, IR , { e i } ) be a s-space over M . Note that if T is λ − natural and( f, τ ) : λ −→ λ is a morphism of s-spaces then T ∈ I ( f,τ ) . In the other hand, if T ∈ I ( f,τ ) f, τ ) automorphism of λ , then λ T is constant in each fiber of N . A necessary andsufficient condition for a tensor T to have a constant matrix representation in each fiber isthat T ∈ I ( f a ,τ a ) for all a ∈ O , where ( f a , τ a ) is the morphism defined by f a ( z ) = R a ( z ) and τ a ( b ) = a − b.a . Let us see some facts about the relationship between the natural tensorsand the morphisms of s-spaces. The next two proposition follow from Proposition 22. Proposition 47
Let λ and λ ′ be two s-spaces over M and ( f, τ ) : λ −→ λ ′ be a morphismwith linking map C . If T is a λ ′ − natural tensor with λ ′ T = A ∈ IR n × n , then T is λ − natural if and only if ( C ( z ) − ) t .A.C ( z ) − is a constant map. Proposition 48
Let ( f, τ ) : λ −→ λ ′ be a morphism of s-spaces with linking map C and T a tensor on M that is λ and λ ′ − natural . Let A and B ∈ R n × n such that λ T = A and λ ′ T = B , then C ( z ) t .A.C ( z ) = B for all z ∈ N . In particular, if λ = λ ′ the image of the linking map of any automorphism have to beincluded in the group of invariance of all the λ − natural tensors. For example, if λ =( LM × GL ( n ) , ψ, GL ( n ) , R, { H i , V ij } ) and ( f, τ ) is an automorphism of λ with linking map C , then C ( z ) = Id ( n + n ) × ( n + n ) for all z ∈ LM × GL ( n ). Proposition 49
Let λ = ( N, ψ, O, R, { e i } ) and λ ′ = ( N ′ , ψ ′ , O ′ , R ′ , { e ′ i } ) be two s-spacesover M , ( f, τ ) : λ −→ λ ′ be a morphism of s-space, T a λ ′ − natural tensor and let A ∈ R n × n such that λ ′ T = A . Then λ ′ T ◦ f comes from a tensor on M if and only if ( L ( a )) t .A.L ( a ) = A for all a ∈ O .Proof. Since T is λ ′ − natural , ( L ′ ( a ′ )) t .A.L ′ ( a ′ ) = A for all a ′ ∈ O ′ , then the Propositionfollows from Proposition 19. Remark 50
There are tensors on M that are not λ − natural for any λ s-space over M .Let T be a not null tensor on M , then there exists p ∈ M such that T ( p ) : M p × M p −→ IR isnot the null bilinear form. Let f be a differentiable function on M that satisfies f ( p ) = 1 and f ( q ) = 0 for a point q different of p and consider the tensor e T defined by e T ( ξ ) = f ( ξ ) .T ( ξ ) .If ˜ T is λ − natural , then λ ˜ T ≡ A and since e T ( q ) = 0 , A must be the zero matrix. Butfor z ′ ∈ ψ − ( p ) , we have that λ e T ( z ′ ) = [ e T ( q )( e i ( z ′ ) , e j ( z ′ ))] = f ( p )[ T ( p )( e i ( z ′ ) .e j ( z ′ ))] = 0 ,hence T is not λ − natural . Proposition 51
Let T be a symmetric tensor on M with index and rank constant, thenthere is a s-space λ over M such that T is λ − natural .Proof. If rank ( T ) = 0 then T is the null tensor and T is λ − natural for all λ . Sup-pose that rank ( T ) = r ≥ index ( T ) = r − s . For every p ∈ M there is a base { v , . . . , v s , v s +1 , . . . , v r , v r +1 , . . . , v n } of M p that diagonalizes the matrix of T ( p ), that is15 T ( p )( v i , v j )] = Id s × s − Id ( r − s ) × ( r − s )
00 0 0 = I sr . Let λ = ( N, π, O, · , { π i } ) where N = { ( q, v ) ∈ LM : [ T ( q )( v i , v j )] = I sr } , O = O ( s ) 0 00 O ( r − s ) 00 0 GL ( n − r ) and theaction, the projection and the map { π i } are similar to those of LM . Then λ T = I sr Let λ = ( N, ψ, O, R, { e i } ) and λ ′ = ( N ′ , ψ ′ , O ′ , R ′ , { e ′ i } ) be s-spaces over M and N respec-tively and h : M −→ M ′ be a differentiable function. Let f : N −→ N ′ be a differentiablefunction and τ : O −→ O ′ a group morphism. Definition 52
We said that ( f, τ ) is a morphism of s-spaces over h if f ( z.a ) = f ( z ) .τ ( a ) for all z ∈ N and a ∈ O and ψ ′ ◦ f = h ◦ ψ . This definition generalize the concept of morphism of s-spaces. If λ and λ ′ are s-spacesover M , and ( f, τ ) : λ −→ λ ′ is a morphism of s-spaces then ( f, τ ) is a morphism over Id M . Example 53
Let ( M, g ) be a Riemannian manifold and let λ = ( O ( M ) × R n , ψ, O ( n ) , R, { e i } ) the s-space over T M where the projection is defined by ψ ( p, u, ξ ) = ( p, P ni =1 u i ξ i ) and theaction of the orthonormal group on O ( M ) × IR n is given by R a ( p, u ) = ( p, u.a, ξ.a ) . For ≤ i ≤ n , let e i ( p, u, ξ ) = ( π ∗ ψ ( p,u,ξ ) × K ψ ( p,u,ξ ) ) − ( u i , and e n + i ( p, u, ξ ) = ( π ∗ ψ ( p,u,ξ ) × K ψ ( p,u,ξ ) ) − (0 , u i ) , where K is the connection map induced by the Levi-Civita connectionof g . Before we see an example of subs-space let us make a brief comment. The ten-sor on T M that are λ natural with respect to T M agree with the ones of Calvo-Keilhauer[2]. The Sasaki G S and the Cheeger-Gomoll G cg metric are λ − natural with respect to T M . The matrix representation of the Sasaki metric and the Chegeer-Gromoll metric are λ G S ( p, u, ξ ) = (cid:18) Id n × n Id n × n (cid:19) and λ G cg ( p, u, ξ ) = (cid:18) Id n × n | ξ | ( Id n × n + ( ξ ) t .ξ ) (cid:19) re-spectively .Consider the s-space λ ′ = ( O ( M ) , ψ ′ , O ( n − , R ′ , { e ′ i } ) over the unitary tangent T M bundle of M , where ψ ′ ( p, u ) = ( p, u n ) , The action of O ( n − on O ( M ) is given by R ′ a ( p, u ) = ( p, P n − i =1 u i a i , . . . , P n − i =1 u i a in − , u n ) . The maps { e ′ i } are defined by e ′ i ( p, u ) =( π ∗ ψ ( p,u ) × K ψ ( p,u ) ) − ( u i , if ≤ i ≤ n and by e ′ n + i ( p, u ) = ( π ∗ ψ ( p,u ) × K ψ ( p,u ) ) − (0 , u i ) if ≤ i ≤ n − . Let f : O ( M ) −→ O ( M ) × IR n and τ : O ( n − −→ O ( n ) defined by f ( p, u ) = ( p, u, v ) where v is the n th vector of the canonic base of R n , and τ ( a ) = (cid:18) a
00 1 (cid:19) . ( f, τ ) : λ −→ λ ′ is a morphism of s-spaces over the inclusion map of T M in T M . Let M and M ′ be manifolds of dimension n and n ′ respectively. Let λ = ( N, ψ, O, R, { e i } )and λ ′ = ( N ′ , ψ ′ , O ′ , R ′ , { e ′ i } ) be s-spaces over M and M ′ and ( f, τ ) : λ −→ λ ′ a morphism16f s-space over an inmersion h : M −→ M ′ . For every z ∈ N , h ∗ ψ ( z ) ( M ψ ( z ) ) is a sub-space of dimension n of M ′ ψ ( f ( z )) and it is generated by { h ∗ ψ ( z ) ( e ( z )) , . . . , h ∗ ψ ( z ) ( e n ( z )) } .As { e ′ i ( f ( z )) } is a base of M ′ ψ ′ ( f ( z )) , for every z ∈ N there exists a matrix A ( z ) ∈ R n ′ × n ′ with rank ( A ( z )) = n that satisfies { h ∗ ψ ( z ) ( e ( z )) , . . . , h ∗ ψ ( z ) ( e n ( z )) , n ′ − n z }| { , . . . , } = { e ′ ( f ( z )) , . . . , e ′ n ′ ( f ( z )) } .A ( z )In the previous example, A ( p, u ) = (cid:18) Id (2 n − × (2 n −
00 0 (cid:19) . If M = M ′ and h is the identitymap then ( f, τ ) is a morphism of s-spaces and A ( z ) = C − ( z ) is C is the linking map of( f, τ ).In this situation, we have the following definition: Definition 54 λ is a subs-space of λ ′ if there exists a morphism of s-spaces ( f, τ ) over aninjective inmersion h : M −→ M ′ such that f is an inmersion and the map A induced by ( f, τ ) is constant. In this case, we said that λ is a subs-space of λ ′ with morphism ( f, τ ) over h . A s-space λ = ( N, ψ, O, R, { e i } ) is included in λ ′ = ( N ′ , ψ ′ , O ′ , R ′ , { e ′ i } ) if N ⊆ N ′ . Example 55
Let M be a parallelizable manifold , V a vectorial space and V ′ a subspaceof V . Let GL ( V ) the group of linear isomorphisms of V and GL ( V, V ′ ) the subgroup oflinear isomorphisms of V with the property that T ( V ′ ) = V ′ . Consider the s-space λ =( M × V, pr , GL ( V ) , R f , { e i } ) over M , where the action is defined by R f ( p, z ) = ( p, f ( z )) for ( p, z ) ∈ M × V and f ∈ GL ( V ) , and e i = ¯ e i ◦ pr where { ¯ e , · · · , ¯ e n } are the vector fieldsthat trivialized the tangent bundle of M . If λ ′ = ( M × V ′ , pr , GL ( V, V ′ ) , R f , { e i } ) , then λ ′ is a subs-space of λ . Proposition 56
Let λ = ( N, ψ, O, R, { e i } ) and λ ′ = ( N ′ , ψ ′ , O ′ , R ′ , { e ′ i } ) be s-spaces over M such that λ is a subs-space of λ ′ with morphism ( f, τ ) over the identity map of M . If atensor T on M is λ ′ − natural then T is λ − natural .Proof. [ λ T ( z )] ij = T ( ψ ( z ))( e i ( z ) , e j ( z )) = T ( ψ ′ ( f ( z )))( P nl =1 e ′ l ( z ) A li , P ns =1 e ′ s ( z ) A sj ) == P nls A li .A sj [ λ ′ T ] ij , then λ T is a constant map. Remark 57
The converse statement does not holds in general. Let ( M, g ) be a Riemannianmanifold and O ( M ) be the s-space induced by the principal bundle of orthonormal frames.If i O ( M ) : O ( M ) −→ LM and i O ( n ) : O ( n ) −→ GL ( n ) are the respective inclusion, then O ( M ) is a subs-space of LM with morphism ( i O ( M ) , i O ( n ) ) over the identity map of M . Weknown that there are O ( M ) − natural tensors that are not LM − natural . Let T be a tensor on M and let LM T : LM −→ R n × n the matrix map induced bythe s-space LM . Given a s-space λ = ( N, ψ, O, R, { e i } ) over M we have a morphism17Γ , L ) : λ −→ LM , see Example 16. It is clear that λ T = LM T ◦ Γ, thus if T is λ − natural then there exists a matrix A ∈ IR n × n such that Img Γ ⊆ ( LM T ) − ( A ). Proposition 58
Let T be a tensor on M . There exists λ a s-space over M such that T is λ − natural if and only if there exist a matrix A ∈ R n × n and a subs-space of LM includedin ( LM T ) − ( A ) .Proof. Suppose that T is λ − natural ( λ = ( N, ψ, O, R, { e i } )) and let A ∈ R n × n such that λ T = A . Let λ ′ = (Γ( N ) , π, L ( O ) , R ′ , { π i } ), where π , R ′ and { π i } are induced by LM . Themap π : Γ( N ) −→ M is a submersion. Since π (Γ( N )) = ψ ( N ) = M , π is surjective. Let p ∈ M and z ∈ ψ − ( p ), then π (Γ( z )) = p . We are going to see that π ∗ Γ( z ) : N Γ( z ) −→ M p issurjective. Given v ∈ M p there exists w ∈ N z such that ψ ∗ z ( w ) = v . Let α be a curve on N that satisfies α (0) = z and ˙ α (0) = w , then for β ( t ) = Γ( α ( t )) we have that β (0) = Γ( z )and π ∗ Γ( z ) ( ˙ β (0)) = D | ( π ( β ( t ))) = ψ ∗ z ( w ) = v . In the other hand, it is clear that L ( O )acts transitively on Γ( N ), so λ ′ is a s-space and it is a subs-space of LM with morphism( i Γ( N ) , i L ( O ) ) over the identity map of M .Conversely, suppose that there exist A ∈ R n × n and λ = ( N, ψ, O, R { e i } ) a s-space over M that is also a subs-space of LM with morphism ( f, τ ) over the identity map, and itholds that f ( N ) ⊆ ( LM T ) − ( A ). Since { e i ( z ) } = { π i ( f ( z )) } .B for B ∈ GL ( n ), [ λ T ( z )] =[ T ( ψ ( z ))( e i ( z ) , e j ( z ))] = B t . [ T ( ψ ( z ))( π i ( f ( z )) , π j ( f ( z )))] .B = B t .A.B Definition 59
Let M be a manifold and let A : { λ i = ( N i , ψ i , O i , R i , { e l } ) } i ∈ I be a col-lection of s-spaces over M . The collection A is called an Atlas of s-spaces if for each pair ( i, j ) ∈ I × I there is a morphism of s-spaces ( f ij , τ ij ) : λ i −→ λ j such that f ij : N i −→ N j is a diffeomorphism. We said that the s-spaces λ and β are compatible if there exists a morphism ( f λβ , τ λ,β ) : λ −→ β and ( f βλ , τ β,λ ) : β −→ λ such that f λβ and f βλ are diffeomorphisms. Hence, anatlas is a set of compatible s-spaces over M . If A satisfies that for an atlas B , A ⊆ B implies A = B , we called it a maximal atlas . In other words, if λ is a s-space compatible with thes-spaces of A then λ ∈ A . If λ is a s-space over M let us notate with A = < λ > themaximal atlas generated by λ . Let A be a maximal atlas, it follows from the definition that A = < λ > for every λ ∈ A . Note that there are different maximal atlases over a manifold.Consider a metric on M , then < LM > and < O ( M ) > are maximal s-spaces but they aredifferent because LM and O ( M ) are not compatible.Let λ be a s-space over M , then A = { λ } is an atlas. Therefore the concept of atlas isa generalization of the notion of s-space. 18 xample 60 Let λ = ( N, ψ, O, R, { e i } ) be a s-space over M and let A : N −→ GL ( n ) be adifferentiable function. Consider λ A = ( N, ψ, O, R, { e Al } ) where e Al ( z ) = P ni =1 e i ( z ) A il ( z ) .The collection A = { λ A } A ∈F ( M ) is an atlas of s-spaces. Example 61
Let M be a parallelizable manifold and { H i } ni =1 the vector fields that trivial-ized the tangent bundle of M . Let ( N, g ) be a Riemannian manifold such that its isometrygroup I ( N,g ) acts transitively on N . Let λ ( N,g ) = ( M × N, pr , I ( N,g ) , R f , { H i ◦ pr } ) where theaction of I ( N,g ) on M × N is given by R f ( z, p ) = ( z, f ( p )) . If ( N ′ , g ′ ) is isometric to ( N, g ) then λ ( N,g ) is compatible with λ ( N ′ ,g ′ ) . If N ′ is not diffeomorphic to N then < λ ( N,g ) > and < λ ( N ′ ,g ′ ) > are different maximal atlas of s-spaces over M . Definition 62
Let A and B be two atlases of s-spaces over M and F a collection of mor-phisms of s-spaces from a s-space of A to a s-space of B . F will be called a morphismbetween the atlas A and B if for every λ ∈ A and β ∈ B there exist ( f, τ ) ∈ F such that ( f, τ ) : λ −→ β . Remark 63
Let A and B be two atlas over M , λ ∈ A , β ∈ B and ( f , τ ) : λ −→ β .Consider F = { f β β ◦ f ◦ f λλ , τ β β ◦ τ ◦ τ λλ , } λ ∈A , β ∈B where ( f β β , τ β β ) : β −→ β and ( f λλ , τ λλ ) : λ −→ λ are the morphisms that show the compatibility between β and β andbetween λ and λ . F is morphism of atlases between A and B . Remark 64 If λ is a s-space over M we have a canonical morphism (Γ λ , L λ ) : λ −→ LM (see Example 16), hence for every s-space λ we have a morphism between the atlases < λ > and < LM > . It seems interesting to ask if this property characterized < LM > . In otherwords, if a s-space β satisfies that for every λ there exists a morphism ( f λ , τ λ ) : λ −→ β ithas to be necessarily compatible with LM ? λ (Γ λ ,L λ ) } } |||||||| ( f λ ,τ λ ) (cid:30) (cid:30) ======= LM ( f LM ,τ LM ) β (Γ β ,L β ) r r The answer is no. Consider a parallelizable Riemannian manifold ( M, g ) . Let { H i } ni =1 be orthonormal fields that trivialized the tangent bundle of M . If λ = ( N, ψ, O, R, { e i } ) is as-space over M let ( f λ , τ λ ) : λ −→ O ( M ) defined by f ( z ) = ( ψ ( z ) , H ( ψ ( z )) , . . . , H n ( ψ ( z ))) and τ ( a ) = Id n × n . Therefore, for every maximal atlas A there is a morphism between itand O ( M ) , and we just know that O ( M ) it is not compatible with LM . A F A ,LM z z vvvvvvvvvv F A ,O ( M ) % % JJJJJJJJJJ < LM > F LM,O ( M ) < O ( M ) > F O ( M ) ,LM q q ut there are more atlases with this property. If ( M, g ) is an oriented manifold, themaxi-mal atlas generated by the s-space induced by the principal fiber bundles of orthonormaloriented bases SL ( M ) have this property. The atlas < ( M, Id M , { } , R , { H i } ) > , where R is the trivial action, is another example. Definition 65
Let A be an atlas of s-spaces over M . A tensor T on M will be called A − natural if T is λ − natural for all λ ∈ A . Note that the concept of
A − naturality generalized the notion of λ − naturality . If weconsiderer the atlas A = { λ } then T is A − natural if and only if T is λ − natural . Example 66
Let λ be a s-space over M and consider the subatlas of the atlas given inthe Example 60 defined by A = { λ A } A ∈ GL ( n ) . Then T is A − natural if and only if T is λ − natural . Let T be a λ − natural tensor on M and A ′ = { λ A } A ∈F ( N,G T ) , then T is A ′ − natural and it has the same matrix representation in all the s-spaces of the atlas. Remark 67 If A is a maximal atlas then the unique A − natural tensor is the null tensor.Let λ = ( N, ψ, O, R, { e i } ) ∈ A and f : N −→ IR be a differentiable function such that f ( z ) = 0 for all z ∈ N and f is not constant. If λ ′ = ( N, ψ, O, R, { f.e i } ) , hence λ ′ ∈ A ,but the null tensor is the only λ − natural and λ ′ − natural at the same time, therefore T ≡ is the unique A − natural tensor.
Definition 68
Let A be an atlas of s-spaces over M and T a tensor on M . T is called A − weak natural if there exists λ ∈ A such that T is λ − natural . If A = { λ } or A is the atlas of Example 66, then the concept of A − natural and
A − weak natural coincide.For study the naturality of tensors on a fibration α = ( P, π,
IF) it will be useful considerthe atlases A such that all its s-spaces are trivial over α . An atlas with this property willbe called a trivial atlas over α . The following definition is a generalization of the conceptof naturality with respect to a fibration: Definition 69
Let A be a trivial atlas over a fibration α = ( P, π,
IF) and T a tensor on P ,then T is A − natural with respect to α if T is λ − natural with respect to α for all λ ∈ A . Example 70
Let α = ( P, π, G, · ) be a principal fiber bundle on ( M, g ) endowed witha connection ω . For every W = { W · · · , W k } base of g let λ W = ( N, ψ, O, R, { e Wi } ) where N = { ( p, u, b ) : p ∈ P, u is an orhonormal base of M π ( p ) , b ∈ G } , ψ ( q, u, b ) = q.b , O = O ( n ) × G and the action R is defined by R ( h,a ) ( q, u, b ) = ( qa, uh, a − b ) . For ≤ i ≤ n , e Wi ( p, u, g ) is the horizontal lift of u i with respect to ω at p.g and for ≤ j ≤ k , e n + j ( p, u, g ) is the only one vertical vector on P p.g such that ω ( p )( e n + j ( p, u, g )) = W j . A = { λ W } W ∈ L g is a trivial atlas over α . An easy computation shows that the set of A − natural tensor withrespect to α are all of those that there exists λ W such that T has a matrix representation f the form λ W T ( q, u, a ) = (cid:18) f ( a ) .Id n × n B ( a ) (cid:19) , where f : G −→ IR and B : G −→ IR k × k are differentiable functions. As above, if A is a maximal trivial atlas over α the only A − natural tensor with respectto α is the null tensor. So we have a weak definition of naturality for this case too. We saidthat T is A − weak natural with respect to α if T is λ − natural with respect to α for some λ ∈ A . We conclude showing some examples of s-spaces:
Let G be a Lie group of dimension k . We notated with e the unit of G . If v = { v , . . . , v n } is a base of g , let H vi be the unique left invariant vector field on G such that H vi ( e ) = v i .Then { H v ( g ) , . . . , H vn ( g ) } is base of the tangent space of G at g . Example 71
Given v a basis of g , let λ v = ( N, ψ, G, R, { e vi } ) be the s-space over G definedby N = G × G , ψ ( g, h ) = g.h , R a ( g, h ) = ( g.a, a − .h ) and e vi ( g, h ) = H vi ( g.h ) for ≤ i ≤ k .Like e vi ◦ R a ( g, h ) = e vi ( g, h ) , the base change morphism L v is constantly the identity matrixof R k × k . Therefore, if T is a tensor on G it satisfies that λ v T ◦ R a = λ v T .
For this reason, all constant matricial maps come from a tensor, hence the λ v − natural tensors are in a one to one relation with the matrices of R k × k .Suppose that λ v T depends only of one parameter, for example λ v T ( g, h ) = λ v T ( h ) ,then [ λ v T ( g ′ , h ′ )] ij = [ λ v T ( g ′ hh ′− , h ′ )] ij = T ( g ′ h )( H vi ( g ′ h ) , H vj ( g ′ h )) = [ λ v T ( g ′ , h )] ij =[ λ v T ( g, h )] ij , that is T is λ v − natural . Therefore, T is λ v − natural if and only if T is λ v T depends only of one parameter. The left invariant metrics are tensors of this type.Let v ′ be another base of g and consider λ v ′ . If a vv ′ ∈ GL ( k ) is the matrix that satisfies v ′ = a vv ′ v , then we have that e v ′ i ( g, h ) = e vi ( g, h ) .a vv ′ and λ v ′ T = ( a vv ′ ) t . λ v T.a vv ′ for atensor T on M . Thus the set of λ v − natural tensors is independent of the choice of thebase v . We can observe that ( Id G × G , Id G ) is a morphism of s-spaces with constant linkingmap a vv ′ , so T ∈ I ( Id G × G ,Id G ) if and only if a vv ′ ∈ G T ( g, h ) . Example 72
Let λ = { N, ψ, O, R, { e i }} be the s-space over G defined by N = G × L g = { ( g, v ) : g ∈ G and v is a base of g } , ψ ( g, v , . . . , v n ) = g , O = GL ( n ) , R ξ ( g, v ) = ( g, v.a ) and e i ( g, v ) = H vi ( g ) . Since { e i } ◦ R ξ = { e i } .ξ , λ T ◦ R ξ = ξ t . λ T.ξ for all ξ ∈ GL ( k ) . Therefore, there is only one λ − natural and is the null tensor. he left invariant metrics on G are not λ − naturals but for a metric T on G we havethat T is a left invariant metric if and only if λ T ( g, v ) = λ T ( v ) . If T is a left invariantmetric, then [ λ T ( g, v )] ij = T ( g )(( L g ) ∗ e ( v i ) , ( L g ) ∗ e ( v j )) = T ( e )(( L g − ) ∗ g (( L g ) ∗ e ( v i )) , ( L g − ) ∗ g (( L g ) ∗ e ( v i ))) = T ( e )( v i , v j ) = [ λ T ( e, v )] ij . Suppose that the matrix representation induced by T depends only of the parameter of g .Let g, h ∈ G and w, v ∈ T g G we have to see that T ( g )( v, w ) = T ( hg )(( L h ) ∗ g ( v ) , ( L h ) ∗ g ( w )) .Let { u , . . . , u n } be a base of g . If v = P ni =1 v i ( L g ) ∗ e ( u i ) and w = P ni =1 w i ( L g ) ∗ e ( u i ) , then ( L h ) ∗ g ( v ) = P ni =1 v i ( L hg ) ∗ e ( u i ) and ( L h ) ∗ g ( w ) = P ni =1 w i ( L hg ) ∗ e ( u i ) . Hence, T ( hg )(( L h ) ∗ g ( v ) , ( L h ) ∗ g ( w )) = ( v , . . . , v n ) . λ T ( hg, u ) . w ... w n = T ( g )( v, w ) . Let T be a tensor such that λ T ( g, v ) depends only of v . We know that λ T ( g, v.ξ ) =( ξ ) t . λ T ( e, v ) .ξ for all ξ ∈ GL ( k ) . Fixed v ∈ L g and let F : L g −→ GL ( k ) be defined by v = v .F ( v ) . Then, λ T ( g, v ) = ( F ( v )) t . λ T ( e, v ) .F ( v ) for all ( g, v ) ∈ G × L g . So we havethat λ T depends only of the parameter of L g if and only if there exist A ∈ R k × k and adifferentiable function F : L g −→ GL ( k ) that satisfies F ( w.ξ ) = F ( w ) .ξ , such that λ T ( g, w ) = ( F ( w )) t .A.F ( w ) Example 73
Fixed v ∈ L g and consider λ v = ( G × O ( k ) , ψ, O ( k ) , R, { e vi } ) where ψ ( g, ξ ) = g , R a ( g, ξ ) = ( g, ξa ) , e vi ( g, ξ ) = H v.ξi ( g ) . λ is a s-space over G with base change morphism L = Id O ( k ) . If T is a tensor of M , then λ T ◦ R a = a t . λ T.a . Therefore, T is λ − natural ifand only if λ T ( g, ξ ) = f ( g ) .Id k × k with f : G −→ IR a differentiable function. Is easy to seethat λ T (( g, ξ ) .a ) = ( ξa ) t . λ T ( g, Id ) . ( ξa ) , hence the matrix representation of T depends onlyof the parameter of O ( k ) if and only if λ T ( g, ξ ) = ξ t .A.ξ with A ∈ IR n × n . Let α = ( P, π, G, · ) be a principal fiber bundle endowed with a connection ω on a Riema-nnian manifold ( M, g ). Let us denote with M ad ( g ) the set of metrics on g that are invariantby the adjoint map ad . Consider the metric on P defined by h ( p )( X, Y ) = g ( π ( p ))( π ∗ p ( X ) , π ∗ p ( Y )) + ( l ◦ π )( p )( ω ( X ) , ω ( Y )) (1)where l : M −→ M ad ( g ). If G is compact, M ad ( g ) = ∅ , and if g is also a semisimplealgebra, then essentially there is (unless scalar multiplication) only one positive defined ad -invariant metric [14]. If l is a constant function, h is called a bundle metric . It is easyto see that π : ( P, h ) −→ ( M, g ) is a Riemannian submersion.22et l be an ad -invariant map on g . We are going to consider the s-space λ = ( N, ψ, O, R, { e i } )over P given by N = { ( q, u, v, g ) : q ∈ P, u is an orthonormal base of M π ( q ) , v is an or-thonormal base of g with respect to l and g ∈ G } , ψ ( q, u, v, g ) = q.g , O = O ( n ) × O ( k ) × G and the action is defined by R ( a,b,h ) ( q, u, v, g ) = ( qh, ua, vb, h − g ). For 1 ≤ i ≤ n , e i ( q, u, v, g ) is the horizontal lift with respect to ω of u i at q.g and, for 1 ≤ j ≤ k , e n + j ( q, u, v, g ) is the unique vertical vector on P p.g such that ω ( q.g )( e n + j ( q, u, v, g )) = v j . λ is a trivial s-space over α .Let G be a compact Lie group with g a semisimple algebra and h a metric on P of thetype of (1). Then, we have the following proposition: Proposition 74 h is λ − natural with respect to α if and only if h is a bundle metric.Proof. By definition λ h ( q, u, v, g ) is the matrix of h ( q.g ) with respect to de base { e i ( q, u, v, g ) , e n + i ( q, u, v, g ) } . For 1 ≤ i, j ≤ n , we have that: h ( q.g )( e i ( q, u, v, g ) , e j ( q, u, v, g )) = g ( u i , u j ) + 0 = δ ij For 1 ≤ i ≤ n and 1 ≤ j ≤ k : h ( qg )( e i ( q, u, v, g ) , e n + j ( q, u, v, g )) = 0 = h ( qg )( e n + j ( q, u, v, g ) , e i ( q, u, v, g ))and for 1 ≤ i, j ≤ k : h ( q.g )( e n + i ( q, u, v, g ) , e n + j ( q, u, v, g )) = l ◦ π ( qg )( v i , v j ) = f ( π ( q )) .δ ij because g has essentially one ad − invariant metric . Since λ h ( q, u, v, g ) = (cid:18) Id n × n f ( π ( q )) .Id k × k (cid:19) h is λ − natural with respect to α if and only if f is a constant map, that is to say that h is a bundle metric. Remark 75 If g has different ad − invariant metrics, and h is a metric of the type of(1), then λ h : N −→ IR ( n + k ) × ( n + k ) only depends of the parameter of G if l = δ.l with δ aconstant. In general, the metrics of type (1) that are λ − natural with respect to α are thebundle metrics induced by the ad − invariant metric l . Remark 76
The s-space λ depends of the metric l and of the connection ω . Let ω ′ beanother connection on α and consider the s-space λ ′ induced by it. The difference be-tween the connection are the horizontal subspaces that each one determine and the dif-ference between λ ω and λ ω ′ are the maps e i : N −→ T P and e ′ i : N −→ T P . Let A ( p, u, v, g ) = (cid:18) a ( p, u, v, g ) a ( p, u, v, g ) a ( p, u, v, g ) a ( p, u, v, g ) (cid:19) ∈ GL ( n + k ) be the matricial map that satisfies { e ′ i , e ′ n + j } = { e i , e n + j } .A where a ( p, u, v, g ) ∈ R n × n , a ( p, u, v, g ) ∈ IR n × k , a ( p, u, v, g ) ∈ k × k and a ( p, u, v, g ) ∈ IR k × n . Since e n + j ( p, u, v, g ) = e ′ n + j ( p, u, v, g ) , we have that a ≡ and a ≡ Id k × k . If T is a tensor, then λ ω ′ T ( p, u, v, g ) = (cid:18) a t ( p, u, v, g ) a t ( p, u, v, g )0 Id k × k (cid:19) . λ ω T ( p, u, v, g ) . (cid:18) a ( p, u, v, g ) 0 a ( p, u, v, g ) Id k × k (cid:19) Suppose as in the proposition above that there is essentially one ad − invariant metric.Then if h is a metric of type ( 1) we have that λ ω ′ h ( p, u, v, g ) = (cid:18) a t ( p, u, v, g ) a ( p, u, v, g ) + f ( π ( p )) a t ( p, u, v, g ) .a ( p, u, v, g ) f ( π ( p )) .a t ( p, u, v, g ) f ( π ( p )) a ( p, u, v, g ) f ( π ( p )) .Id k × k (cid:19) Therefore, if the connections satisfy that a ∈ O ( n ) and a is a constant map, then h is λ − natural with respect to α if and only if h is λ ′ − natural with respect to α . In thissituation h is a bundle metric. References [1] Araujo, J. and Keilhauer, G.R.:
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