{'fraction_non_alphanumeric': 0.05398963338504029, 'fraction_numerical': 0.01380954182588627, 'mean_word_length': 3.7124264574790513, 'pattern_counts': {'":': 0, '<': 2, '<?xml version=': 0, '>': 0, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 41, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
On second-order, divergence-free tensors

This paper deals with the problem of describing the vector spaces of divergence-free, natural tensors on a pseudo-Riemannian manifold that are second-order; i.e., that are defined using only second derivatives of the metric. The main result establishes isomorphisms between these spaces and certain spaces of tensors (at a point) that are invariant under the action of an orthogonal group. This result is valid for tensors with an arbitrary number of indices and symmetries among them and, in certain cases, it allows to explicitly compute basis, using the theory of invariants of the orthogonal group. In the particular case of tensors with two indices, we prove the Lovelock tensors are a basis for the vector space of second-order tensors that are divergence-free, thus refining the original Lovelock's statement.


Introduction
Divergence-free tensors appear in a variety of places; among them, let us highlight that: -In a relativistic theory of gravitation, the field equation is assumed to be of the form: where T 2 is the energy-momentum tensor of matter, g is the Lorentz metric of space-time that measures proper time and G 2 (g) is a suitable 2-tensor intrinsically constructed from the metric g; i.e., it is a natural 2-tensor.
By physical considerations (infinitesimal conservation laws), T 2 is assumed to be divergencefree, so the field equation is determined by the choice of a divergence-free, natural tensor G 2 . The paradigmatic example is the Einstein tensor, that fits into the field equation of General Relativity.
-By the second Noether's Theorem, the Euler-Lagrange tensor of a natural variational principle in the bundle of metrics is divergence-free. It is still an open problem to know whether any divergencefree, natural 2-tensor is the Euler-Lagrange tensor of a variational principle ( [2], [21]).
In a pseudo-Riemannian manifold (X, g), examples of natural tensors include tensor products and contractions among the metric, the dual metric, the curvature tensor or its covariant derivatives. The simplest natural tensors are those defined using only second derivatives of the metric, also called second-order tensors. This paper deals with the problem of describing those second-order tensors that are divergence-free.
The major breakthrough in the area remains the work done by D. Lovelock regarding symmetric tensors with 2 indexes. In a couple of famous papers ( [15], [16]), he wrote down a basis for the R-vector space of second-order 2-tensors that are symmetric and divergence-free. These Lovelock tensors are closely related to the existence of universal curvature identities ( [11]) or the Chern-Gauss-Bonnet theorem ( [10]), and have, since then, received much attention both by mathematical and physical communities (see, v. gr., [6] or [14]).
Nevertheless, a similar analysis for tensors with different symmetries or with a higher number of indices revealed difficult ( [1], [3], [4], [7]). Some progress has been made with the aid of computer programs ( [4]), but there is still a lack of general results.
In this paper, we prove several statements, valid for tensors with any number of indices of symmetries, that allow to describe divergence-free tensors as tensors invariant under the action of an orthogonal group.
Firstly, we prove that divergence-free, second-order tensors are "algebraic", in the sense that their local expressions are polynomial functions of g ab,cd : Theorem 3.5: Any second-order tensor that is divergence-free is polynomial.
Its proof relies on some techniques introduced by Lovelock, that we develop so as to show the vanishing of derivatives of sufficiently large order. In the process, we make use of simple facts of graph theory, that allows to avoid lengthy calculations using symmetries of indices.
As a consequence, for example, it follows that any non-zero, second-order tensor that is divergencefree has an even number of indices.
Secondly, it is easy to see that a polynomial tensor is divergence-free if and only if its homogeneous components are divergence-free (Proposition 4.1).
Then, we introduce certain spaces of tensors, Div k , depending on the symmetries and the number p of indices under consideration, and we prove: where g x is any pseudo-Riemannian metric with the fixed signature.
There exists an inclusion: Second-order, natural p-tensors homogeneous of degree k and divergence-free ⊂ Div k Og x .
That is to say, the problem is reduced to computing invariant tensors under the action of the orthogonal Lie group O gx , where the classical theory of invariants can be applied.
In the last Sections of the paper, we apply this technique to compute basis of divergence-free tensors in some particular cases.
In the simplest situation, that of tensors with 2 indices, we recover Lovelock's result. Indeed, we prove a stronger statement, as we show that the Lovelock's tensors are also a basis for the vector space of second-order 2-tensors with null-divergence, but not necessarily symmetric: Theorem 5.3: The Lovelock tensors L 0 , . . . , L p , where 2p ≤ n − 1, are a basis for the R-vector space of second-order, natural 2-tensors that are divergence-free.
In dimension 4, this refined version was already established by Lovelock himself ( [16]), but the situation in higher dimensions remained open.
On the other hand, as regards to tensors with some of their indices symmetric, we recover some of the results of and [3] and [4], and we prove a new statement for totally symmetric tensors: Theorem 6.2: Any second-order 2k-tensor that is divergence-free is a constant multiple of the symmetrization of g * ⊗ k . . . ⊗g * .
Finally, for differential forms we obtain a non-existence result: Theorem 6.4: There are no skew-symmetric, second-order tensors with zero-divergence, but the zero tensor.
The paper is structured as follows: The first preliminary Section collects together several classical definitions and results that are used later on.
Our later development heavily relies on the natural derivative of second-order tensors, originally developed by Lovelock. In Section 2, we present a new geometrical construction of this derivative, showing that naturalness corresponds to certain symmetries of the derivative. Sections 3 and 4 are devoted to the proof of Theorems 3.5 and 4.2, respectively. The last Sections contain the rest of the announced results.

Invariant theory of the orthogonal group
Let E be a finite dimensional R-vector space and let g be a non-singular metric of signature (p, q) on it. Let O p,q be the orthogonal algebraic group of linear isometries of (E, g).
Later on, we make a strong use of the following classical theorem ( [13], [19]): The vector space (⊗ r E * ) Op,q = Hom Op,q (⊗ r E, R) of invariant linear forms on ⊗ r E vanishes if r is odd, while, for r even, it is spanned by contractions of the type: where σ is a permutation of 1, . . . , r.
Throughout the paper, this theorem is applied in conjunction with the semi-simplicity of the orthogonal group: if V ⊂ ⊗ r E is a sub-representation of the group O p,q , then any O p,q -equivariant linear map f : V → R is the restriction of a O p,q -equivariant linear map f : ⊗ r E → R.

Remark 1.2
In the standard formulation of this result, the orthogonal group is considered as an algebraic group; that is, if the invariance of linear forms is understood as invariance under the action of all the points (real or imaginary) of the algebraic variety O p,q .
In this paper the orthogonal group is considered as a Lie group; correspondingly, the invariance condition is understood as invariance under the action of the rational (i.e., real) points of the algebraic variety O p,q only.

Second-order tensors
Let X be a smooth manifold of dimension n, and let M → X be the bundle of pseudo-Riemannian metrics, with a fixed signature. To fix notations, let us also recall that the divergence of a p-contravariant tensor T on a pseudo-Riemannian manifold (X, g) is the (p − 1)-tensor: where c − 1 p denotes the contraction of the first covariant with the p th -contravariant indices. Its local expression, using summation over repeated indices, is: For simplicity, from now on we will only consider contravariant tensors: Definition 1.1 A second-order p-tensor (not necessarily natural) is a second-order differential operator T : M ⊗ p T X , that is, a morphism of bundles: where J 2 M stands for the bundle of 2-jets of sections of M .
In local coordinates, the components of a second-order p-tensor are smooth functions T (g) j 1 ...jp = T j 1 ...jp (x i , g ab , g ab,c , g ab,cd ) .

Definition 1.2
The divergence of a second-order p-tensor T : M ⊗ p T X is the third order differential operator div T : M ⊗ p−1 T X defined as: (div T )(g) := div g (T (g)) .
A second-order p-tensor T is divergence-free if div T is the zero map. Proof: In local coordinates: for some smooth function F on J 2 M . Reordering this sum, we obtain: and the thesis follows because g ab,cdk are elements of a chart on J 3 M .

Second-order natural tensors
The bundle J 2 M of jets of metrics is a natural bundle; that is, diffeomorphisms τ : U → V between open sets of X act on their sections: That is, T is natural if, for any diffeomorphism τ : U → V between open sets of X, it holds: Any metric jet j 2 x g defines a normal tensor g 2 x ∈ N 2 : simply choose normal coordinates for g at x and write (g 2 x ) ijkl := g ij,kl , for the identity of the Gauss Lemma guarantees that the symmetrization of the last three indices of g 2 x is zero.
Lemma 1.4 Second-order normal tensors have the following symmetries: 1. They are symmetric under the interchange of the first pair with the second pair of indices: 2. The cyclic sum of the last three indices is zero.
Proof: Let us only check the first one, for the other is trivial. If G ∈ N 2 , then: Local diffeomorphisms act transitively on J 1 M ; therefore, a natural tensor T : J 2 M → ⊗ p T X is determined by its restriction to any fibre A j 1 x g of the projection J 2 M → J 1 M . To compute the value of T on an element p ∈ A j 1 x g we can take normal coordinates; that is, there exist a smooth map t such that the following triangle commutes: where π : A j 1 x g → N 2 is the map that sends a 2-jet to its normal tensor. These ideas lead to the following classical result (v. gr. [9]): Theorem 1.5 Let g x be a pseudo-Riemannian metric at x, and let O gx be the orthogonal group of (T x X, g x ).
There exists an isomorphism of R-vector spaces: Via this isomorphism, zeroth-order natural tensors corresponds with invariant elements in ⊗ p T x X: 2 Derivative of second-order tensors x g at any point. Let d denote the flat connection of the affine space A j 1 x g . Let T : J 2 M → ⊗ p T X be a morphism of bundles. The restriction of T to any of these fibres is a smooth map: x g is the tensor: More generally, the m th -covariant derivative defines the tensor: The derivative of T : J 2 M −→ ⊗ p T X is the morphism of bundles: Analogously, the higher derivatives are In local coordinates, the coefficients of T are and, analogously, the coefficients of the m th -derivative T m) are: The local expression of the derivative (2.0.3) together with Proposition 1.3 imply: If a second-order p-tensor T is divergence-free, then its derivative T satisfies the following linear symmetry: T j 1 ...jp;abcd := T j 1 ...jp;abcd + T j 1 ...d;abjpc + T j 1 ...c;abdjp .

Natural tensors
The naturalness of a second-order tensor is inherited by its derivative: The derivative of a second-order, natural tensor is also a natural tensor.
Proof: At any point j 1 x g ∈ J 1 M , the naturalness of T implies the commutativity of: Hence, the corresponding tangent linear maps at any point j 2 x g ∈ A j 1 x g satisfy the commutative diagram: that amounts to the naturalness of T .
of contravariant normal tensors of order 2 is defined as the kernel of the symmetrization in the last 3 indices.
Let j 1 x g ∈ J 1 M and consider the map that sends a 2-jet to its normal tensor: This map is affine and the equations of its tangent linear map π * are: Moreover, it is a retract of the inclusion N 2 ⊂ S 2,2 , and hence S 2,2 = N 2 ⊕ Ker (π * ).
Proposition 2.4 (Symmetries of the derivative of natural tensors) If a second-order tensor T : J 2 M → ⊗ p T X is natural, then its derivative T takes its values in ⊗ p T X ⊗ N 2 : (2.0.5) Proof: Let j 2 x g ∈ J 2 M and let j 1 x g be its 1-jet, so that j 2 x g ∈ A j 1 x g .
As T is a natural tensor, there exists a O gx -equivariant smooth map t : x for any j 2 xg with the prefixed value g x at x. In other words, the following triangle commutes: . (2.0.6) Hence, their tangent linear maps at j 2 x g also commute: proving that T j 2 x g annihilates Ker π * ⊂ S 2,2 . Therefore, via Lemma 2.3 and the following isomorphisms 3 Polynomial character of second-order, divergence-free tensors Let us begin this section with a short digression on graphs, that will be understood as finite CWcomplexes of dimension 1: Definition 3.1 A graph is a compact Hausdorff topological space K , together with a finite subset K 0 ⊂ K , whose elements will be called vertices, such that: Edges with equal endpoints will be called loops. Also, there can possibly be several edges between the same pair of vertices, v. gr: Lattin letters k, l, . . . will represent vertices, and an edge joining the vertices k and l will be denoted as hyphened pair k − l. The number of edges and vertices of a graph will be denoted e and v, respectively .
A tree is a connected graph with no cycles.
Any cycle satisfies the relation e = v , and any tree, the relation e = v − 1 . A connected graph is a hairy cycle if it contains a cycle whose removal produces a disjoint union of hairs, v. gr: In particular, any cycle is a hairy cycle (the bald case, so to speak). Any hairy cycle also satisfies the relation e = v .
Finally, let us say a vertex is simple if there is only one edge arriving to it (the vertex of a loop is not considered to be simple), and that a vertex is connected to a cycle if there is a cycle in its connected component. -after the removal of the edge k − l , the vertex k is still connected with a cycle.

The graph is a disjoint union of hairy cycles.
Proof: If 1 is not satisfied, let us prove that the connected component of a cycle is a hairy cycle.
Any vertex h connected to the cycle, and not inside the cycle, has to be on an edge whose opposite endpoint is inside the cycle; otherwise there exists an edge k − l satisfying 1 (see figure). Moreover, h has to be simple; otherwise, the previous edge k − h satisfies 1 . Finally, there does not exist an edge k − l between two vertices of the cycle, different from the edges of the cycle, because such an edge would satisfy 1 .
Therefore, the connected components of the graph are hairy cycles or trees.
As any hairy cycle satisfies e = v , and any tree e = v − 1 , the hypothesis e ≥ v for the graph implies no connected component is a tree.

Vanishing of derivatives
Definition 3.4 A second-order p-tensor T : J 2 M → ⊗ p T X is polynomial (in the second derivatives g ab,cd ) of degree < m if its restrictions to the affine spaces A j 1 are polynomial maps of degree < m, for any fibre A j 1 x g .
This condition is equivalent to saying that the m th -derivative T m) is null.
Let g x be a pseudo-Riemannian metric at a point x ∈ X. By Theorem 1.5, a second-order, polynomial natural tensor is defined by a polynomial O gx -equivariant map t : N 2 → ⊗ p T x X; that is, by an element in As these spaces of equivariant maps are spanned by total contractions of indices (Theorem 1.1) and tensors in S d i N 2 have an even number of indices, for any d i , it readily follows: Proposition 3.2 If p is odd, there are no polynomial, natural p-tensors T : J 2 M → ⊗ p T X, but the zero tensor. Definition 3.5 For m ≥ 1, let Div m x ⊂ ⊗ p T x X ⊗ S m N 2 be the vector subspace whose elements satisfy: Due to the symmetries of S m N 2 (any quatern a i b i c i d i can be put in the first position, and, by Lemma 1.4, a 1 b 1 can be interchanged with c 1 d 1 ), any element in Div m x satisfies: As a consequence, if the three indices j p a i b i are equal, then T j 1 ...l ; ... ll ... = 0 .
Due to Corollary 2.1 and Theorem 2.4, if a natural tensor T : J 2 M → ⊗ p T X is divergence-free, then its m th -derivative takes value in this subspace:

Proposition 3.3
If there exist m ∈ N such that Div m x = 0, then any divergence-free, natural ptensor T : J 2 M −→ ⊗ p T X has to be polynomial (in the second derivatives of the metric), of degree less than m.

Consider a component of an element in Div m
x : that we understand as a linear function on Div m x .
Definition 3.6 Its associated graph is defined as follows: • It has n vertices, labelled by k ∈ {1, . . . , n}, where n = dim X. The indexes j 1 . . . j p are irrelevant to construct the graph. As an illustration, consider the following example, in which n = 6 and m = 3 : (3.0.9) whose associated graph is: Consider a component of an element in Div m x : If, in the associated graph, the vertex j p ∈ {1, . . . , n} is connected to a cycle, then this component is zero. As a consequence, any divergence-free, natural tensor T : J 2 M → ⊗ p T X is polynomial (in the second derivatives of the metric), of degree less or equal than (n − 1)/2. Among all the elements equivalent to T , let be one with the highest number of loops (i.e., edges k − k with equal endpoints) in the associated graph. As 2m ≥ n, the graph associated to T R satisfies e ≥ v , and we can invoke Proposition 3.1.
If the graph associated to T R is a disjoint union of hairy cycles, then any vertex is connected with a cycle, and hence T R = 0 (Lemma 3.4), in contradiction with the hypothesis.
Otherwise, let k − l be an edge as in 1 in Proposition 3.1. Then:

Computation of divergence-free tensors
Let us explain how the results in the precedent Section allow to reduce the computation of secondorder p-tensors that are divergence-free to a problem of invariants for the orthogonal group.
Let T : J 2 M −→ ⊗ p T X be a second-order, natural p-tensor that is divergence-free. By Theorem 3.5, this tensor is polynomial on the second derivatives of the metric, of degree m ≤ (n − 1)/2. Let us decompose it as where the tensors T k are homogeneous polynomials of degree k in the second derivatives of the metric. As any diffeomorphism of the base manifold X acts linearly on the coordinates g ab,cd of J 2 M , it is trivial to check that each tensor T k is natural.
Let g x be a fixed metric at a point x ∈ X. By Theorem 1.5, the natural tensor T : J 2 M → ⊗ p T X corresponds with a polynomial O gx -equivariant map t : N 2 → ⊗ p T x X, satisfying T (j 2 x g) = t (g 2 x ) on each metric jet j 2 x g having the prefixed value at x.
If t = t 0 + . . . + t m is the decomposition into homogeneous components, then it is easy to check that each addend t k corresponds to the p-tensor T k . This shows that the natural p-tensor T k is homogeneous of weight −2 − 2k respect to the metric (i.e., it satisfies T k (λ 2 g) = λ −2−2k T k (g)) and, consequently, so does its divergence div T k .
Therefore, in the equality 0 = div T = div T 0 + · · · + div T m each addend has a different weight, so each div T k is zero. All in all, we have proved: where each addend T k is a divergence free, second-order, natural p-tensor which is a homogeneous polynomial of degree k in the second derivatives of the metric.
As regards to the computation of homogeneous tensors, consider the commutative triangle: , where π(j 2 x g) = g 2 x , and j 1 x g is any jet with the prefixed value at x. As π is a surjective affine map, T | is a homogeneous polynomial of degree m if and only if so it is t. Therefore, the following bijections hold: We use the symbol N 2 to denote the space of contravariant normal tensors at x, as well as to denote the bundle of such tensors.
This sequence of maps sends each natural tensor T : J 2 M → ⊗ p T X, homogeneous of degree m in the second derivatives of g, to its m th -derivative T m) : Observe that, after differentiating m times, the tensor T m) does no longer depend on the second derivatives of the metric; hence, by naturalness, nor does it on the first derivatives.
The divergence-free condition can now be imposed, resulting: Proof: If the tensor T is divergence-free, then its m th -derivative (at x), T m) takes its values inside the subspace Div m x ⊂ ⊗ p T x X ⊗ S m N 2 , by formula (3.0.8). And the vector space of zeroth-order natural tensors T : M → ⊗ p T X ⊗ S m N 2 that take values inside Div m x is isomorphic to (Div m x ) Og x , by Theorem 1.5.

Remark 4.3
Although it will not be needed in the rest of the paper, let us remark that this inclusion is indeed an isomorphism. This is due to the fact that a second-order, natural tensor T is divergence-free if and only if its local components satisfy (1.2.1). This statement can, in turn, be proved using the formula: which can be found, for example, in [8].

Lovelock tensors
In this Section, let us consider tensors with p = 2 indices. In this case, is the vector subspace defined by the equations: Let g x be a pseudo-Riemannian metric at a point x ∈ X. The key computation is the following: Proof: The vector space (Div m x ) Og x = Hom Og x (R , Div m x ) is isomorphic to: which, in turn, is spanned by iterated contraction of indices (Theorem 1.1). Let us prove that any total contraction of indices is proportional to: where equal letters denote contraction of the corresponding positions. We argue by descendent induction on the number of contracted pairs (i.e., contraction of an index in an odd position with the index in the following position).
Given a total contraction, if the first index is not contracted with the second one, then: ... jk... ∼ T ik ; ... jj... and the induction hypothesis applies.
Otherwise, we can assume the third index is not contracted with the fourth, and hence: This two addends have the same number of contracted pairs as the original one, so we are reduced to the previous case.
As a consequence, the vector space of divergence-free 2-tensors, homogeneous of degree k ≤ (n − 1)/2, has dimension at most one. In the following subsection, let us define explicit generators for these spaces.

Definition of the Lovelock tensors
Let g be a pseudo-Riemannian metric and let us consider it as a one-form with values on one-forms.
Its Riemann-Christoffel tensor R can also be understood as a 2-form with values on 2-forms that is symmetric, i.e., a section of S 2 Λ 2 T * X ⊂ Λ 2 T * X ⊗ Λ 2 T * X.
With this language, the differential Bianchi identity and the torsion-free property of the Levi-Civita connection ∇ amount to the equations: (5.0.10) With respect to the wedge product of forms, consider the following (n − 1)-forms with values on (n − 1)-forms: where k runs from 0 to the integer part of (n − 1)/2. These L k are clearly symmetric, i.e., sections of S 2 (Λ n−1 T * X) ⊂ Λ n−1 (T * X) ⊗ Λ n−1 (T * X), and also satisfy d ∇ L k = 0, in virtue of (5.0.10).
The following statement is easy to prove (see, v. gr., [17]): Proposition 5.2 Contraction with a volume form, ω X , defines a linear isomorphism: and symmetric 2-tensors correspond with sections of S 2 (Λ n−1 T * X). Moreover, if T and Π are a 2-tensor and a valued (n−1)-form corresponding via this isomorphism, then: Definition 5.1 The Lovelock's tensors L k are the 2-contravariant tensors on X corresponding to the vector-valued forms L k via the isomorphism above.
Hence, they are symmetric and divergence-free 2-tensors.
Apart from the trivial case of the dual metric, L 0 = g * , the simplest Lovelock tensor, L 1 , is proportional to the contravariant Einstein tensor; i.e., via the isomorphism above, where q stands for the number of −1 in the signature (p, q) of g.
In general, it is easy to check that the k th -Lovelock tensor L k is a homogeneous polynomial of degree k on the second-derivatives of the metric. Therefore, L k generates the vector space of divergence-free tensors T : J 2 M → ⊗ 2 T X that are homogeneous polynomials of degree k on the second derivatives of the metric.
To be precise: The Lovelock tensors L 0 , . . . , L m , where 2m ≤ dim X −1, are a basis for the R-vector space of second-order, natural 2-tensors that are divergence-free. That is, Natural tensors T : J 2 M → ⊗ 2 T X that are divergence-free = L 0 , . . . , L m .
Proof: By Theorem 3.5, any tensor T of the type under consideration is polynomial, of degree m ≤ (n − 1)/2 in the second derivatives of the metric. By Proposition 4.1, T = T 0 + · · · + T m , where each T k is divergence-free and is homogeneous of degree k in the second derivatives of the metric.
As the space of such tensors has dimension ≤ 1, due to Theorem 4.2 and Lemma 5.1, the tensor T k coincides, up to a constant factor, with L k .

Remarks 5.4
Our proof also characterizes the k th -Lovelock tensor L k as the only, up to a constant factor, second-order, natural 2-contravariant tensor which is divergence-free and homogeneous of weight w = −2 − 2k.

Other computations
Let p ≥ 4 (recall that, due to Theorem 3.5 and Proposition 3.2, there are no second-order, divergencefree natural tensors with an odd number of indices, but the zero tensor).
As an illustration of the techniques explained above, let us firstly consider tensors that are symmetric in 3 indices; i.e., sections of: Definition 6.1 For m ≥ 1, let SDiv m x ⊂ ⊗ p−3 T x X ⊗ S 3 T x X ⊗ S m N 2 be the vector subspace whose elements satisfy: SDiv m x = 0 .
As a consequence, any natural tensor T : J 2 M → ⊗ p−3 T X ⊗ S 3 T X that is divergence-free (in one of the symmetric indices) is indeed a zeroth-order tensor.
Theorem 6.2 If k ≥ 2, any divergence-free, natural tensor T : J 2 M → S 2k T X is a constant multiple of S 2k .
Proof: Due to the previous Proposition, any totally symmetric, divergence-free tensor has to be zeroth order. By Theorem 1.5, the space of zeroth-order natural tensors is isomorphic to Hom Og x (R, S 2k T x X) = (S 2k T x X) Og x , so the statement follows from: which is a trivial computation, using Theorem 1.1.
Non-existence of divergence-free, differential forms Definition 6.2 For m, p ≥ 1, let ΛDiv m x ⊂ Λ p T x X ⊗ S m N 2 be the vector subspace whose elements satisfy: Proof: The proof is similar to that of Lemma 5.1: let us prove that any total contraction of indices is proportional to: T iijj... ; kk...