^{1}

^{2}

^{1}

In this paper, as a new contribution to the tensor-centric warfare (TCW) series [1] [2] [3] [4], we extend the kinetic TCW-framework to include non-kinetic effects, by addressing a general systems confrontation [5], which is waged not only in the traditional physical Air-Land-Sea domains, but also simultaneously across multiple non-physical domains, including cyberspace and social networks. Upon this basis, this paper attempts to address a more general analytical scenario using rigorous topological methods to introduce a two-level topological representation of modern armed conflict; in doing so, it extends from the traditional red-blue model of conflict to a red-blue-green model, where green represents various neutral elements as active factions; indeed, green can effectively decide the outcomes from red-blue conflict. System confrontations at various stages of the scenario will be defined by the non-equilibrium phase transitions which are superficially characterized by sudden entropy growth. These will be shown to have the underlying topology changes of the systems-battlespace. The two-level topological analysis of the systems-battlespace is utilized to address the question of topology changes in the combined battlespace. Once an intuitive analysis of the combined battlespace topology is performed, a rigorous topological analysis follows using (co)homological invariants of the combined systems-battlespace manifold.

The principal objective of the Modeling Complex Warfighting (MCW) Strategic Research Investment (SRI) is to better enable dealing with uncertainty, meaning achieving reliable decision-making in environments that are non-ergodic. Such systems are incomprehensible, in a sense, through observation of past data, because they lack stability and manifest unique transient states. Phenomena associated with war and battle are inherently non-ergodic, a fact that has been observed at least since the birth of modern military thinking (see, for instance, [^{1}). This paper attempts to approach the MCW problem from the multilayered nature of warfare and look at Blue, Red and Green entities. The aim is to undertake analysis which closely represents the realities of modern warfare. When modeling complex modern combined battlefields it is therefore important to consider neutral forces―which we label “Green”―since they are much more now than in the past even the central feature of the strategic situation. Indeed, in modern asymmetric confrontations neutral or non-engaging groups have been known to side with one side or the other or even to engage actively in conflict, which rapidly changes the dynamics of the situation. Hence, social and psychological domains play an increasingly significant role in understanding the dynamics of modern armed conflict.

The dynamics of the combined effects of these and other factors mean that the statistical properties of such problem environments are non-stationary. As a result, the outcomes of battles are not predictable, since the battle is inherently non-ergodic; yet, it is possible to deal with such systems nonetheless by establishing conditions that are weaker than ergodicity, which have essentially topology-changing nature.

ω = d α + γ

The tensor-centric warfare (TCW) series of papers (see [

ω (1)

where γ and the Red and Blue forces are defined as vectors d α and ω , defined on their respective configuration n-manifolds γ (with local coordinates [ ω ] ∈ H p ( M ) , for ∫ C ω = ∫ C γ ⇔ 〈 C , ω 〉 = 〈 C , γ 〉 , ) and γ (with local coordinates ω ). The Red and Blue vectorfields, . and ω , include the following terms (placed on the right-hand side of Equation (1)):

• Linear Lanchester-type terms, γ and ω , with combat tensors [ ω ] ∈ H p ( M ) and γ ∈ H Δ p ( M ) defined via bipartite and tripartite adjacency matrices, respectively defining Red and Blue aircraft formations (according to the aircraft-combat scenario from [

• Quadratic Lanchester-type terms, H Δ p ( M ) and H Δ p ( M ) ≅ H d p ( M ) , with the 4th-order tensors γ and M Red representing strategic, tactical and operational capabilities of the Red and Blue forces (see [

• Entropic Lie-dragging of the opposite side terms, M Blue and T M Red , where T M Blue and n = 30 . In case of resistance, the Lie derivatives are positive, ⊙ and f RB : W → ℝ , so that the non-equilibrium battlefield entropy grows, p ∈ W ; in case of non-resistance, the Lie derivatives vanish, f RB ( p ) = c and f RB − 1 ( p ) = { p ∈ W ( M Red , M Blue ) : f RB ( p ) = c } . , so that the battlefield entropy is conserved, μ ( W ) (see [

• Entropic Red-Blue commutators, C λ and S 2 , for modeling warfare symmetry (see [

• Hamilton-Langevin delta strikes, μ ( S 2 ) = 2 and S 1 × I , including (on both sides) discrete striking spectra (slow-fire missiles) and continuous striking spectra (rapid-fire missiles), as well as bidirectional random strikes, Hamiltonian vectorfields, self-dissipation, opponent-caused dissipation and non-delta random forces (see [

In the present paper, we extend the above kinetic Red-Blue framework to include non-kinetic effects, by addressing the general systems-confrontation [

Dynamically speaking, the basic Red-Blue pair of vectorfields (1) is extended with the following Green vectorfield:

μ ( S 1 × I ) = 0 (2)

where g ∈ W is the Green flow on the Green manifold ( f RB , g ) ; f RB and x 1 represent non-kinetic effects in scalar and vector form, respectively, and all the other terms are the same as in Equation (1). Notice that on the right-hand side of Equation (2) al the tensors are the sums of the corresponding Red and Blue tensors. This insures the fact that the Green force includes both Red and Blue forces and the covariance is preserved. Also, the kinetic delta strikes are missing, which gives the highest importance to entropic Lie-dragging ( x 2 , with or without resistance) of the Red-and-Blue cyberspace, electromagnetic, psychological and social-network domains (encaptured in the tensors W x 1 u and W x 2 s ).

x 1

Topological motivation for the present paper is inherited from the influence on modern physics by John Wheeler from Princeton^{2}. Although we might have some well-defined and (numerically) solvable local TCW Equations (1)-(2), we lack a picture of the global topology of the systems-battlespace-the environmental configuration manifold in which the combat happens-with its dramatic spacial (eliptic/hyperbolic) changes. In (1) both the Red and Blue vectorfields, ( f RB α , f RB β ) and ∂ t ϕ ( x , t ) = − ∇ f RB t α β , , and their corresponding flows, f RB t α β and f RB α , which consist of the integral lines of the vectorfields f RB β and λ obtained by their numerical integration starting from the chosen initial conditions, λ and Δ t , respectively-are defined on their respective configuration n-manifolds^{3}, d and w . To give the global picture of the battlefields governed by local Equation (1)-(2) we need to perform the topological analysis of the joint manifold including all three submanifolds: d w , d and d ∘ d ≡ d 2 = 0 .

Global topological analysis is an extension of local geometric analysis. To utilize the geometric framework most suitable for the present topological analysis^{4}, we will assume that Red and Blue (as well as Green) configuration manifolds, ℝ 3 , f = f ( x , y , z ) and d f = w , are endowed with the pseudo-Riemannian geometry, which is both elliptic (positive metric) and hyperbolic (negative metric; see, e.g. [^{5} from [

d

The non-equilibrium phase transitions occurring at the battlefield at various stages of warfare, can be superficially characterized by sudden entropy growth. However, these rapid changes of the systems-battlespace always have underlying structural topology changes (see [^{6}.

In this section we develop the basic differential topology of the systems-battlespace, mainly following the work of the Fields medalist John Milnor [

To start with the systems-battlespace topology, we introduce an important concept from differential topology (and its gravitational-physics applications): the so-called cobordism^{7}. Briefly, the cobordism relation, denoted d ( g ( f ) ) = g ′ ( f ) d f , between two compact (i.e., closed and bounded) n-manifolds, ∂ C and ∂ C , means that their disjoint union ∂ ( ∂ C ) = 0 is the boundary ∂ ∘ ∂ = ∂ 2 = 0 of a compact { x i , i = 1 , ⋯ , n } -manifold W. In other words, cobordism T M between two compact n-manifolds T ∗ M and { ∂ i ≡ ∂ / ∂ x i } ∈ T M is the compact { d x i } ∈ T ∗ M -manifold W whose boundary a ∈ Ω p ( ℝ n ) is the disjoint union of b ∈ Ω q ( ℝ n ) and a ∧ b = ( − 1 ) p q b ∧ a ∈ Ω p + q ( ℝ 4 ) . This is an equivalence relation and the equivalence class of cobordisms is denoted by a = a i d x i ^{8} (for technical details, see e.g. [

a

a = a ∧ b = a i b j d x i ∧ d x j = − b j a i d x j ∧ d x i = − b ∧ a .

The two-party battlespace cobordism from ^{9}:

∂ C

which in our case of an extended three-party systems-battlespace manifold, ∂ C = ∑ i c i ∂ N i , reads:

β

In the language of abstract algebra, we say that the following diagram commutes:

In plain English, this commutative diagram reads: if we have a cobordism between the Red and Blue manifolds, and a cobordism between the Blue and Green manifolds, then we also have a cobordism between the (initial) Red and (final) Green manifolds. That is, d = ∂ i d x i (see

Clearly, this operation can be extended to any number of parties, producing the so-called chain cobordism, between the first and the last manifold; e.g., in case of four parties/manifolds (Red, Blue, Green, Yellow), we have the following commutative chain cobordism:

That is, d β = 0 , etc.

Closely-related to the systems-battlespace cobordisms are the Morse functions of the systems-battlespace. Namely, on the Red-Blue systems-battlespace (

β

we can define the real-valued Red-Blue Morse function, α (see [

β = d α

which can be seen as the Red-Blue landscape. Its gradient vectorfield:

d β = d d α = 0

according to the Morse lemma, defines the Red-Blue level set^{10} of equipotential contour lines (of equal altitude) at the critical points p of d 2 = 0 where the gradient vanishes: curl F = 0 . The finite set F = grad f of m critical points of div F = 0 is denoted by Crit( F = curl g ).

( A , B , C )

In addition, we need to consider only nondegenerate critical points of f i : A i → A i + 1 , that is, only those critical points p of the vanishing gradient, ( A i ) , which have the nondegenerate Hessian Im ( f i ) = Ker ( f i + 1 ) or, non-singular Hessian matrix:

( G , ∗ )

By definition, the function ( H , ⋅ ) is Morse if all critical points are nondegenerate. All nondegenerate critical points in Crit( ( G , ∗ ) ) are isolated in the Red-Blue systems-battlespace W.

The index ( H , ⋅ ) of each critical point p is the number of negative eigenvalues of the Hessian matrix h : G → H . In other words, each critical point p of the Red-Blue Morse function x , y ∈ G has its own index h ( x ∗ y ) = h ( x ) ⋅ h ( y ) . , which is the number of independent directions around p in which e G decreases. Therefore, we have natural indices of e H for the minima of h ( x − 1 ) = h ( x ) − 1 , Ker ( h ) for the saddles of h : G → H , and e H for the maxima of Ker ( h ) = { x ∈ G : h ( x ) = e H } .

Topology change of the Red-Blue systems-battlespace Im ( h ) happens as an abrupt change of the shape of the level sets of the Morse function h : G → H whenever it passes through the critical values Im ( h ) = { h ( x ) : x ∈ G } where Ω p , otherwise the topology of W does not change. The mechanism of topology change is attaching a Z p ( M ) = Ker d p -cell (the so-called “handlebody”) to B p ( M ) = Im d p − 1 , completely determined by the index d p − 1 ∘ d p = d 2 = 0 ⇒ Im ( d p − 1 ) = B p ( M ) ⊂ Z p ( M ) = Ker ( d p ) . , at the critical points p of B p . Therefore, the index Z p determines the topology changes of the Red-Blue systems-battlespace W (from ^{11}.

According to the Morse-cobordism theorem (see [^{12}, C ∈ Z n ∂ .

The fundamental topological invariant of the Red-Blue systems-battlespace ω ∈ Z d n is its Euler characteristic, H n ∂ , defined as the alternating sum of the critical points H d n :

b n = b n

d ( i ∂ ∂ ¯ ) = i ( ∂ + ∂ ¯ ) ∂ ∂ ¯ = i ( ∂ 2 ∂ ¯ − ∂ ∂ ¯ 2 ) = 0.

which is equivalent to the alternating sum of the Betti numbers^{13} d ⋆ F = − ⋆ J of the systems-battlespace W:

div E = 4 π ρ

Based on the sign of the quadratic forms ∂ t E − curl B = − 4 π j and g = g i j , we can distinguish the following four principal cases of the Red-Blue landscape topology, or four critical points, of the Morse function ∇ 2 f = 1 det ( g ) ∂ i ( det ( g ) g i j ∂ j f ) . ^{14}, with its corresponding indices^{15}:

d − δ = ∂ D

Case 1 (depicted in

Case 2 (depicted in

Case 3 (depicted in

Case 4 (depicted in

Now, we can introduce the third player into our wargame, the Green force C b a , represented by its own pseudo-Riemannian quadratic form:

k b E c d a b B c R d ∈ M Red

where the social-network type, system-confrontation tensor | L B N b a | = 0 is defined as combinations of kink (Tanh) and bell (Sech) functions applied to Green force adjacency matrix. In this way, we obtain the Red-Blue-Green systems-battlespace ∂ t S = 0 , which is also a pseudo-Riemannian | [ R a , B a ] | ≥ 0 ∈ M Red -manifold.

On the triple configuration manifold | [ B a , R a ] | ≥ 0 ∈ M Blue , depicted in

δ B a ( H − L )

which represents the Red-Blue-Green landscape. Its gradient vectorfield:

Green : ∂ t G a ︷ G r e e n . v e c f i e l d = γ ( A b a + C b a ) ( R b + B b ) ︷ G r e e n . l i n . L a n c h a s t e r + γ b ( E c d a b + F c d a b ) G c ( R d + B d ) ︷ G r e e n . q u a d . L a n c h a s t e r + G b L G ( N b a + N b a ) ︷ G r e e n . L i e . d r a g g i n g + [ G a , ( R b + B b ) ] ︷ G r e e n . w a r . s y m m e t r y ,

G a ( x , t ) ∈ M Green

according to the Morse lemma, defines the Red-Blue-Green level set^{16} of equipotential contour lines (see

In addition, we need to consider only nondegenerate critical points of B a = B a ( x , t ) , that is, only those critical points p of the vanishing gradient, R ˙ a , which have the nondegenerate Hessian B ˙ a or, non-singular Hessian matrix:

R 0 a

By definition, the function B 0 a is Morse if its all critical points are nondegenerate. All nondegenerate critical points in Crit( M Red ) are isolated in the Red-Blue systems-battlespace W.

As before, the index M Blue of each critical point p of the Morse function M Green is the number of negative eigenvalues of the Hessian matrix M Red . Topology change of the Red-Blue-Green systems-battlespace M Blue happens as an abrupt change of the shape of the level sets of the Morse function M Green whenever it passes through the critical values A a b R a R b where C a b B a B b , otherwise the topology of γ ( A a b + C a b ) ( R a R b + B a B b ) does not change. The mechanism of topology change is attaching a A a b R a R b -cell/handlebody, completely determined by the index C a b B a B b , at the critical points p of γ ( A a b + C a b ) ( R a R b + B a B b ) . Therefore, the index A a b determines the topology changes of the Red-Blue-Green systems-battlespace C a b (from

As in the case of cobordisms, this 3-party Morse function can be extended to address more players (e.g., various groups within Green, various coalition partners within Blue and Red, or even a third conflicting Yellow force) in the wargame.

In this section, we move to the realm of (co)homology, which can be summarized by Wheeler’s BBZ dictum: “the boundary of a boundary is zero”. We explore the systems-battlespace topology changes, using Morse (co)homology techniques. We will apply Morse (co)homology to the systems-battlespace-cobordism n-manifold W using two approaches, classical approach of Morse homology and modern approach of Morse cohomology (see the Appendix for the basic (co)homology definitions, all rooted in the BBZ dictum).

γ ( A a b + C a b )

The basic Morse theory was further developed into the Morse homology^{17} by three Fields Medalists: R. Thom, S. Smale and J. Milnor. In this section we give a brief overview of of Morse homology, applied to the systems-battlespace-cobordism manifold W, using the abbreviated Morse-Smale approach (for a detailed technical review of Morse homology, see [

As a background, we summarize and make the qualitative concepts from the previous section more precise and, for simplicity, restricted to the Red-Blue systems-battlespace. Let G a b ≡ γ ( A a b + C a b ) represent a Cob f -smooth Morse function on the systems-battlespace-cobordism n-manifold W, equipped with the pseudo-Riemannian metric tensor: M 1 . The point M 2 is the critical point of M 1 ⊔ M 2 if ∂ W . In local coordinates in a neighborhood of ( n + 1 ) on W,

Cob f this means M 1 for

M 2 . The (finite) set of critical points of ( n + 1 ) is denoted by Crit( ∂ W = M 1 ⊔ M 2 ). The Hessian of the Morse function M 1 at a critical point M 2 defines a symmetric bilinear form:

W ( M 1 , M 2 ) ∈ Cob f ( M 1 , M 2 )

on the tangent space ( n + 1 ) to the systems-battlespace-cobordism manifold at the point ∂ W = M Red ⊔ M Blue , which is in local coordinates W ( M Red , M Blue ) ∈ Cob f ( M Red , M Blue ) . represented by the matrix

of second partial derivatives, W ( M Red , M Blue , M Green ) . Index c ( M 1 , M 2 , M 3 ) and nullity of the matrix

W ( M 1 , M 2 , M 3 ) are called the index and nullity of the critical point c : Cob f ⊙ Cob g → Cob h of the Morse function ⊙ . Since W is a compact n-manifold, it is always possible to alter a given Morse function c ( M 1 , M 2 , M 3 ) : Cob f ( M 1 , M 2 ) ⊙ Cob g ( M 2 , M 3 ) → c Cob h ( M 1 , M 3 ) , into a self-indexing Morse function, which has: W ( M Red , M Blue , M Green ) , for every critical point c ( M Red , M Blue , M Green ) : Cob f ( M Red , M Blue ) ⊙ Cob g ( M Blue , M Green ) → c Cob h ( M Red , M Green ) . (for the proof, see [

We develop the Morse homology of the systems-battlespace-cobordism n-manifold W in the following three steps:

1) On the systems-battlespace manifold W we define the negative gradient flow, ( < 1 ` 1 ` 3 ; 700 > ( 0,0 ) [ M Red ` M Blue ` M Green ; Cob f ` Cob h ` Cob g ] ), as a map Cob h = Cob f ⊙ Cob g such that:

< 1 ` 1 ` 1 ` − 1 ; 800 ` 600 > ( 200,0 ) [ M Red ` M Blue ` M Yellow ` M Green ; Cob f ` Cob k ` Cob g ` Cob h ] (3)

Cob k = Cob f ⊙ Cob g ⊙ Cob h

From the work of Smale [

2) Using the negative gradient flow (3), we can decompose the systems-battlespace manifold W into a disjoint union of unstable submanifolds,^{18} f RB , (or equivalently, a disjoint union of stable submanifolds, ∇ f RB ( p ) = 0 ), using the prescription due to R. Thom. Let x be a critical point of the Morse function { p 1 , p 2 , ⋯ , p m } ∈ W . We define the unstable submanifold, f RB , of the point x under the negative gradient flow ( f RB ), to be the set of all points flowing from the critical point x, formally:

f RB

so ∇ f RB = 0 is an embedded open disk in W with dimension equal to ∇ 2 f RB .

Similarly, we define the stable submanifold, H RB = ( ∂ 2 f RB ∂ R a ∂ R b | p + ∂ 2 f RB ∂ B a ∂ B b | p ) , with det ( H RB ) ≠ 0. , of the point x under the negative gradient flow ( f RB : W → ℝ ), to be the set of all points flowing into the critical point x, formally:

f RB

so λ an embedded open disk in W with dimension equal to H RB .

A function is said to be Morse-Smale if the unstable and stable submanifolds intersect transversely for any two critical points, x and y of f RB . Here comes the Morse--Smale condition: for a generic metric λ the intersection: f RB is transverse^{19}.

1) We can now define the boundary operator λ = 0 (see Appendix A.2), as:

f RB

where λ = 1 is the number of points in the quotient manifold: f RB . The proof of the BBZ-condition: λ = 2 is based on gluing and cobordism arguments (see [

f RB

states that, for two generic Morse functions W RB , their homology groups f RB and f RB ( p ) ∈ W RB are isomorphic^{20}. Furthermore, for a generic ∇ f RB ( p ) = 0 they are isomorphic [

W RB

Apart from the “classical” Thom-Smale-Milnor approach to Morse homology, in 1980s Ed Witten from Princeton (the only physicist who become the Fields Medalist) rediscovered in [

Witten’s approach (see [

C a b B a B b < 0

Since every harmonic p-form is closed ( f RB ), we have a linear map: ∇ f RB = 0 , by taking the de Rham cohomology class A a b R a R b < 0 . The de Rham theorem states that the de Rham cohomology C a b B a B b > 0 is isomorphic to the singular homology f RB , as well as to any other cohomology with real coefficients, ∇ f RB = 0 . In addition, the Hodge theorem states that an arbitrary de Rham cohomology class A a b R a R b < 0 , C a b B a B b < 0 of an oriented compact Riemannian manifold M can be represented by a unique harmonic form f RB , which means that the natural map: ∇ f RB = 0 is actually an isomorphism: G a = G a ( x , t ) .

We derive the Morse-Witten cohomology for the Red-Blue systems-battlespace cobordism n-manifold W in the following four steps:

1) To start with, we take the Red-Blue Morse function D a b G a G b = γ ( A a b + C a b ) ( R a R b + B a B b ) , along with the pseudo-Riemannian metric D a b , and consider the long de Rham complex on W (i.e., the long exact sequence of exterior vector-spaces W ( M Red , M Blue , M Green ) ; see Appendix A.2):

( n + 1 )

The complex W ( M Red , M Blue , M Green ) can be decomposed into the direct sum of finite-dimensional eigenspaces f RBG : W → ℝ of the Hodge Laplacian f RBG = A a b R a R b + C a b B a B b + D a b G a G b , as:

∇ f RBG = ∇ A a b R a R b + ∇ C a b B a B b + ∇ D a b G a G b ,

The Hodge-de Rham theory (see Appendix A.3) implies the following isomorphisms:

f RBG

2) Next, the very definition of the Hodge Laplacian, ∇ f RBG = 0 , implies that its product { p 1 , p 2 , ⋯ , p m } ∈ W with the exterior de Rham differential d (as well as with the codifferential f RBG ) is commutative:

f RBG

Therefore, we can restrict the de Rham differential d to f RBG by acting on the subcomplex: ∇ f RBG = 0 , and obtain the ∇ 2 f RBG -restricted de Rham complex:

H RBG = ( ∂ 2 f RBG ∂ R a ∂ R b | p + ∂ 2 f RBG ∂ B a ∂ B b | p + ∂ 2 f RBG ∂ G a ∂ G b | p ) , with det ( H RBG ) ≠ 0.

To prove that the restricted de Rham complex f RBG : W → ℝ is exact, we note that if any p-form f RBG is in the kernel of λ , Ker( f RBG ), then H RBG ; therefore we have:

W RBG

Since f RBG commutes with both d and f RBG ( p ) ∈ W RBG , we see that ∇ f RBG ( p ) = 0 ,

which means that the complex W RBG is exact. From the exactness of the restricted de Rham complex λ , it follows that the a-parameterized curve of complexes: λ has as its cohomology the set/group f RBG of all harmonic p-forms on W, for any real λ .

3) We can now introduce Witten’s main idea from [

f RB

(compare with Equations (11)-(12) in Appendix A.3). The deformed differential ∇ f RB ( x c ) ≡ ∇ f RB [ ( R , B ) ] = 0 yields the deformed de Rham cohomology, also called the Witten cohomology:

x c

( x 1 , ⋯ , x n ) c = ( R 1 , ⋯ , R n , B 1 , ⋯ , B n ) c

which is also isomorphic to f RB : f RB because we are only conjugating the de Rham differential d with x c .

4) The deformed cohomology, ∇ 2 f RB ( x ) : T x W × T x W → ℝ , such that : ∇ 2 f RB ( x c ) = ( ∂ 2 f RB ∂ x a ∂ x b ) , is computed using the Hodge theory, by considering the Witten Laplacian: T x W and the decomposition: x c , where x = ( x i ) is the eigenspace of ( ∂ 2 f RB ∂ x a ∂ x b ) , along with the t-parameterized curve of the chain complexes: λ , spanned by all eigenforms of ∇ 2 f RB ( x c ) with eigenvalues x c . The t-parameterized curve of the chain complexes: f RB , generated by the Witten Laplacian f RB gives both the Morse-Witten cohomology index ( x c ) = f ( x c ) and its dual, the deformed homology x c of the Red-Blue systems-battlespace cobordism n-manifold W, as follows. Namely, Witten stated in [

• The dimension, ∇ 2 f RB ( x ) = number of critical points of W u ( x ) of index p, i.e., the subset of Crit( W s ( x ) ) of index p^{21};

• Any deformed boundary operator f RB induced as a dual by W x u ( x ) on − ∇ f is carried by the connecting orbits of the negative gradient flow W x u ( x ) = { p ∈ W | lim t → − ∞ ϕ ( p , t ) = x } , (see previous subsection) from the critical points of W x u ( x ) of index p down to those of index ( index ( x ) ).

In this way, Witten’s deformed cohomology, W x s ( x ) , generated by the deformed Laplacian, − ∇ f , induces its dual, the deformed homology W x s ( x ) = { p ∈ W | lim t → ∞ ϕ ( p , t ) = x } , of the Red-Blue systems-battlespace cobordism n-manifold W.

Modern warfare, compared with its historical precedents, is marked by a shift from large-scale annihilation along defined fronts, and relatively little regard for neutral parties caught in the situation, to aims of causing system failure that undercuts an opposition’s ability or willingness to fight, simultaneous conflict occurring across multiple domains without definable lines, and foundational international and national legal and social expectations about human, environmental and social consequences of armed conflict. Indeed, in contemporary conflict, social and humanitarian concerns can often both motivate confrontation and decide operational success. Arguably, this shift has been driven by a complex interwoven web of technological developments, social change, and legal, moral and ethical constraints, which first came to the fore in the modern sense during the soul-searching in post-Napoleonic Europe that simultaneously yielded the basis for both the modern professional military force and international humanitarian law. The combined effect of these factors is extreme nonlinearity, which makes approaches to modeling war and battle that represent simple attrition largely obsolete.

In this paper, we have extended the previously developed kinetic TCW-framework, to include non-kinetic effects, by addressing the general systems-confrontation, which means that our modeling of armed conflict includes interaction not only in the traditional physical Air-Land-Sea domains, but also in non-physical cyberspace, electromagnetic, psychological and social-network domains. In addition, we extend the TCW framework with the ability to represent “Green” neutral parties as richly as the main “Blue” and “Red” adversaries, and extend this to many factions, including coalition partners in Blue and Red and factions within Green, or even to situations with three or more main adversaries. In our formulation, Green may hold the ability to decide operational success from conflict between Blue and Red. This paper attempts to address this generic scenario representative of modern war and battle conditoins using rigorous methods and techniques from modern topology, specifically, by extending the kinetic Red-Blue scenario into this more general kinetic + non-kinetic Red-Blue-Green scenario. In particular, we have focussed here on the question of dramatic changes in the topology of the systems-battlespace, which appears as non-equilibrium phase transitions occurring at the battlefield at various stages of warfare, and is usually superficially characterized by sudden entropy growth. Such sudden changes have been long recognised as central features of war and battle; we thus have new modeling machinery with which to study their occurrence and effects.

We have performed a two-level topological analysis of the systems-battlespace. We have started gently with a largely intuitive analysis of the systems-battlespace topology using visual cobordisms and Morse functions. Then, we performed a rigorous topological analysis of the systems-battlespace by deriving its (co)homological invariants. Specifically, we derived the Morse-Smale homology and the Morse-Witten cohomology of the systems-battlespace manifold. All the necessary geometrical and topological background is given in the self-content and comprehensive Appendix, which provides the Hodge-de Rham theory based on the Stokes theorem.

The authors are grateful to Dr. Tim McKay, Joint and Operations Analysis Division, Defence Science & Technology Group, Australia-for his support the research work presented in this paper.

The authors declare no conflicts of interest regarding the publication of this paper.

Ivancevic, V., Pourbeik, P. and Reid, D. (2019) Tensor-Centric Warfare V: Topology of Systems Confrontation. Intelligent Control and Automation, 10, 13-45. https://doi.org/10.4236/ica.2019.101002

Here we give a brief introduction to the Stokes-de Rham theory on arbitrary smooth manifolds, followed by its extension, the Hodge theory on Riemannian manifolds, all three standing at the crossroads of differential geometry, algebraic topology and modern physics, thus enriching all three disciplines (see [

At the core of differential geometry (and its application to algebraic topology) lies the celebrated Stokes theorem. This fundamental result of modern mathematics (see, e.g. [^{22}, defined in the Euclidean f RB -plane g ∈ W (via two smooth functions W f RB , g ( x , y ) = W g u ( x ) ∩ W g s ( y ) ) as:

1-form: ∂ , and

2-form: ∂ : C λ ( f RB ) → C λ − 1 ( f RB ) , such that : ∂ x = ∑ y ∈ Crit λ − 1 ( f RB ) n ( x , y ) y ,

are related by the Green theorem in the closed region n ( x , y ) with the boundary W f RB , g ( x , y ) / ℝ ^{23}:

∂ 2 = ∂ ∘ ∂ = 0

which can be rewritten as the Stokes theorem:

H λ Morse ( f RB ) = Ker ( ∂ ) / Im ( ∂ ) , (4)

( f RB α , f RB β )

The integrands Ω p ( W ) ≃ H p ( W ) ≃ H d p ( W ) ≃ H p ( W , ℝ ) . and Δ = d δ + δ d in the Stokes theorem (4) are the special 1D and 2D cases of general exterior differential p-forms, which are completely antisymmetric covariant tensors of rank p in Δ d (for δ ). Their “exterior calculus” can be introduced in the following “way of physics” where the most frequently used Euclidean Δ d = d δ d + δ d 2 = d δ d + d 2 δ = d Δ . space is d λ . Here in Ω λ ∗ ( W ) = ⊕ p = 1 n Ω λ p ( W ) , given the frame: λ ^{25} and its dual coframe: Ω λ ∗ : 0 → Ω λ 0 ( W ) → d λ Ω λ 1 ( W ) → d λ ⋯ → d λ Ω λ n ( W ) → 0. , we can define the vector space of all p-forms, denoted Ω λ ∗ for ω ∈ Ω λ p ( W ) , using the exterior derivative operator, d λ , which is governed by the BBZ closure-property: d λ ; so that we have the following four p-forms (defined using Einstein’s summation convention over repeated indices d λ ω = 0 ):

1-form-generalizing Green’s 1-form ω = 1 λ λ ω = 1 λ Δ ω = 1 λ ( d δ + δ d ) ω = 1 λ d δ ω = d ( 1 λ δ ω ) . :

Δ

For example, in 4D electrodynamics, δ represents the electromagnetic (co)vector potential.

2-form-generalizing Green’s 2-form d ( 1 λ δ ω ) ∈ Ω λ p ( W ) :

Ω λ ∗ ,

with components:

Ω λ ∗ ,

or

Ω a ∗ ( W ) = ⊕ λ ≤ a n Ω λ ∗ ( W ) ,

H p (W)

so that

Δ t

where Ω a ∗ ( t , W ) = ⊕ λ ≤ a Ω λ p ( t , W ) represents the exterior product^{26}.

3-form

Δ t ,

with components:

λ ≤ a ,

or

Ω a ∗ ( t , W ) ,

so that

Δ t ,

where H t p ( W ) is the skew-symmetric part of H p t ( W ) .

For example, in the 4D electrodynamics, t → ∞ represents the field strength 2-form Faraday (usually denoted by Ω a ∗ ( ∞ , W ) ), which satisfys the sourceless magnetic Maxwell’s equation,

Bianchi identity: f RB , in components:

f RB ,

where the square bracket dim [ Ω a p ( ∞ , W ) ] denotes the antisymmetric part of the covariant tensor f RB :

f RB

4-form

∂ t ,

with components:

d t ,

or

Ω a p ( ∞ , W ) ,

so that

− ∇ f RB

These are all possible p-forms in f RB and p + 1 is called the top-ranked form.

Generalization to higher-dimensions is straightforward: for H t p ( W ) , we have the Kaluza-Klein-type Euclidean space Δ t = d t δ t + δ t d t , in which the top-ranked form is:

H p t (W)

with components:

A ,

or

d A ,

so that

( x , y ) ,

etc.

In such a way introduced exterior calculus of p-forms enables generalization of the Green theorem (and all other integral theorems from vector calculus) to the general Stokes theorem for any p-form ℝ 2 , defined in an oriented domain C in the Euclidean space P , Q ∈ ℝ 2 as:

A = P d x + Q d y (5)

Furthermore, a nonlinear generalization of the Stokes theorem (5) to any oriented smooth manifold provides the general machinery for integration on smooth manifolds. It is based on the fundamental de Rham’s duality between p-forms and p-chains, described in the dual language of (co)cycles and (co)boundaries, as follows.

Notation change: to improve the flow of the paper, we drop boldface letters from now on.

On a smooth n-manifold M, a cycle is a finite p-chain^{27} d A = ( ∂ Q ∂ x − ∂ P ∂ y ) d x d y such that C ∈ ℝ 2 and a boundary is a p-chain B such that ∂ C for some (p + 1)-chain ∮ ∂ C P d x + Q d y = ∬ C ( ∂ Q ∂ x − ∂ P ∂ y ) d x d y , . Its dual, a cocycle (i.e., a closed form) is a p-cochain ∫ ∂ C A = ∫ C d A such that A and a coboundary^{28} (i.e., an exact form) is a p-cochain ℝ n such that A , for some (p − 1)-cochain d A . All exact forms are closed, i.e., all coboundaries are cocyles ( ℝ n ) and all boundaries are cycles ( p ≤ n ). Converse is true only locally (by the Poincaré lemma^{29}); it holds globally only for contractible manifolds (including ℝ n and star-shaped spaces).

Integration on a smooth manifold M should be thought of as a nondegenerate bilinear pairing ℝ 4 between p-forms and p-chains (spanning a finite domain on M). The duality of p-forms and p-chains on M is based on the de Rham period, the ℝ 4 -pairing:

{ ∂ i }

{ d x i }

where C is a cycle, C i j k = − 6 ∂ k B [ i j ] is a cocycle, and B [ i j ] is their inner product B i j (see [

Naturally, this fundamental topological duality is rooted in the Stokes theorem (5), as:

d B = 0 , symbolically written as: ∂ k B [ i j ] = 0 (6)

where [ i j ] is the boundary of the p-chain C oriented on M coherently with C. While the boundary operator B i j is a global operator, the coboundary operator d is local, and thus more suitable for applications. The dual BBZ-closure property:

B [ i j ] = 1 2 ( B i j − B j i ) .

is proved using the Stokes’ theorem (6), in period notation as:

D = d C ( = d d B ≡ 0 ) ∈ Ω 4 (ℝ4)

or, in integral notation as:

D = 1 4 ! D i j k l d x i ∧ d x j ∧ d x k ∧ d x l

A.2. De Rham’s (Co)chain Complex and (Co)homologyIn the Euclidean 3D space D = ∂ l C [ i j k ] d x l ∧ d x i ∧ d x j ∧ d x k we have the following short exact sequence^{30}, called short de Rham cochain complex:

D i j k l = − 24 ∂ l C [ i j k ] .

ℝ 4

Using the BBZ-closure property: ℝ n , we obtain the standard identities from vector calculus:

∫ ∂ C w = ∫ C d w .

As a duality in C ∈ C p ( M ) , we also have another short exact sequence, called short chain complex:

∂ C = 0

Its own BBZ-closure property: B = ∂ C implies the following three boundaries:

C ∈ C p + 1 (M)

where ω ∈ Ω p ( M ) is a 0-boundary (or, a point), d ω = 0 is a 1-boundary (or, a line), ω is a 2-boundary (or, a surface), and ω = d θ is a 3-boundary (or, a hypersurface). Similarly, the de Rham complex implies the following three coboundaries:

θ ∈ Ω p − 1 (M)

where ω = d θ ⇒ d ω = d d θ = 0 is 0-form (or, a function), B = ∂ C ⇒ ∂ B = ∂ ∂ C = 0 is a 1-form, ℝ n is a 2-form, and 〈 , 〉 is a 3-form.

These two short (co)homological constructions are, according to de Rham [

• The de Rham cochain complex 〈 cycle , cocycle 〉 given by (see [

Period : = ∫ C ω : = 〈 C , ω 〉 ,

satisfying the closure property on M: ω , where 〈 C , ω 〉 = ω ( C ) is the vector space over 〈 C , ω 〉 : Ω p ( M ) × C p ( M ) → ℝ of all finite cochains ω on the manifold M and 〈 C , ω 〉 = 0 .

• The chain complex ∫ ∂ C ω = ∫ C d ω given by (see [

〈 ∂ C , ω 〉 = 〈 C , d ω 〉

∂ C

satisfying the closure property on M: ℝ , where ω is the vector space over d n = d : Ω n ( M ) → Ω n + 1 ( M ) of all finite chains C on the manifold M and A • .

The de Rham cochain complex A • : 0 ← C 0 ( M ) ← ∂ C 1 ( M ) ← ∂ C 2 ( M ) ← ∂ C 3 ( M ) ← ∂ ⋅ ⋅ ⋅ ← ∂ C n ( M ) ← 0, generates the de Rham cohomology, the functional space of closed p-forms modulo exact (p-1)-forms on a smooth manifold. More precisely, the subspace of all closed p-forms (or, cocycles) on a smooth manifold M, denoted by ∂ ∘ ∂ ≡ ∂ 2 = 0 , is the kernel, A n = C n ( M ) ∈ A • , of the exterior derivative d-operator (also called the de Rham d-homomorphism^{31}); the sub-subspace of all exact p-forms (or, coboundaries) on M is the image, ℝ , denoted by ∂ n = ∂ : C n + 1 ( M ) → C n ( M ) . The quotient vector space^{32} A • , defined as^{33}:

Z d p ( M ) ⊂ Ω p ( M ) (7)

is called the pth de Rham cohomology group of a manifold M, which is a topological invariant of M. Two p-cocycles Ker ( d ) are cohomologous, or belong to the same cohomology class, Im ( d ) , if they differ by a (p − 1)-coboundary, B d p ( M ) ⊂ Z d p ( M ) . The dimension Ker ( d ) / Im ( d ) of the de Rham cohomology group H d p ( M ) : = Z d p ( M ) B d p ( M ) = { p -cocycles } { ( p − 1 ) -coboundaries } = Ker [ d : Ω p ( M ) → Ω p + 1 ( M ) ] Im [ d : Ω p − 1 ( M ) → Ω p ( M ) ] , of the manifold M is called the Betti number α , β ∈ Ω p ( M ) .

Its dual, the chain complex [ α ] ∈ H d p ( M ) generates the chain homology, the functional space of p-cycles modulo (p + 1)-boundaries on a smooth manifold. The subspace of all p-cycles on a smooth manifold M is the kernel, α − β = d θ ∈ Ω p − 1 ( M ) , of the b p = dim [ H d p ( M ) ] -operator, denoted by H d p ( M ) , and the sub-subspace of all p-boundaries on M is the image, b p , of the A • -operator (also called the Ker ( ∂ ) -homomorphism), denoted by ∂ . Two p-cycles Z p ∂ ( M ) ⊂ C p ( M ) are homologous, if they differ by a (p + 1)-boundary Im ( ∂ ) . Then ∂ and ∂ belong to the same homology class, B p ∂ ( M ) ⊂ C p ( M ) , where C 1 , C 2 ∈ C p is the homology group of the manifold M, the quotient vector space C 1 − C 2 = ∂ B ∈ C p + 1 ( M ) , defined as:

C 1

C 2

where b p is the vector space of cycles and χ ( M ) = ∑ p = 0 n ( − 1 ) p b p . is the vector space of boundaries on M. The dimension H d p ( M ) of the dual homology group K is, by the de Rham theorem^{34}, the same Betti number d = ∂ + ∂ ¯ .

If we know the Betti numbers for all (co)homology groups of the manifold M, then we can calculate the Euler-Poincaré characteristic of M as:

H ∂ ¯ p , q (K)

The de Rham cohomology (7) serves as the “model” for all other cohomologies (see [^{35}, (see Appendix in [

α

A.3. Hodge Theory BasicsSpecialization of the exterior Stokes-de Rham theory, from arbitrary smooth manifolds to a compact (i.e., closed and bounded), oriented Riemannian n-manifold M with the metric tensor ( n − p ) ^{36}, enables definition of the Hodge operators (star, inner product, codifferential, Laplacian and adjoints) and the subsequent formulation of the Hodge decomposition theorem, as follows.

Hodge star ⋆ α . The Hodge star operator { e i d x i } ∈ M maps any p-form ⋆ ( e i ∧ e j ) = e k , ( ⋆ ) 2 = 1. into its dual ⋆ -form ∧ on an n-manifold M. It is the linear operator defined locally in the coframe g = g i j as:

α , β ∈ Ω p (M)

The star ⋆ commutes with the exterior product ⋆ ⋆ α = ( − 1 ) p ( n − p ) α and depends on the Riemannian metric α ∧ ⋆ α = 0 ⇒ α ≡ 0 on M, as well as on the orientation (reversing orientation would change the sign). For any two p-forms ⋆ ( c 1 α + c 2 β ) = c 1 ( ⋆ α ) + c 2 ( ⋆ β ) , the star α ∧ ⋆ β = β ∧ ⋆ α is defined by the following four properties [

• 〈 α , β 〉 μ ;

• μ ;

• { x i } ; and

• μ = det ( g i j ) d x 1 ∧ ⋯ ∧ d x n = ⋆ ( 1 ) , , which can be written as a pairing: vol ( M ) .

Here, vol ( M ) = ∫ M ⋆ ( 1 ) . is the volume form, defined in local coordinates ℝ 3 on an n-manifold M as:

( x , y , z ) (8)

so that the total volume, ⋆ d x = d y ∧ d z , ⋆ d y = d z ∧ d x , ⋆ d z = d x ∧ d y , on M, is given by:

ℝ 3

⋆ F

For example, in Euclidean ( α , β ) space with global Cartesian ( α , β ) = ∫ M α ∧ ⋆ β = ∫ M 〈 α , β 〉 ⋆ ( 1 ) = ∫ M β ∧ ⋆ α = ( β , α ) , coordinates, we have:

( α , α ) ≥ 0 and ( α , α ) = 0 iff α = 0 ; ( ⋆ α , ⋆ β ) = ( α , β ) .

so that the Hodge dual here corresponds to the standard cross-product in Ω p ( M ) .

Also, in 4D electrodynamics (expanded below), the 2-form Faraday F has the dual 2-form Maxwell α ∈ Ω p ( M ) (see [

dual Bianchi identity: ‖ α ‖ 2 = ( α , α ) = ∫ M α ∧ ⋆ α = ∫ M 〈 α , α 〉 ⋆ ( 1 ) , ,

where Δ α = 0 is the 3-form dual of the current 1-form J.

Hodge inner product. For any two p-forms F = d A with compact support on an n-manifold M, the bilinear, positive-definite and symmetric Hodge A = A i d x i -inner product L ( A ) = 1 2 ( F ∧ ⋆ F ) is defined as:

S ( A ) = 1 2 ∫ F ∧ ⋆ F (9)

L 2

Thus, operation (9) turns the space S ( A ) = 1 2 ( F , F ) into an infinite-dimensional inner-product space. From (9) it follows that for every p-form δ we can define the norm functional:

d : Ω p ( M ) → Ω p + 1 (M)

for which the Euler-Lagrange equation becomes the Laplace equation: δ : Ω p ( M ) → Ω p − 1 ( M ) .

For example, the free Maxwell electromagnetic field, δ = ( − 1 ) n ( p + 1 ) + 1 ⋆ d ⋆ , so that d = ( − 1 ) n p ⋆ δ ⋆ . (where δ = − ⋆ d ⋆ is the electromagnetic potential 1-form) has the standard Lagrangian (see, e.g. [

ω ∈ Ω p ( M ) , with the corresponding action: δ ,

which can be rewritten, using the Hodge δ ω = ( − 1 ) n ( p + 1 ) + 1 ⋆ d ⋆ ω , δ d ω = ( − 1 ) n p + 1 ⋆ d ⋆ d ω . -inner product (9), as:

ω = f .

Hodge codifferential δ f = 0 .

The Hodge dual (or, formal adjoint) to the exterior derivative α on a Riemannian n-manifold M is the linear codifferential operator β , a generalization of the standard divergence, defined by [

α = δ β

That is, if the dimension n of the manifold M is even, then β .

Applied to any p-form α , the codifferential δ α = 0 gives:

⋆ α

If d ⋆ α = 0 is a 0-form (i.e., a scalar function) then δ . If a p-form δ ∘ δ = δ δ = δ 2 = 0 is a codifferential of a (p + 1)-form d ∘ d = d d = d 2 = 0 , that is δ ⋆ = ( − 1 ) p + 1 ⋆ d , then ⋆ δ = ( − 1 ) p ⋆ d is called the coexact form. A p-form d δ ⋆ = ⋆ δ d is called coclosed if ⋆ d δ = δ d ⋆ ; then α is closed (i.e., α ∈ Ω p ( M ) ) and conversely.

The Hodge codifferential β satisfies the following three rules:

• β ∈ Ω p + 1 ( M ) , the same as: L 2 ;

• ( d α , β ) = ( α , δ β ) . ; U ( 1 ) ;

• A = A μ d x μ = A μ d x μ + d f ; F = d A .

In addition, if F = 1 2 F μ ν d x μ ∧ d x ν , with F μ ν = ∂ ν A μ − ∂ μ A ν . is a p-form (i.e., d F = 0 and δ F = − 4 π J ) and F [ μ ν , η ] = 0 and F μ ν , μ = − 4 π J μ , is a (p + 1)-form (i.e., J = J μ d x μ ) then the following identity holds for the Hodge δ J = 0 -inner product:

J μ , μ = 0

The standard application of (co)differentials is classical electrodynamics, in which the gauge field is an electromagnetic potential 1-form (which is a connection on a Δ -bundle):

δ , (f is an arbitrary scalar field);

with the corresponding electromagnetic field 2-form (the curvature of the connection A) Δ : Ω p ( M ) → Ω p ( M ) , in components given by (see Appendix A.1)

Δ = δ ∘ d + d ∘ δ = δ d + d δ = ( d + δ ) 2 .

Electrodynamics is governed by the Maxwell equations^{37}, which in exterior formulation read:

Δ

which in tensor components reads:

δ Δ = Δ δ = δ d δ ;

where the comma-subscript denotes the partial derivative and the electric current 1-form d Δ = Δ d = d δ d ; is conserved, by the electric continuity equation:

⋆ Δ = Δ ⋆ . , in components: d t .

Hodge Laplacian δ t .

e t f

The codifferential { d x j } can be coupled with the exterior derivative d to construct the Hodge Laplacian operator t → ∞ , which is a harmonic generalization of the Laplace-Beltrami operator^{38}, given by^{39}:

d f = 0 (10)

The Laplacian Δ t satisfies the following three rules:

• t → ∞

• α

• Δ α = 0 ⇔ ( d α = 0 , δ α = 0 )

We remark here that Ed Witten considered in [

( β , β ) ≥ 0 , with adjoints: β , (11)

and deformed Laplacian: ( δ α , δ α ) .

For ( d α , d α ) , d α = 0 is the Hodge Laplacian (10), whereas for δ α = 0 , one has the following expansion (in a flat neighborhood on an oriented compact Riemannian manifold M with local coordinates H Δ p ( M ) ):

ℝ 3 (12)

where → represents the Hessian of the Morse function f and → is

the commutator of the frame ⋆ d and the coframe → ^{40} in M. This becomes very large for δ , except at the critical points of f, i.e., where → . Therefore, the eigenvalues of Δ will concentrate near the critical points of f for → , and we get an interpolation between de Rham cohomology and Morse cohomology. Witten’s deformation is considered in the subsection 3.2 above.

A p-form Δ is called harmonic iff: α = d β .

Thus, ω = δ β is harmonic in a compact domain L 2 iff Δ γ = 0 is both closed and coclosed in D. Every harmonic form is both closed and coclosed; as a proof, we have:

α

Since β for any form ( d α , β ) = ( α , δ β ) and ( δ α , β ) = ( α , d β ) . then δ and μ must vanish separately;

Hence, d μ = 0 and ∫ M d ( α ∧ ⋆ β ) = 0 ^{41}.

All harmonic p-forms on a smooth manifold M form the vector space Δ .

As an example, to translate the notions from standard vector calculus in ( Δ α , β ) = ( α , Δ β ) , we first identify scalar functions with 0-forms, field intensity vectors with 1-forms, flux vectors with 2-forms and scalar densities with 3-forms. Then we have the following correspondence:

Grad ( d α , d β ) + ( δ α , δ β ) d: on 0-forms; curl ( Δ α , α ) ≥ 0 ( Δ α , α ) = 0 : on 1-forms;

Div Δ α = 0 Δ : on 1-forms; div grad n ≥ p ω ∈ Ω p ( M ) : on 0-forms;

Curl curl-grad div α ∈ Ω p − 1 ( M ) β ∈ Ω p + 1 ( M ) : on 1-forms.

γ ∈ Ω p (M)

We remark here that exact and coexact p-forms ( ω and d ω = 0 ) are mutually orthogonal with respect to the δ β -inner product (9). The orthogonal complement consists of forms that are both closed and coclosed, i.e., of harmonic forms ( ω = d α + γ ).

Hodge adjoints and self-adjoints. If ω is a p-form and γ is a (p + 1)-form then we have [

d α (13)

This relation is usually interpreted as saying that the two exterior differentials, d and ω , are mutually adjoint (or, dual). This identity follows from the fact that for the volume form γ given by (8) we have [ ω ] ∈ H p ( M ) and thus: ∫ C ω = ∫ C γ ⇔ 〈 C , ω 〉 = 〈 C , γ 〉 , .

Relation (13) also implies that the Hodge Laplacian γ is self-adjoint (or, self-dual), formally: ω , which is obvious, since either side is . . Furthermore, since ω , with γ only when ω , the Laplacian [ ω ] ∈ H p ( M ) is a positive-definite, self-adjoint elliptic operator.

Hodge decomposition theorem. Now we have all the necessary ingredients to formulate the celebrated Hodge decomposition theorem (HDT), which states: on a compact orientable Riemannian n-manifold M, any exterior p-form (with γ ∈ H Δ p ( M ) ) can be written as a unique sum of an exact form, a coexact form, and a harmonic form. Formally, for any p-form H Δ p ( M ) , there is a unique exact (p − 1)-form H Δ p ( M ) ≅ H d p ( M ) , a unique coexact (p + 1)-form γ and a harmonic p-form f RBG , such that:

∇ f RBG = 0

For the proof, see [

In physics community, the exact form M Red is called longitudinal, while the coexact form M Blue is called transversal, so that they are mutually orthogonal. Thus any form can be orthogonally decomposed into a sum of: 1) a harmonic form, 2) a longitudinal form, and 3) a transversal form^{42}.

Since T M Red is harmonic, T M Blue . Also, by Poincaré lemma, n = 30 . In case ⊙ is a closed p-form: f RB : W → ℝ , then the coexact term p ∈ W in HDT is absent, so we have the short Hodge decomposition: f RB ( p ) = c , hence f RB − 1 ( p ) = { p ∈ W ( M Red , M Blue ) : f RB ( p ) = c } . and μ ( W ) differ by C λ . In de Rham’s terminology, S 2 and μ ( S 2 ) = 2 belong to the same cohomology class S 1 × I . Now, by the de Rham theorems it follows that if C is any p-cycle, then we have:

μ ( S 1 × I ) = 0

that is, T 2 and μ ( T 2 ) = 4 have the same periods f RBG : W → ℝ More precisely, if p ∈ W is any closed p-form, then there exists a unique harmonic p-form f RB ( p ) = c with the same periods as those of f RBG − 1 ( p ) = { p ∈ W ( M Red , M Blue , M Green ) : f RBG ( p ) = c } . (see [

W u (x)

Our final statement in this section is the Hodge-Weyl theorem [