Betti and Tachibana numbers

ISSN 0001-4346, Mathematical Notes, 2014, Vol. 95, No. 6, pp. 856–864. © Pleiades Publishing, Ltd., 2014.Original Russian Text © S. E. Stepanov, 2014, published in Matematicheskie Zametki, 2014, Vol. 95, No. 6, pp. 926–936.

Betti and Tachibana Numbers

S. E. Stepanov*

Financial University under the Government of the Russian Federation,Moscow, Russia

Received May 4, 2012; in final form, March 11, 2013

Abstract—The Tachibana numbers tr(M), the Killing numbers kr(M), and the planarity numberspr(M) are considered as the dimensions of the vector spaces of, respectively, all, coclosed, andclosed conformal Killing r-forms with 1 ≤ r ≤ n− 1 “globally” defined on a compact Riemanniann-manifold (M, g), n ≥ 2. Their relationship with the Betti numbers br(M) is investigated. Inparticular, it is proved that if br(M) = 0, then the corresponding Tachibana number has theform tr(M) = kr(M) + pr(M) for tr(M) > kr(M) > 0. In the special case where b1(M) = 0and t1(M) > k1(M) > 0, the manifold (M, g) is conformally diffeomorphic to the Euclidean sphere.

DOI: 10.1134/S0001434614050307

Keywords: compact manifold, Tachibana number, Killing number, planarity number, Bettinumber, conformal Killing form, conformal Killing (co)closed form.

1. INTRODUCTION AND RESULTS

It is known (see [1, Note 11]) that the Lie algebra S(M,g) of the group C(M,g) of infinitesimalconformal transformations of a connected Riemannian n-manifold (M,g), n ≥ 3, can be realized as thespace of conformal Killing vector fields and has dimension dimS(M,g) ≤ (1/2)(n + 1)(n + 2). Theequality is attained for a conformally flat Riemannian manifold (M,g).

A generalization of conformal Killing vector fields are the conformal Killing r-forms, 1 ≤ r ≤ n− 1,or conformal Killing–Yano tensors of rank r. For r = 1, such forms are dual to conformal Killingvector fields. During the whole almost half-century history, beginning with Tachibana’s paper [2] andending with recent works [3]–[5], conformal Killing forms have caused extensive interest of researchers,partly because of their numerous physical applications [6], [7], [8, pp. 414, 426], [9].

In particular, it was proved in [4] that the dimension tr of the space of conformal Killing r-forms ona connected n-manifold (M,g) for 1 ≤ r ≤ n− 1 is subject to the constraint

tr ≤(n+ 2)!

(r + 1)!(n − r + 1)!,

and that the equality is attained for conformally flat Riemannian manifolds.Let us refine this result. Consider the hyperbolic n-space, n ≥ 2, which is a Riemannian manifold

of constant negative curvature. As shown later on, it follows from the assertions stated above that,in this space, the dimension tr of the space of conformal Killing r-forms, 1 ≤ r ≤ n− 1, is equalto (n+ 2)!/((r + 1)!(n − r + 1)!), and the dimension of the space of conformal Killing vector fieldsis (1/2)(n + 1)(n + 2). The quotient of the hyperbolic space by an appropriate discrete group ofmotions [10, Sec. 4], [11, Chap. 8, Sec. 12], [12, Chap. 2, Sec. 2.4] is a compact manifold of constantnegative curvature. On such a manifold, there exist no nonzero globally defined conformal Killingforms [13] and, in particular, conformal Killing vector fields [14, Chap. II, Sec. 12].

Thus, the known statements are concerned with the dimensions of the vector spaces of conformalKilling r-forms, 1 ≤ r ≤ n− 1, and vector fields defined in a neighborhood of an arbitrary point of theRiemannian manifold (M,g).

*E-mail: [email protected]

856

BETTI AND TACHIBANA NUMBERS 857

This paper studies the dimension of the space of conformal Killing r-forms, 1 ≤ r ≤ n− 1, and, inparticular, the space of conformal Killing vector fields globally defined on a compact (without boundary)Riemannian n-manifold (M,g), n ≥ 2. In [13], this dimension was called the Tachibana number tr(M),by analogy with the Betti number br(M) of the compact manifold (M,g), which equals the dimension ofthe space of harmonic r-forms on (M,g) [15, Chap. 4, Sec. 2]. Moreover, as shown in [16], the Tachibananumbers have the duality property tr(M) = tn−r(M), which is an analog of the Poincare duality for theBetti numbers br(M) = bn−r(M), and are scalar conformal invariants of the manifold (M,g).

The group I(M,g) of infinitesimal isometries is a subgroup of the group C(M,g). Its Lie algebraT(M,g) has dimension at most n(n+ 1)/2 and can be realized as a space of Killing vector fields [1,Chap. VI, Sec. 3]. These fields can be defined as coclosed conformal Killing vector fields.

In turn, closed conformal Killing vector fields form the space £(M,g) of concircular vectorfields [17], which has dimension dim£(M,g) ≤ n+ 1.

The equalities in the two estimates of dimensions given above are attained for a manifold (M,g) ofconstant curvature, where

dimC(M,g) = dimT(M,g) + dim£(M,g). (1.1)

A generalization of Killing vector fields is Killing r-forms, 1 ≤ r ≤ n− 1, i.e., coclosed conformalKilling r-forms, or Killing–Yano tensors of rank r [14, Chap. III, Sec. 3]; [18, Chap. 31, Sec. 3].For r = 1, such forms are dual to Killing vector fields. In turn, a generalization of concircular vectorfields is planar r-forms, 1 ≤ r ≤ n− 1, i.e., closed conformal Killing r-forms, or closed conformalKilling–Yano tensors of rank r [3], [9]. For r = 1, such forms are dual to concircular vector fields.

It was proved in [19] and [20] that, on a connected Riemannian n-manifold (M,g), the dimensions krand pr of the spaces of Killing and planar r-forms, 1 ≤ r ≤ n− 1, have the bounds

kr ≤(n+ 1)!

(r + 1)!(n − r)!, pr ≤

(n+ 1)!

r! (n− r + 1)!. (1.2)

Moreover, the equality is attained only for a Riemannian manifold of constant curvature, on which, ascan be verified directly,

tr = kr + pr. (1.3)

Inequalities (1.2) and equality (1.3) are of local character.In [13], the dimensions of the spaces of Killing and planar r-forms, 1 ≤ r ≤ n− 1, globally defined

on a connected compact (without boundary) Riemannian n-manifold (M,g), n ≥ 2, were called Killingnumbers kr(M) and planarity numbers pr(M), respectively. As shown in [3], [13], and [16], thesenumbers have the following duality property: kr(M) = pn−r(M) and are scalar projective invariants ofthe manifold (M,g).

The following theorem determines a relationship between the numbers tr(M), kr(M), and pr(M)under the assumption that the corresponding Betti number br(M) vanishes.

Theorem 1. Let (M,g) be a connected compact oriented Riemannian n-manifold. Suppose thatthe Betti number br(M) vanishes for some 1 ≤ r ≤ n− 1 and the corresponding Tachibana numbertr(M) and Killing number kr(M) are related by the inequalities tr(M) > kr(M) > 0. Then thedifference tr(M)− kr(M) is equal to the planarity number pr(M) of the given manifold.

In the case r = 1, a similar assertion was proved in [21] for the sphere Sn in Euclidean space R

n+1.We supplement this assertion by the following statement.

Corollary 1. Suppose that, for a connected compact oriented Riemannian n-manifold (M,g), thefirst Betti, Tachibana, and Killing numbers satisfy the conditions

b1(M) = 0 and t1(M) > k1(M) > 0.

Then the manifold (M,g) is conformally diffeomorphic to the hypersphere Sn in Euclidean space

Rn+1 and, therefore,

b2(M) = · · · = bn−2(M) = 0.

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858 STEPANOV

The duality property of the Betti and Killing numbers and the planarity property imply the followingassertion.

Corollary 2. Suppose that, for a connected compact oriented Riemannian n-manifold (M,g),the Betti number bn−1(M) vanishes and the Killing number kn−1(M) does not vanish. Then themanifold (M,g) is conformally diffeomorphic to the hypersphere S

n in Euclidean space Rn+1

and, therefore,

b2(M) = · · · = bn−2(M) = 0.

The following statement supplements the “existence theorems” for the Tachibana numbers tr(M)known from [13].

Theorem 2. Let (M,g) be a connected compact oriented Riemannian n-manifold, n ≥ 2. Supposethat one of the following two conditions holds:

1) the Ricci tensor Ric is nonpositive and the first Tachibana number t1(M) is equal to p �= 0;

2) the Ricci tensor Ric is nonnegative and the first Betti number b1(M) is equal to p �= 0.

Then the Tachibana and Betti numbers satisfy the relations

tr(M) = br(M) ≥ p!

r! (p− r)!for all r ≤ p.

Finally, consider conformally flat Riemannian manifolds mentioned at the beginning of the paper inrelation to the dimension of the space of conformal Killing forms. First, we recall the statement, whichhas already become classical, that the Betti numbers b1(M), . . . , bn−1(M) of a compact Riemannianmanifold (M,g) with positive curvature operator vanish [22, p. 212]. In [13], we supplemented thisstatement by the “dual theorem” about the vanishing of the Tachibana numbers t1(M), . . . , tn−1(M) ofa compact Riemannian manifold (M,g) with negative curvature operator. The following theorem is ananalog of these two statements for a compact conformally flat Riemannian manifold.

Theorem 3. Let (M,g) be a connected compact oriented conformally flat Riemannian n-manifold, n ≥ 2, with curvature of definite sign. Then the Tachibana numbers tk(M) and theBetti numbers bl(M) satisfy the condition

tk(M) · bl(M) = 0 for all k, l = 1, . . . , n− 1.

In particular, if (M,g) is a connected compact oriented conformally flat Riemannian 2r-manifoldwith nonvanishing scalar curvature s, then

tr(M) · br(M) = 0.

We conclude the introduction where we started, namely, by a bound for the dimension of the space ofconformal Killing forms. The following theorem is valid.

Theorem 4. If the first Betti number b1(M) of a connected compact conformally flat Riemanniann-manifold (M,g) vanishes, then

tr(M) =(n + 2)!

(r + 1)!(n − r + 1)!and b2(M) = · · · = bn−2(M) = 0.



2. PROOFS OF THEOREM 1 AND COROLLARIES 1 AND 2

Consider a compact Riemannian n-manifold (M,g), n ≥ 2, with Levi–Civita connection ∇. Wedenote the algebra of C∞ functions on M by C∞M and the module of differential r-forms on M overthe algebra C∞M by Ωr(M) [11, Chap. 8, Sec. 10]. The Riemannian metric g on the compact orientedmanifold M makes it possible to endow Ωr(M) with the global inner product

〈ω, ω′〉 =ˆM

1

r!g(ω, ω′) dv (2.1)

for any ω, ω′ ∈ Ωr(M), where dv is the volume element of the manifold (M,g). This makes it possible todefine the codifferentiation operator

d∗ : Ωr+1(M) → Ωr(M)

formally adjoint to the exterior differentiation operator

d: C∞Ωr(M) → C∞Ωr+1(M)

(which means that 〈dω, ω′〉 = 〈ω,d∗ω′〉 for any ω ∈ Ωr(M) and ω′ ∈ Ωr+1(M)) and the operator

∇∗ : Ω1(M)⊗ Ωr(M) → Ωr(M)

formally adjoint to the covariant derivative

∇ : Ωr(M) → Ω1(M)⊗ Ωr(M)

(which means that 〈∇ω, θ〉 = 〈ω,∇∗θ〉 for any θ ∈ Ω1(M)⊗Ωr(M) and ω ∈ Ωr(M)); see [11, Chap. 8,Sec. 10], [22, pp. 203–204], [25, Sec. A, Sec. S].

On the basis of the first-order differential operators defined above, the Hodge–de Rham LaplacianΔ = dd∗ + d∗d and the rough Bochner Laplacian ∇∗∇ are constructed, which are second-orderelliptic differential operators [10, Sec. 7]; [23, Chap. 1, Sec. I]; [24, Report XVI, Sec. 3].

Given a compact Riemannian manifold (M,g), let Ωr(M,R) denote the vector space over the realfield R which consists of differential r-forms, 1 ≤ r ≤ n− 1, defined on (M,g). Then

Hr(M,R) = {ω ∈ Ωr(M,R) | Δω = 0}is the finite-dimensional vector space of harmonic r-forms [11, Chap. 8, Sec. 11]; [22, p. 205]. Itsdimension coincides with the Betti number br(M) of the manifold (M,g), i.e., br(M) = dimRKerΔ.

In [25] (see also [13]), in order to study conformal Killing forms on a compact Riemannian manifold(M,g), we constructed the second-order elliptic differential operator

D∗D =1

r(r + 1)

(∇∗∇− 1

r + 1d∗d− 1

n− r + 1dd∗

)(2.2)

and proved that its kernel is the finite-dimensional vector space

Tr(M,R) = {ω ∈ Ωr(M,R) | D∗Dω = 0}of conformal Killing r-forms, 1 ≤ r ≤ n− 1. The numbers tr(M) = dimRKerD∗D were called theTachibana numbers of the manifold (M,g) in [13] and [16].

The Killing, or coclosed conformal Killing r-forms, 1 ≤ r ≤ n− 1, form the vector space

Kr(M,R) = {ω ∈ Ωr(M,R) | D∗Dω = d∗ω = 0},which is a subspace of Tr(M,R). In [13] and [16], the numbers

kr(M) = dimR(KerD∗D ∩Ker d∗)

were called the Killing numbers of the manifold (M,g).The flat, or closed conformal Killing r-forms, 1 ≤ r ≤ n− 1, constitute the vector space

Pr(M,R) = {ω ∈ Ωr(M,R) | D∗Dω = dω = 0},


860 STEPANOV

which is a subspace of Tr(M,R). In [13] and [16], the numbers

pr(M) = dimR(KerD∗D ∩Ker d)

were called the planarity numbers of the manifold (M,g).

It should be mentioned that the space Kr(M,R) ∩Pr(M,R) consists of parallel or, in other words,covariantly constant r-forms, 1 ≤ r ≤ n− 1; these forms are harmonic [22, p. 212], and hence

Kr(M,R) ∩Pr(M,R) ⊂ Hr(M,R).

Proof of Theorem 1. Recall that, for the compact manifold (M,g), the Hodge–de Rham decomposi-tion [15, Chap. 4, Sec. 2]

Ωr(M) = Imd⊕ Imd∗ ⊕KerΔ (2.3)

orthogonal with respect to the global scalar product (2.1) holds; here

Δ = dd∗ + d∗d, KerΔ = Imd ∩Ker d∗.

In addition to (2.3), the orthogonal decompositions [15, Chap. 4, Sec. 2]

Ker d∗ = Imd∗ ⊕KerΔ, (2.4)

Ker d = Imd⊕KerΔ (2.5)

also hold. According to the assumptions of the theorem, we can assume that, for the compactRiemannian n-manifold (M,g), the Betti number br(M) vanishes for some 1 ≤ r ≤ n− 1. ThenKerΔ = 0 in all decompositions (2.3)–(2.5), and, as a consequence,

Ωr(M) = Imd⊕ Imd∗ (2.6)

for 1 ≤ r ≤ n− 1; moreover,

Ker d∗ = Imd∗, (2.7)

Ker d∗ = Imd∗. (2.8)

Over the compact oriented manifold (M,g), consider the vector space Tr(M,R) of finite nonzerodimension tr(M) with inner product (2.1). By the assumptions of the theorem, Tr(M,R) has anonzero subspace Kr(M,R) of dimension kr(M) satisfying the condition tr(M) > kr(M) > 0. Byvirtue of (2.7), the subspace Kr(M,R) consists of coexact conformal Killing r-forms.

Since tr(M) �= kr(M) > 0, it follows that the vector subspace

Kr(M,R) = {ω ∈ Ωr(M,R) | ω ∈ KerD∗D ∩ Imd∗}

in Tr(M,R) must have a complement orthogonal with respect to the inner product (2.1). Accordingto (2.6) and (2.8), such a complement is the subspace

Pr(M,R) = {ω ∈ Ωr(M,R) | ω ∈ KerD∗ D∩ Imd}of exact conformal Killing r-forms, for which we have

Kr(M,R) ∩Pr(M,R) = {0},

because there are no nonzero harmonic r-forms on (M,g). Therefore, we have the orthogonaldecomposition

Tr(M,R) = Kr(M,R)⊕Pr(M,R). (2.9)

Finally,

tr(M) = kr(M) + pr(M).

This completes the proof of Theorem 1.



Remark 1. In [27], it was proved that, for a compact conformally flat Riemannian 2r-manifold (M,g)with constant scalar curvature s > 0, the orthogonal decomposition

Tr(M,R) = Kr(M,R) +Pr(M,R)

holds. Moreover, according to [14, Chap. IV, Sec. 2], we have br(M) = 0 for such a manifold. In [13], asimilar decomposition was obtained for a compact n-manifold (M,g) with positive curvature operator.As is known [28], such a manifold is diffeomorphic to a spherical space form, and in the case where Mis simply connected, to the sphere S

n in Euclidean space Rn+1. In both of these cases, we have

b1(M) = · · · = bn−1(M) = 0.

Now, let us prove Corollaries 1 and 2.

Proof of Corollary 1. According to Theorem 1, for a compact Riemannian manifold (M,g) withb1(M) = 0 and t1(M) > k1(M) > 0, the decomposition

T1(M,R) = K1(M,R) +P1(M,R) (2.10)

orthogonal with respect to the inner product (2.1) holds. It follows from t1(M) �= k1(M) > 0 thatp1(M) ≥ 1; therefore, the space P1(M,R) contains at least one exact conformal Killing 1-form ω, whichequals grad f for some function f ∈ C∞M satisfying the equations ∇∇f = −(1/n)Δf · g. Accordingto [26], the existence of such a nonconstant function f ∈ C∞M means that the compact manifold (M,g)is conformally diffeomorphic to the sphere S

n in Euclidean space Rn+1.

Proof of Corollary 2. The duality properties imply

b1(M) = bn−1(M) = 0, p1(M) = kn−1(M) �= 0;

hence there exists at least one exact conformal Killing 1-form on (M,g). The rest of the proof repeatsthe concluding part of the proof of Corollary 1.

3. PROOF OF THEOREM 2

Let (M,g) be a compact connected Riemannian n-manifold with t1(M) = p �= 0. Then thereexist p linearly independent globally defined conformal Killing 1-forms ω1, . . . , ωp on (M,g). LetX1, . . . ,Xp denote the conformal Killing vector fields dual to them, i.e., such that ωα(Y ) = g(Y,Xα)for α = 1, . . . , p. Suppose that the Ricci tensor Ric is nonpositive. Then X1, . . . ,Xp are parallel [14,Chap. II, Sec. 12], i.e., ∇X1 = 0, . . . , ∇Xp = 0. It follows that the 1-forms ω1, . . . , ωn are parallel aswell. Therefore, all forms

θi1i2...ir = ωi1 ∧ ωi2 ∧ · · · ∧ ωir for 1 ≤ i1 < · · · < ir ≤ p.

are parallel. These forms are conformal Killing and, simultaneously, harmonic, because any parallelr-form is conformal Killing and harmonic for any r = 1, . . . , n− 1. There are p!/(r! (p − r)!) linearlyindependent forms among the θi1i2...ir . Since any parallel form is completely determined by its initialvalues at some fixed point of the manifold [22, p. 212], it follows that

tr(M) = br(M) ≥ p!

r! (p− r)!for all r ≤ p.

To prove item 2), it suffices to mention a theorem in [22, p. 208], according to which any harmonic1-form on a compact oriented Riemannian manifold with Ric ≥ 0 is parallel.

Remark 2. Adding t1(M) = n to condition (1) in Theorem 2 (where Ric ≤ 0) and b1(M) = n tocondition (2) (where Ric ≥ 0), we obtain, as a corollary, the existence of n linearly independentparallel vector fields on the compact Riemannian n-manifold (M,g). In this case, the manifold (M,g)degenerates into a flat Riemannian torus T n [22, p. 208], for which

tr(Tn) = br(T

n) =n!

r! (n− r)!.


862 STEPANOV


The Bochner and Hodge–de Rham Laplacians are related by the classical Bochner–Weitzenbockformula [10, Sec. 7], [23, Chap. 1, Sec. I]

Δ = ∇∗∇+ Fr, (4.1)

where Fr can be algebraically (and even linearly) expressed in terms of the tensor curvature R of themanifold (M,g). Taking into account (4.1), we obtain a new expression of the differential operator (2.2):

D∗D =1

r(r + 1)

(r

r + 1d∗d +

n− r

n− r + 1dd∗ − Fr

).

Therefore,

〈Fr(ω), ω〉 =r

r + 1〈dω,dω〉+ n− r

n− r + 1〈d∗ω,d∗ω〉 − r(r + 1)〈Dω,Dω〉 (4.2)

for any nonzero r-form ω.Suppose that (M,g) is a compact conformally flat manifold and the quadratic form Ric(X,X) for

the Ricci tensor Ric and any nowhere vanishing vector field X ∈ C∞TM is negative definite. Then,provided that n ≥ 2r, we have [14, Chap. IV, Sec. 2]

g(Fr(ω), ω) ≤ −n− r

n− 1λ · g(ω, ω),

where −λ is the maximum (negative) eigenvalue of the matrix ‖Ric ‖, for all 1 ≤ r ≤ n− 1 and anynonvanishing form ω ∈ Ωr(M). Thus, any conformal Killing r-form ω must satisfy the inequality

−n− r

n− 1λ〈ω, ω〉 ≥ r

(n − r)(r + 1)〈dω,dω〉+ 1

n− r + 1〈d∗ω,d∗ω〉 ≥ 0.

This is possible only if any conformal Killing r-form ω, r ≤ (1/2)n, vanishes at each point of themanifold (M,g). Taking into account the duality property of the Tachibana numbers, we conclude thatt1(M) = · · · = tn−1(M) = 0.

Now, suppose that the quadratic form Ric(X,X) is positive definite for any nowhere vanishing vectorfield X ∈ C∞TM . In this case, we have b1(M) = · · · = bn−1(M) = 0 [14, Chap. IV, Sec. 2]. This provesthe first assertion of Theorem 1.

For a conformally flat 2r-manifold (M,g), an expression for Fr is known [14, Chap. IV, Sec. 2];namely, Fr = s/(2r − 1). Thus, Eq. (2.4) takes the form

1

2(2r − 1)〈sω, ω〉 = r

r + 1〈dω,dω〉+ n− r

n− r + 1〈d∗ω,d∗ω〉 − r(r + 1)〈Dω,Dω〉 (4.3)

for any nonzero r-form ω.It remains to consider the case where the scalar curvature s does not vanish on the compact manifold

(M,g).Suppose that the scalar curvature s is a positive function. Let smin denote its minimum value on

(M,g). Then any harmonic r-form ω must satisfy the inequality

−r(r + 1)〈Dω,Dω〉 ≥ 1

2(2r − 1)smin · 〈ω, ω〉 ≥ 0.

This is possibly only if any harmonic r-formω vanishes at each point of (M,g) and, therefore, br(M) = 0.Now, suppose that the scalar curvature s is a negative function and −smax is its maximum value on

(M,g). Then any conformal Killing r-form ω must satisfy the inequality

0 ≤ r

r + 1〈dω,dω〉+ n− r

n− r + 1〈d∗ω,d∗ω〉 ≤ − 1

2(2r − 1)smax · g(ω, ω) ≤ 0.

This is possible only if any conformal Killing r-form ω vanishes at each point of (M,g) and, therefore,tr(M) = 0.



Remark 3. We emphasize that the conformal Killing r-forms, 1 < r < n, considered in the book [14]have nothing in common (except the name) with those considered in the modern literature and, inparticular, in this paper; the notion was introduced by Yano as a generalization conformal Killing vectorfields but has not been developed further.


As is known [29], any simply connected compact conformally flat Riemannian manifold (M,g) ofdimension n ≥ 2 is conformally diffeomorphic to the hypersphere Sn in Euclidean space Rn+1 and, there-fore, b1(M) = · · · = bn−1(M) = 0. Moreover, the condition that a compact connected manifold issimply connected is equivalent to the vanishing of its first Betti number b1(M).

Now, taking into account the conformal invariance of the Tachibana numbers tr(M), we must findthe dimension of the space of conformal Killing r-forms defined globally on the Euclidean n-sphere.

First, we refer to results of [31], where it was proved, following Tachibana [30], that the r-form ω withcomponents ωa1...ar = Aba1...arx

b for any antisymmetric constants Aba1...ar in a Cartesian coordinatesystem x1, . . . , xn, xn+1 in Euclidean space Rn+1 is a Killing r-form defined globally on the hypersphere

Sn : (x1)2 + · · ·+ (xn+1)2 = 1.

This means, in particular, that

kr(Sn) ≥ (n+ 1)!

(r + 1)!(n − r)!.

On the other hand, it is known [31] that a system of differential equations determining such a Killingr-form ω is completely integrable only in the case of a Riemannian n-manifold (M,g) of constantcurvature under the initial conditions

ωi1...ir(x0) = Bi1...ir , (∇ir+1ωi1...ir)(x0) = Ci1...irir+1

for any point x0 ∈ M and constants Bi1...ir and Ci1...irir+1 antisymmetric with respect to all indices.These constants are easy to count; their number is (n+ 1)!/((r + 1)!(n − r)!). Thus, kr(Sn) = (n+1)!/((r + 1)!(n − r)!). Next, using the duality of the Killing and planarity numbers, we obtain

pr(Sn) = kn−r(S

n) =(n + 1)!

r! (n− r + 1)!.

It remains to refer to Theorem 1, which implies

tr(Sn) = kr(S

n) + pr(Sn) =

(n+ 2)!

(r + 1)!(n − r + 1)!.

REFERENCES1. S. Kobayashi and K. Nomizu, Foundations of Differential Geometry (Interscience, New York, 1969;

Nauka, Moscow, 1981), Vol. 1.2. S. Tachibana, “On conformal Killing tensor in a Riemannian space,” Tohoku Math. J. 21, 56–64 (1969).3. S. E. Stepanov, “The vector space of conformal Killing forms on a Riemannian manifold,” Zap. Nauchn. Sem.

POMI 261, 240–265 (1999); [J. Math. Sci. 110 (4), 2892–2906 (2002)].4. U. Semmelmann, “Conformal Killing forms on Riemannian manifolds,” Math. Z. 245 (3), 503–627 (2003).5. G. A. Rod and J. Silhan, “The conformal Killing equation on forms—prolongations and applications,”

Differential Geom. Appl. 26 (3), 244–266 (2008).6. I. M. Benn and P. Chalton, “Dirac symmetry operators from conformal Killing–Yano tensors,” Classical

Quantum Gravity 14 (5), 1037–1042 (1997).7. S. E. Stepanov, “On conformal Killing 2-form of the electromagnetic field,” J. Geom. Phys. 33 (3-4), 191–

209 (2000).8. V. P. Frolov and A. Zelikov, Introduction to Black Hole Physics (Oxford Univ. Press, Oxford, 2011).9. T. Houri, T. Oota, and Yu. Yasui, “Closed conformal Killing–Yano tensor and the uniqueness of generalized

Kerr–Nut–de Sitter,” Classical Quantum Gravity 26 (045015) (2009).


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10. M. Gromov, “Sign and Geometric Meaning of Curvature (Izd. Udmurt. Univ., Izhevsk, 1999) [Rend. Sem.Mat. Fis. Milano 61 (1), 9–123 (1991)].

11. D. V. Alekseevskii, A. M. Vinogradov, and B. B. Lychagin, “Basic ideas and notions of differential geometry,”in: Current Problems of Mathematics. Fundamental Directions, Itogi Nauki i Tekhniki (VINITI,Moscow, 1988), Vol. 28, pp. 5–289 [in Russian].

12. J. A. Wolf, Spaces of Constant Curvature (Amer. Math. Soc., Providence, RI, 2011; Nauka, Moscow,1982).

13. S. E. Stepanov, “Curvature and Tachibana numbers,” Mat. Sb. 202 (7), 1059–1069 (2011) [Sb. Math. 202(7), 135–146 (2011)].

14. K. Yano and S. Bochner, Curvature and Betti Numbers (Princeton Univ. Press, Princeton, 1953; Inostran-naya Literatura, Moscow, 1957).

15. S. P. Novikov, “Topology,” in: Current Problems of Mathematics. Fundamental Directions, Itogi Nauki iTekhniki (VINITI, Moscow, 1986), Vol. 12, pp. 5–252 [in Russian].

16. S. E. Stepanov, “Some conformal and projective scalar invariants of Riemannian manifolds,” Mat. Zametki80 (6), 902–907 (2006) [Math. Notes 80 (6), 848–852 (2006)].

17. W. Kuhnel and H.-B. Rademacher, “Conformal transformations of pseudo-Riemannian manifolds,” in RecentDevelopments in Pseudo-Riemannian Geometry, ESI Lect. Math. Phys. (European Math. Soc., Zurich,2008), pp. 261–298.

18. D. Kramer, H. Stephani, E. Herlt, M. A. H. MacCallum, and E. Schmutzer, Exact solutions of EinsteinТsfield equations (DVW, Berlin, 1980; Energoizdat, Moscow, 1982).

19. S. E. Stepanov and I. I. Tsyganok, “On the dimension of the vector space of conformal Killing forms,” inDifferential Geometry of Manifolds of Figures (Kaliningrad. Univ., Kaliningrad, 2011), Vol. 42, pp. 134–140 [in Russian].

20. J. Mikes, S. Stepanov, and I. Hintrleitner, “Projective mappings and dimensions of vector spaces of threetypes of Killing–Yano tensors on pseudo-Riemannian manifolds of constant curvature,” AIP Conf. Proc.1460, 202–205 (2012).

21. J.-P. Bourguignon and J.-P. Ezin, “Scalar curvature functions in a conformal class of metrics and conformaltransformations,” Trans. Amer. Math. Soc. 301 (2), 723–736 (1987).

22. P. Petersen, Riemannian Geometry, in Grad. Texts in Math. (Springer-Verlag, New York, 1998), Vol. 171.23. A. Besse, Einstein Manifolds (Springer, Berlin, 1987; Mir, Moscow, 1990).24. Four-Dimensional Riemannian Geometry. Arthur Besse’s Seminar, Ed. by A. N. Tyurin (Mir, Moscow,

1985) [in Russian].25. S. E. Stepanov, “A new strong Laplacian on differential forms,” Mat. Zametki 76 (3), 452–458 (2004) [Math.

Notes 76 (3), 420–425 (2004)].26. Y. Tashiro, “Complete Riemannian manifolds and some vector fields,” Trans. Amer. Math. Soc. 117, 251–275

(1965).27. S. Tachibana, “On the proper space of Δ for m-forms in 2m-dimensional conformal flat Riemannian

manifolds,” Natur. Sci. Rep. Ochanomizu Univ. 29 (2), 111–115 (1978).28. C. Bohm and B. Wilking, “Manifolds with positive curvature operators are space forms,” Ann. of Math. (2)

167 (3), 1079–1097 (2008).29. R. Schoen and S.-T. Yau, “Conformal flat manifolds, Kleinian groups and scalar curvature,” Invent. Math.

92 (1), 47–71 (1988).30. Sh. Tachibana, “On Killing tensor in a Riemannian space,” Tohoku Math. J. (2) 20, 257–264 (1968).31. Sh. Tachibana and T. Kashiwada, “On the integrability of Killing–Yano’s equation,” J. Math. Soc. Japan 21

(2), 259–265 (1969).


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Betti and Tachibana numbers