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1 Introduction

1.1 The history of conformal Killing forms has started almost a half of century ago by works of Tachibana and Kashiwada [7, 26]. During this long time, this topic has given rise to an active interest (see for example [8, 16, 17, 22]) because of great number of its applications (see e.g. [4, 10, 20]). This paper is devoted to the study of dimensions of vector spaces of Killing forms (see [17]).

1.2 In the second section, we investigate conformal Killing differential r-forms \((1\,{\le } \,r\,{\le }\, n {-} 1)\) on local coordinates of an arbitrary neighbourhood \(U\) of \(n\)-dimensional Riemannian manifold \((M, g)\). We consider a vector space \(T^r\) of these forms and deal with two subspaces—the subspace \(P^r\) of planar r-forms (i.e. closed conformal Killing forms) and subspace \(K^r\) of Killing r-forms (coclosed conformal Killing r-form). For \(r=1\) we obtain the following three vector spaces of \(1\)-forms: dual to conformal Killing vector field, concircular vector field and Killing vector field. In this section there are found dimensions \(t_r, k_r\) and \(p_r\) of these free “local” vector spaces on the manifold \((M, g)\) with constant curvature.

1.3 In the third section there are studied “complete” conformal Killing \(r\)-forms \((1\,{\le }\, r\,{\le }\, n {-} 1)\) on \(n\)-dimensional closed Riemannian manifold \((M,g)\), vector space \(T^r(M, \mathbb {R})\) of these forms and two subspaces of it \(K^r(M, \mathbb {R})\) of Killing forms and \(P^r(M, \mathbb {R})\) of planar \(r\)-forms; dimensions of these spaces are denoted by \(t_r(M)\), \(k_r(M)\) and \(p_r(M)\), respectively. In the case of closed manifold \((M,g)\) with zero Betti numbers \(b_r(M)=0\), we show the orthogonal decomposition of \(T^r\) in the form \(T^r(M, \mathbb {R})=K^r(M, \mathbb {R})+P^r(M, \mathbb {R})\), which implies the relation \(t_r(M)=k_r(M)+p_r(M)\). In the case \(b_1(M)=0\) and \(t_1(M) \ne k_1(M)\), we will prove that \((M,g)\) is globally conformal to the \(n\)-dimensional sphere \(S^n\) of the Euclidean space \(R^{n+1}\).

2 Definitions and Notations

2.1 Let us consider \(n\)-dimensional Riemannian manifold \((M,g)\) with Levi-Civita connection. Denote by \(C^{\infty }(M)\) a vector space of \(C^{\infty }\)-function on \(M\) and by \(\varOmega ^r(M)\) a vector space of differentiable \(r\)-forms on \(M\). Taking local orientation of \(M\) we introduce the Hodge operator \(*\) defining an isomorphism \(*\) : \(\varOmega ^r \rightarrow \varOmega ^{n-r}\) such that \(g(\omega ,*\varTheta )=(-1)^{r(n-r)}g(*\omega ,\varTheta )\), for any \(\omega \in \varOmega ^r(M), \varTheta \in \varOmega ^{n-r}(M)\), and \(*^2=(-1)^{r(n-r)}Id_{\varOmega ^r(M)}\) (see [2, definition 1.51], [13, p. 203]).

For the exterior differential operator \(\mathrm{d}: C^\infty \varLambda ^r(M) \rightarrow C^\infty \varLambda ^{r+1}(M)\) there exists a formal adjoint operator \(\delta : \varOmega ^{r+1}(M) \rightarrow \varOmega ^{r}(M)\) which is called codifferential operator (see [2, definition 1.56], [13, p. 203, 204], [15, § 25]) and it is defined by

$$\begin{aligned} \delta =(-1)^{(n-r)(r+1)}*\mathrm{d}* \end{aligned}$$
(1)

2.2 Let us remind well known definitions of three types of Killing vector fields in Riemannian geometry.

A vector field \(Z\) on a Riemannian manifold \((M,g)\) is called infinitesimal conformal transformations or conformal Killing vector field (see [5, § 69], [11, p. 120]) if \(L_Zg=2\sigma g\) for Lie derivative \(L_Z\) with respect to the vector field \(Z\) and some \(\sigma \in C^{\infty }M\).

Defining for a vector field \(Z\) a dual 1-form \(\omega \) by the relation \(\omega = g(Z, \cdot )\) we introduce a denotation \(\omega ^{\#}=Z\) (see [2, denotation 1.38]). Now, the identity \(L_Zg=2\sigma \cdot g\), by which an infinitesimal conformal transformation is defined, may be written by

$$\begin{aligned} (L_Zg)(X,Y)\mathop {=}\limits ^{\text {def}}(\nabla _X)Y+(\nabla _Y\omega )X=-\frac{2}{n}(\delta \omega )g(X,Y) \end{aligned}$$
(2)

or, equivalently, by

$$\begin{aligned} \nabla \omega =-\frac{1}{2}\mathrm{d}\omega - \frac{1}{n}g\cdot \delta \omega = 0. \end{aligned}$$
(3)

Any vector field \(Z\) with \(L_Zg=0\) on a Riemannian manifold \((M,g)\) is called infinitesimal isometry or Killing vector field.

Clearly, if \(\sigma =0\) then every infinitesimal conformal transformation \(Z\) is an infinitesimal isometry. Because \(\sigma =n^{-1}(-\delta \omega )\), for 1-form \(\omega \) dual to the vector field \(Z=\omega ^{\#}\), we see that any infinitesimal isometry may be considered as coclosed infinitesimal conformal transformation.

Let us remind that a vector field \(Z\) is called concircular (see [30]) if \(\nabla Z= \rho \, Id_M\) for \(\rho \in C^{\infty }M\). In this case, for every 1-form \(\omega \) with \(\omega ^{\#}=Z\) we have \(\nabla \omega =n^{-1}(-\delta \omega )g\). Therefore, a concircular vector field may be defined as closed infinitesimal conformal transformation.

2.3 Let us deal with a generalization of three types of Killing vector field defined above.

Let an \(n\)-dimensional Riemannian manifold \((M,g)\) be given. A form \(\omega \in \varOmega ^r(M)\) is called conformal Killing \(r\) -form (see [7]) if there exists a form \(\varTheta \in \varOmega ^{r-1}(M)\) with

$$\begin{aligned} \begin{array}{c} (\nabla _Y\omega )(X,X_2 ,\,\dots \,,X_r) + (\nabla _X\omega )(Y,X_2 ,\,\dots \,,X_r)= 2g(X,Y)\varTheta (X_2\!,\,\dots \,,X_r)\\ -\sum \limits _{a=2}^{r} (-1)^a\left( g(Y,X_a)\varTheta (X,X_2\!,\,\dots \,,\! \hat{X}_a \!,\,\dots \,,\! X_r + g(X,X_a)\varTheta (Y,Y_2 \!,\,\dots \,,\! \hat{X}_a \!,\,\dots \,,\! X_r) \right) \end{array} \end{aligned}$$
(4)

for any vector fields \( Y,X,X_2 ,\,\dots \,,X_r \in C^{\infty }({\textit{TM}})\), where \(\hat{X}_a\) means that \(X_a \) is omitted. The form \(\varTheta \in \varOmega ^{r-1}(M)\) is called an associated form of the conformal Killing form \(\omega \in \varOmega ^r(M)\). Moreover, the identity (see [7])

$$\begin{aligned} \delta \omega = -(n-r+1)\varTheta \end{aligned}$$
(5)

holds as the corollary of (3).

Equation (4) are a natural generalization of Eq. (2). Equation (4) may be written in the form (see [20, 22])

$$\begin{aligned} \nabla \omega = (r+1)^{-1}\mathrm{d}\omega + (n-r+1)^{-1}g\wedge \delta \omega . \end{aligned}$$
(6)

These relations constitute necessary and sufficient conditions that an \(r\)-form \(\omega \) is a conformal Killing form (\(1\le r\le n-1\)).

The set of conformal Killing forms on a Riemannian manifold \((T,g)\) forms a vector space \(T^r\) (with real coefficients) (see [19]).

A form \(\omega \in \varOmega ^r(M)\) on an \(n\)-dimensional Riemannian manifold \((M,g)\) is called a Killing r-form if it is a coclosed conformal Killing r-form. Such form \(\omega \in \varOmega ^{r}(M)\) fulfils its definition equations (see [14] and [10], Definition 31.3.1)

$$\begin{aligned} (\nabla _Y\omega )(X,X_2,\,\dots \,,X_r)+ (\nabla _X\omega )(Y,X_2,\,\dots \,,X_r)=0 \end{aligned}$$
(7)

or equivalent equations

$$\begin{aligned} \nabla \omega =(r+1)^{-1}\mathrm{d}\omega . \end{aligned}$$
(8)

This form \(\omega \) is a generalization of an \(1\)-form \(\omega \in \varOmega ^{-1}(M)\), which is dual to the Killing vector field \(Z=\omega ^{\#}\). The set of all Killing \(r\)-forms constitutes a vector space \(K^r \subset T^r\) (see [19]).

A form \(\omega \in C^{\infty }\varLambda ^r(M)\) on an \(n\)-dimensional Riemannian manifold \((M,g)\) is called a planar r-form if it is a closed conformal Killing r-form (see [19]). Such form \(\omega \in C^{\infty }\varLambda ^r(M) \) fulfils its definition equations

$$\begin{aligned} \nabla \omega =(n-r+1)^{-1}g\wedge \delta \omega , \end{aligned}$$
(9)

for \(1\le r\le n-1\).

This form \(\omega \) is a generalization of an \(1\)-form \(\omega \in \varOmega ^1(M)\), which is dual to the concircular vector field \(Z=\omega ^{\#}\). The set of all such \(r\)-forms constitutes a vector space \(P^r \subset T^r\) (see [19]).

3 Dimensions of Vector Spaces of Killing Forms and Vector Fields on Non-compact Riemannian Manifold

3.1 Let \((M,g)\) be an \(n\)-dimensional connected Riemannian manifold. Let us remind some well known facts on dimensions of three types spaces of Killing vector fields on \((M,g)\). It is known that the dimension of a Lie algebra of a group \(C(M,g)\) of infinitesimal conformal transformations of connected Riemannian \(n\)-dimensional manifold \((M, g)\) is not greater than \(\frac{1}{2}(n+1)(n+2)\) and this algebra is a vector space of conformal Killing vector fields (see [11, p. 120]). The equality is obtained in the case of conformally flat Riemannian manifold \((M,g)\).

The dimension of a Lie algebra of a subgroup \(I(M,g)\) of infinitesimal transformations is not greater than \(\frac{1}{2}(n+1)n\) and this algebra is a vector space of Killing vector fields (see [11, p. 101]). The equality is obtained in the case of Riemannian manifold \((M,g)\) with constant curvature.

The dimension of a vector space of concircular vector fields on connected \(n\)-dimensional manifold \((M,g)\) is not greater than \(n+1\) and the equality is obtained for manifold with constant curvature (see [6]).

3.2 To generalize facts presented above we will find precise upper bounds of dimensions of vector spaces of three types of Killing \(r\)-forms \((1\,{\le }\, r\,{\le } \,n {-} 1)\). Let us investigate a connected \(n\)-dimensional Riemannian manifold \((M,g)\). To form \(\omega \in C^{\infty }\varLambda ^r(M)\) the condition of integrability of arbitrary equation of a systems of Eqs. (5), (7) or (8) is the Ricci identity ([5, § 11]). This identity has in any local coordinate system \(x^1 ,\,\dots \,,x^n\) of a manifold \((M,g)\) the following expression:

$$\begin{aligned} \nabla _j\nabla _k\omega _{{i_1}{i_2}\dots {i_r}} - \nabla _k\nabla _j\omega _{{i_1}{i_2}\dots {i_r}} = - \sum _{\alpha =1}^{r} \omega _{{i_1}{i_2}\dots {i_{\alpha -1}} l {i_{\alpha +1}} \dots i_r } R_{i_{\alpha } j k}^{l} \ , \end{aligned}$$
(10)

where \(\omega _{i_1,i_2\dots i_p}=\omega (X_{i_1}, X_{i_2} ,\,\dots \,,X_{i_p} )\) and \(R_{jkl}^i X_i= R(X_k,X_l)X_j\) are local components of a conformal Killing \(r\)-form and curvature tensor \(R\) for \(X_k=\frac{\partial }{\partial x^k}\) and \(\nabla _k=\nabla _{X_k}\).

Ricci identity (10) establishes restrictions not only the choice of components of \(r\)-form \(\omega \) but also on the curvature tensor \(R\) of the manifold \((M,g)\). We have the following theorem.

Theorem 3.1

On an \(n\)-dimensional connected Riemannian manifold \((M,g)\), the dimensions \(t_r, k_r\) and \(p_r\) of vector spaces of conformal Killing \(r\)-form \(T^r\), co-closed confomal Killing (Killing) \(r\)-form \(K^r\) and closed conformal Killing \(r\)-form \(P^r\), \((1\,{\le }\, r\,{\le }\, n {-} 1)\), respectively have the following upper bounds

$$ t_r \le \frac{(n+2)!}{(r+1)!(n-r+1)!}, \ \ k_r \le \frac{(n+1)!}{(r+1)!(n-r)!}, \ \ p_r \le \frac{(n+1)!}{r!(n-r+1)!}. $$

The equalities are obtained in the case of Riemannian manifold \((M,g)\) with constant nonzero curvature.

Proof

The case when \((M,g)\) is locally flat manifold is trivial; in [19, 20] on the basis of (10) there are defined components \(\omega _{i_1 \dots i_r}=A_{k i_1 \dots i_r} x^k\,+\,B_{i_1 \dots i_r}\) of Killing \(r\)-form \(\omega \), for an arbitrary local Cartesian coordinate system \(x^1 ,\,\dots \,,x^n\). Here, \(A_{k i_1 \dots i_r}, \ B_{i_1 \dots i_r}\) are local components of constant skew-symmetric \((r +1)\)-forms and \(r\)-forms, respectively.

With respect to this result, in [18] there for some special coordinate system \(x^1 ,\,\dots \,,x^n\) of manifold \((M,g)\) with constant sectional curvature \(C\ne 0\) were found components \(\omega _{i_1 \dots i_r}=\mathrm{e}^{(r+1)\varphi }(A_{k i_1 \dots i_r}x^k + B_{i_1 \dots i_r}), \ \varphi =\frac{1}{2(n+1)}\mathrm{ln}(\mathrm{det}\ g)\), of Killing \(r\)-form \(\omega \). Therefore the dimension \(k_r\) of a space \(K^r\) of coclosed conformal Killing \(r\)-forms on Riemannian manifold with constant curvature is equal to

$$ k_r={n \atopwithdelims (){r+1}} + {n \atopwithdelims ()r} = {{n+1} \atopwithdelims (){r+1}}= \frac{(n+1)!}{(r+1)!(n-r)!}. $$

In the case of an arbitrary connected manifold \((M,g)\), it is evident that the dimension \(k_r\) of a space \(K^r\) is not greater then this number, i. e. \(k_r \le \frac{(n+1)!}{(r+1)!(n-r)!}\).

It is known (see [8, 19, 20]) that there exists an isomorphism \(*\) : \(K^{n-r} \rightarrow P^r\). It gives the possibility to count the dimension \(p_r\) of a space \(P^r\) of closed conformal Killing \(r\)-forms on a manifold \((M,g)\) with nonzero constant sectional curvature \(C\ne 0\); this dimension is equal to \(p_r=\frac{(n+1)!}{r!(n-r+1)!}.\) For arbitrary connected manifold \((M,g)\), it is evident that the dimension \(p_r\) of a space \(P^r\) is not greater then this number, i. e. \(p_r \le \frac{(n+1)!}{r!(n-r+1)!}\).

For manifold \((M,g)\) with constant sectional curvature \(C\ne 0\), in [7, 26] there by direct calculation was obtained decomposition of any conformal Killing \(r\)-form \(\omega \) into direct sum \(\omega =\omega _1\,+\,\omega _2\) of a coclosed conformal Killing (Killing) \(r\)-form \(\omega _1\) and of closed conformal Killing (planar) \(r\)-form \(\omega _2=\nabla \varTheta \) for Killing \((r-1)\)-form \(\varTheta \).

Clearly, the arbitrariness of choice of an \(r\)-form \(\omega _2=\nabla \varTheta \) is given by the number of parameters on which a Killing \((r-1)\)-form depends. This number is equal to \({{n} \atopwithdelims (){r}} + {{n} \atopwithdelims (){n-1}}={{n+1} \atopwithdelims (){r}}\). In the case of an arbitrary connected manifold \((M,g)\), it is obvious that the arbitrariness of determination of an exact conformal Killing \(r\)-form \(\omega _2=\nabla \varTheta \) does not be greater then this number.

Based on a pointwise decomposition \(\omega =\omega _1+\omega _2\) the expression of an arbitrary conformal Killing \(r\)-form \(\omega \) in some special local coordinate system \(x^1 ,\,\dots \,,x^n\) on a manifold \((M,g)\) with nonzero constant curvature \(C\ne 0\)

$$\begin{aligned} \omega _{i_1 \dots i_r}&=\mathrm{e}^{(r+1)\varphi } \left( A_{k i_1 \dots i_r}x^k+B_{i_1 \dots i_r}-\frac{1}{C}\left( \varphi _{[i_1}C_{k i_2\dots i_r]}x^k \right. \right. \\&\qquad \qquad \qquad \left. \left. +\,\varphi _{[i_1}D_{i_2 \dots i_r]}+ \frac{1}{r}C_{i_1 \dots i_r} \right) \right) , \end{aligned}$$

where \(A_{k, i_1 \dots i_r}, B_{i_1 \dots i_r}, C_{i_1 i_2 \dots i_r}\) and \(D_{i_1 \dots i_{r-1}}\) are local components of constant skew-symmetric forms. Now, we may compute the dimension \(t_r\) of a space \(T^r\) of conformal Killing \(r\)-forms on a manifold with constant curvature \(C\ne 0\) which is equal to the following summ

$$ t_r={{n+1}\atopwithdelims (){r+1}} + {{n+1}\atopwithdelims ()r} = {{n+2} \atopwithdelims (){r+1}}= \frac{(n+2)!}{(r+1)!(n-r+1)!}. $$

In the case of an arbitrary connected manifold \((M,g)\), it is obvious that the dimension of a space \(T^r\) is not greater then this number, i. e.

$$ t_r \le \frac{(n+2)!}{(r+1)!(n-r+1)!}. $$

We have proved the theorem.

Considering the upper bounds of dimension of vector space of conformal Killing forms \(T^r\) and of space of Killing \(r\)-forms \(K^r\) as found in Theorem 3.1 we for \(r=1\) obtain the following well known proposition.

Corollary 3.1

The dimensions of vector spaces of conformal Killing vector field \(T^1\), Killing vector fields \(K^1\) and concircular vector fields \(P^1\) on connected Riemannian manifold \((M,g)\) do not be greater then \(\frac{1}{2}(n+1)(n+2), \frac{1}{2}(n+1)n\) and \(n+1\), respectively. The equalities are obtained in the case of manifold with constant nonzero section curvature.

3.3 Formulated in the theorem and its corollary results on dimensions of vector spaces of Killing and conformal Killing forms and of vector fields are substantially local.

As an example, let us consider an \(n\)-dimensional \((n\ge 2)\) hyperbolic space which is a Riemannian manifold with constant negative curvature. As we have proved above, in this space the dimension \(k_r\) of the space of coclosed conformal Killing (Killing) \(r\)-forms \((1\le r \le n{-}1)\) and, especially, the dimension \(k_1\) of Killing vector spaces is equal to \(\frac{(n+1)!}{(r+1)!(n-r)!}\) and \(\frac{1}{2}(n+1)n\), respectively. Factorizing hyperbolic space according to a suitable discrete group of motions (see [29, § 2.4]) we obtain a compact manifold with constant curvature. On this manifold there exists “generally” no nonzero Killing \(r\)-form (see [32, § 1 of Chap. VI]). Therefore, our result on the dimension of a space of Killing \(r\)-forms as well as well known result on the dimension of a space of Killing vector field deals with “local dimensions” \(k_r\) and \(k_1\), especially.

An analogous conclusion may be obtained for dimensions of spaces of conformal Killing forms and vector fields and also for closed conformal Killing forms and concircular vector field.

4 Dimensions of Spaces of Conformal Killing Forms on a Closed Riemannian Manifold

4.1 Let \((M,g)\) be a closed (i.e. compact without the border \(\partial M\)) oriented Riemannian manifold. Let us denote by \(\langle \cdot , \cdot \rangle \) the global inner product

$$\begin{aligned} \langle \omega ,\omega ' \rangle = \int _M \frac{1}{r!} g(\omega ,\omega ') dv \end{aligned}$$
(11)

for arbitrary compact carriers \(\omega ,\omega '{\in }\varOmega ^r(M)\) of \(r\)-form and volume element \(dv\). Then the exterior differential operator \(\mathrm{d}{:}\ \varOmega ^r(M) \rightarrow \varOmega ^{r+1}(M)\) and the adjoin codifferential operator \(\delta {:}\ \varOmega ^{r+1}(M) \rightarrow \varOmega ^{r}(M)\) are connected by the following equality (see [13, p. 204])

$$\begin{aligned} \langle \mathrm{{d}}\omega , \Theta \rangle =\langle \omega ,\delta \Theta \rangle \end{aligned}$$
(12)

for any \(\omega \in \varOmega ^r(M)\) and \(\varTheta \in \varOmega ^{r+1}(M).\)

On a closed manifold \((M,g)\), it holds the following Hodge-de Ram orthogonal decomposition with respect to the global inner product (11) (see [12]):

$$\begin{aligned} \varOmega ^r(M)= \mathop {\text {Im}}{\text {d}} \oplus \mathop {\text {Im}}\delta \oplus \mathop {\text {Ker}}\Delta , \end{aligned}$$
(13)

where \(\Delta =\mathrm {d}\delta + \delta \mathrm {d}\) is Laplace operator with \(\mathop {\text {Ker}}\Delta =\mathop {\text {Im}}\mathrm {d} \cap \mathop {\text {Ker}}\delta \). In addition, there are following orthogonal decompositions (see [12])

$$\begin{aligned} \mathop {\text {Ker}}\delta =\mathop {\text {Im}}\delta \oplus \mathop {\text {Ker}}\Delta , \end{aligned}$$
(14)
$$\begin{aligned} \mathop {\text {Ker}}\mathrm {d}= \mathop {\text {Im}}\mathrm {d} \oplus \mathop {\text {Ker}}\Delta . \end{aligned}$$
(15)

The kernel of Laplace operator \(\Delta \) on \((M,g)\) is a finite dimension vector space \(H^r(M, \mathbb {R})=\{\omega \in \varOmega ^r(M)| \Delta \omega =0 \}\) of harmonic \(r\)-forms, for \(r=1,\,\dots \,,n-1\) (see [15, § 25], [12]). The dimension of \(H^r(M, \mathbb {R})\) is equal to the Betti number \(b_r(M)\) of a manifold \((M,g)\), i.e. \(b_r(M)=\mathop {\text {dim}}_\mathbb {R} \mathop {\text {Ker}}\Delta \). It is known (see [12]), that Betti numbers of a manifold \((M,g)\) are dual in the sense of \(b_r(M)= b_{n-r}(M)\) and for \(n=2r\) Betti numbers are invariant with respect to the conformal transformation of metric \(\overline{g}=e^{2\,F}g\), because in this case \(\overline{\delta }=\delta \) (see [2, Corollary 1.162]).

4.2 Let an \(n\)-dimensional closed manifold \((M,g)\) be given and let us consider a natural with respect to isometric dipheomorphisms differential operator of the first order \(\mathrm{{D}}{:}\ \Omega ^r(M) \rightarrow \Omega ^1(M) \otimes \Omega ^1(M)\) being define by the following (see [3, pp. 312–313], [16, 20])

$$\begin{aligned} \mathrm{D}=\nabla - (r+1)^{-1}\mathrm{d}-(n-r+1)^{-1}g \wedge \delta , \end{aligned}$$
(16)

where \(\wedge \) denotes multiplication of an \((r-1)\)-form \(\delta \omega \) by a metric tensor which is defined by the following rule

$$ (g \wedge \delta \omega )(X_0,X_1 ,\,\dots \,,X_r)=\sum _{a=1}^r (-1)^{a}g(X_0,X_a)(\delta \omega )(X_1 ,\,\dots \,,\widehat{X_a} ,\,\dots \,,x_r) $$

for arbitrary \(( X_0 ,\,\dots \,,X_r)\in C^{\infty }(TM)\).

Then the condition \(\omega \in \mathop {\text {Ker}}\mathrm {D},\) which is equal to the identity \( \nabla \omega =(r+1)^{-1}\mathrm {d}\omega \,+\,(n-r+1)^{-1} {g} \wedge \delta \omega , \) is a necessary and sufficient condition that an \(r\)-form \(\omega \) is a conformal Killing form, \(r=1,\,\dots \,,n-1\) (see [16, 19, 20]).

Especially, it follows from this that for a conformal Killing vector field \(Z=\omega ^{\#}\) any \(1\)-form \(\omega \) belongs to the kernel of differential operator of the first order

$$ \mathrm {D}\ {:=}\ \nabla \omega - \frac{1}{2}\mathrm {d}\omega +\frac{1}{n} { g} \cdot \delta \omega , $$

which is called Ahlfors operator (see [14]).

In the [23] there is for operator \(\mathrm {D}\), defined by (16), found an adjoint operator \(\mathrm {D}^*\). Moreover, there is also the rough Laplacian constructed by (see [2, Definition 1.135], [3, pp. 316–317])

$$\begin{aligned} \mathrm{D}^*\mathrm{D}=\frac{1}{r(r+1)}\left( \nabla ^*\nabla - \frac{1}{r+1}\delta \mathrm{d} -\frac{1}{n{-}r\,{+}\,1}\mathrm{d}\delta \right) , \end{aligned}$$
(17)

where \(\nabla ^*\nabla \) is the rough Bochner Laplacian ([21]).

It follows from general theory that rough Laplacian on a closed Riemannian manifold \((M,g)\) is positive and elliptic (see [3, pp. 316–317]). Because it is an elliptic operator, its kernel is a finite dimensional vector subspace. Therefore, the kernel of a rough Laplacian \(\mathop {\text {Ker}}\mathrm {D}^*\mathrm {D}\) is a finite dimensional vector space \(T^r(M, \mathbb {R})=\{\omega \in \varOmega ^r(M)\ | \ \mathrm {D}^*\mathrm {D}\omega =0\}\). Using the identity \(\langle \mathrm {D}^*\mathrm {D}\omega , \omega \rangle = \langle \mathrm {D}\omega , \mathrm {D}\omega \rangle \), we have that this kernel consists of conformal Killing \(r\)-forms, for all \(r=1 ,\,\dots \,,n{-}1\) (see [16]).

In [17] and [24], the dimension \(t_r(M)=\mathop {\text {dim}}_\mathbb {R}\mathop {\text {Ker}}\mathrm {D}^*\mathrm {D}\), for all \(r{=}1 ,\,\dots \,,n{-}1\) is called Tachibana number of closed Riemannian manifold \((M,g)\) as an analogy to the Betti number \(b_r(M)=\mathop {\text {dim}}_\mathbb {R}\mathop {\text {Ker}}\varDelta \). Tachibana numbers as well as Betti numbers are dual in the sense of \(t_r(M)=t_{n-r}(M)\) (see [17, 22, 24].) Evidently,

$$t_r(M)\le \frac{(n+2)!}{(r+1)!(n-r+1)!}.$$

One of the most important properties of conformal Killing forms is their conformal invariance (see [4]), i.e. for an arbitrary conformal Killing \(r\)-form \(\omega \) the form \(\bar{\omega }=e^{(r+1)f}\omega \) is a conformal Killing form with respect to the conformally equivalent metric \(\,\bar{g}=\mathrm {e}^{2\,f} g\). This implies that Tachibana numbers \(t_r(M)\) for \(r=1 ,\,\dots \,,n{-}1\) are conformal scalar invariants of a closed Riemannian manifold (see [17, 24]).

Coclosed conformal Killing (Killing) \(r\)-forms \((1\,{\le }\,r\,{\le }\,n-1)\) form the vector space \(K^r(M, \mathbb {R})=\{\omega \in \varOmega ^r(M)\ |\ \mathrm {D}^*\mathrm {D}\omega =\delta \omega =0\}\). The dimension \(k_r(M)=\mathop {\text {dim}}_\mathbb {R}(\mathop {\text {Ker}}\mathrm {D}^*\mathrm {D}\cap \mathop {\text {Ker}}\delta )\) was in [17] and [24] called Killing number of closed Riemannian manifold \((M,g)\). Evidently,

$$k_r(M)\le \frac{(n+1)!}{(r+1)!(n-r)!}.$$

Closed conformal Killing (planar) \(r\)-forms \((1\,{\le }\,r\,{\le }\,n-1)\) form the vector space \(P^r(M, \mathbb {R})=\{\omega \in \varOmega ^r(M)\ |\ \mathrm {D}^*\mathrm {D}\omega =\mathrm {d}\omega =0\}\). Its dimension \(p_r(M)=\mathop {\text {dim}}_\mathbb {R}(\mathop {\text {Ker}}\mathrm {D}^*\mathrm {D}\cap \mathop {\text {Ker}}\mathrm {d})\) was in [17] and [24] called planar number of closed Riemannian manifold \((M,g)\). Evidently,

$$p_r(M)\le \frac{(n+1)!}{r!(n-r+1)!}.$$

One of the most important properties of Killing and planar \(r\)-forms is their conformal invariance (see [4]), i.e. for an arbitrary conformal Killing (or planar) \(r\)-form \(\omega \) the form

$$\bar{\omega }=\mathrm{e}^{-(r+1)f}\omega \ \ \mathrm{for} \ \ f=(n+1)^{-1}\ln \sqrt{\frac{\det g}{\det \bar{g}}}$$

is a Killing (respectively, planar) \(r\)-form with respect to the projectively equivalent metric \(\bar{g}\) (see [24]).This implies that the Killing numbers \(k_r(M)\) and planar numbers \(p_r(M)\) for \(r=1 ,\,\dots \,,n-1\) are projective scalar invariants of the Riemannian manifold \((M,g)\).

To summarize, we may formulate

Proposition 4.1

On an \(n\)-dimensional closed Riemannian manifold \((M,g)\) the following hold for all r, r \(=\) \(1\) , ... , \(n-1\)

  1. 1.

    Tachibana numbers \(t_r(M)\) are conformal scalar invariants, they are dual in the sense \(t_r(M)=t_{n-r}(M)\) and they fulfill the relation

    $$t_r(M)\le \frac{(n+2)!}{(r+1)!(n-r+1)!};$$
  2. 2.

    Killing numbers \(t_r(M)\) and planar numbers \(p_r(M)\) are projective scalar invariants, they are dual in the sense \(k_r(M)=p_{n-r}(M)\) and they fulfill the relations

    $$k_r(M)\le \frac{(n+1)!}{(r+1)!(n-r)!}\qquad \hbox {and}\qquad p_r(M)\le \frac{(n+1)!}{r!(n-r+1)!}.$$

4.3 We establish a connection between Betti numbers and Tachibana numbers. Two following theorems hold.

Theorem 4.1

If Ricci tensor \(Ric\) of an \(n\)-dimensional compact and oriented conformal planar Riemannian manifold \((M,g)\), \(n\ge 2\), is definite, then Tachibana \(t_k(M)\) and Betti \(b_l(M)\) numbers cannot be different from zero for arbitrary pair of indices \(k,l=1 ,\,\dots \,,n-1\) .

Proof

Let Ricci tensor \(Ric\) be definite on a compact manifold \((M,g)\), i.e. quadratic form \(Ric(X,X)\) is definite for arbitrary nonzero vector field \(X\in C^{\infty }TM\). Let us suppose that quadratic form \(Ric(X,X)\) is positive definite and manifold \((M,g)\) be compact and oriented conformal flat manifold. Then \(b_1(M)= \dots = b_{n-1}(M)=0\), in accordance with [31].

Further, to get a new expression of the rough Laplacian (17), let us use the classical Bochner-Weitzenböck formula [2], which gives \(\varDelta =\nabla ^*\nabla +F_r\), where \(F_r\) may be a algebraically (even linearly) expressed in terms of the curvature tensor \(R\) of manifold \((M,g)\). Now, the rough Laplacian may be written in the form

$$ \mathrm{D}^*\mathrm{D}=\frac{1}{r(r+1)}\left( \mathrm{d}^*\mathrm{d} - \frac{n-r}{n-r+1}\mathrm{d d}^* - F_r \right) . $$

Then any conformal Killing \(r\)-form \(\omega \) must fulfill the following equation

$$ \int _M g(F_r(\omega ),\omega )dv=\frac{r}{r+1}\langle \mathrm{d}\omega , \mathrm{d}\omega \rangle + \frac{n-r}{n-r+1}\langle \mathrm{d}^*\omega , \mathrm{d}^*\omega \rangle . $$

If we suppose that quadratic form \(Ric(X,X)\) is negative definite, then on a compact conformal flat manifold we have the following inequality [31]

$$ g(F_r(\varTheta ), \varTheta ) \le -\frac{n-r}{n-1}\lambda \cdot g(\varTheta ,\varTheta ), $$

for the greatest (negative) eigenvalue \(-\lambda \) of the matrix \(\Vert Ric \Vert \), for all \(1\le r \le n-1\) and any nonzero form \(\varTheta \in \varOmega ^r(M).\) Consequently, any conformal Killing \(r\)-form \(\omega \) must fulfill the inequality

$$ \frac{r}{(n-r)(r+1)}\langle \mathrm{d}\omega , \mathrm{d}\omega \rangle + \frac{1}{n-r+1}\langle \mathrm{d}^*\omega , \mathrm{d}^*\omega \rangle \le -\frac{n-r}{n-1}\lambda \langle \omega , \omega \rangle . $$

This is possible only if a conformal Killing \(r\)-form vanishes at any point of manifold \((M,g)\) and then \(t_1(M)= \dots =t_{n-1}(M)=0\). The theorem is proved.

Theorem 4.2

If on an \(n\)-dimensional closed Riemannian manifold \((M,g)\) Betti number \(b_r(M)=0\), for \(1\,{\le }\,r\,{\le }\,n-1\), and Tachibana number \(t_r(M)>k_r(M) \ne 0\), then for Killing numbers \(k_r(M)\) and planar numbers \(p_r(M)\) it holds \(t_r(M)=k_r(M)+p_r(M).\)

Proof

Let on an \(n\)-dimensional closed Riemannian manifold \((M,g)\) Betti number \(b_r(M)=0\), for \(1\,{\le }\,r\,{\le }\,n-1\). Then in decompositions (13–15) there is \(\mathop {\text {Ker}}\Delta =0\) and therefore we have the following orthogonal decomposition

$$\begin{aligned} \varOmega ^r(M)=\mathop {\text {Im}}\mathrm {d}\oplus \mathop {\text {Im}}\delta \end{aligned}$$
(18)

and besides \(\mathop {\text {Ker}}\mathrm {d}=\mathop {\text {Im}}\mathrm {d}\) and \(\mathop {\text {Ker}}\delta =\mathop {\text {Im}}\delta \). Since \(t_r(M)>k_r(M)\ne 0\) we obtain for Killing number \(k_r(M)=\mathop {\text {dim}}_\mathbb {R}(\mathop {\text {Ker}}\mathrm {D}^*\mathrm {D}\, \cap \, \mathop {\text {Im}}\mathrm {d}^*)\) and for planar number \(p_r(M)=\) \(\mathop {\text {dim}}_\mathbb {R}(\mathop {\text {Ker}}\mathrm {D}^*\mathrm {D}\ \cap \ \mathop {\text {Im}}\mathrm {d})\ne 0.\) Consequently, it is clear that decomposition (18) implies the following orthogonal decomposition

$$\begin{aligned} \mathop {\text {Ker}}\mathrm {D}^*\mathrm {D}=(\mathop {\text {Ker}}\mathrm {D}^*\mathrm {D}\ \cap \ \mathop {\text {Im}}\delta )\oplus (\mathop {\text {Ker}}\mathrm {D}^*\mathrm {D}\cap \mathop {\text {Im}}\mathrm {d}). \end{aligned}$$
(19)

It may be rewritten in the form \(T^r(M, \mathbb {R})\,{=}\,K^r(\mathbb {R},M)\oplus P^r(M, \mathbb {R}), 1\,{\le }\,r\,{\le }\,n-1\), where in accordance with (14) a space \(K^r(M, \mathbb {R})\) consists of co-exact conformal Killing \(r\)-forms and in accordance with (15) a space \(P^r(M, \mathbb {R})\) consists of exact ones. It follows from this the equality \(t_r(M)=k_r(M)+p_r(M)\). The summands on the right side represent dimensions of namely these vector spaces. Let us remark that in the case \(t_r(M)=0\) the equality which may be proved turns into an identity.

The proof is finished.

Remark 4.1

In the article [25], an analogical orthogonal decomposition \(T^r(M, \mathbb {R})=K^r(M, \mathbb {R})+P^r(M, \mathbb {R})\) was established on an \(2r\)-dimensional closed conformal flat Riemannian manifold \((M,g)\) with constant positive scalar curvature. Let us remark, that \(b_r(M)=0\) is fulfilled in such manifold in accordance with [31].

In [17] this decomposition is established for a closed manifold with positive curvature operator, where \(b_1(M)= \dots =b_{n-1}(M)=0\) in accordance with [13]. It is important to say that in the both decompositions the space \(P^r(M, \mathbb {R})\) consists of exact conformal Killing \(r\)-forms, \(1{\le }r{\le }n-1\), and the space \(K^r(M, \mathbb {R})\) consists of coexact ones. These facts does not contained in the cited article.

Especially, for \(r=1\) we have the following corollary.

Corollary 4.1

If for an \(n\)-dimensional closed Riemannian the first Betti number \(b_1(M)=0\) and at the same time \(t_1(M)\ne 0\), \(t_1(M)\ne k_1(M)\), then \((M,g)\) is globally conformal to an \(n\)-dimensional sphere \(S^n\) of Euclidean space \(\mathbb {R}^{n+1}\). If \(s=const\), \(s>0\), additionally, then a manifold \((M,g)\) is globally isometric to the sphere \(S^n\). If for an \(n\)-dimensional closed Riemannian manifold \(s=const\) and \(s=0\) or \(s=const\) and \(s<0\), then \(t_1(M)=k_1(M)\).

Proof

For an \(n\)-dimensional closed Riemannian manifold \((M,g)\) with \(b_1(M)=0\), \( t_1(M)\ne 0, \ t_1(M)\ne k_1(M)\) we have the following orthogonal decomposition with respect to the global inner product

$$\begin{aligned} T^1(M, \mathbb {R})=K^1(M, \mathbb {R})+ P^1(M, \mathbb {R}), \end{aligned}$$
(20)

where \(P^1(M, \mathbb {R})\) contains at least one exact conformal Killing \(1\)-form, i.e. a form which may be expressed as \(\omega =\mathrm {grad} f\) with \(\nabla \nabla f=\rho g\), where \(\rho = - \frac{1}{n}\varDelta f.\) In this case, manifold \((M,g)\) is globally conformal to an Euclidean sphere \(S^n\) with the standard metric \(\bar{g}_{can}\) (see [27]). It is known (see [32]), that the presence of orthogonal decomposition (20) on closed Riemannian manifold \((M,g)\) with an additional condition \(s=const\) implies that \((M,g)\) is globally isometric to the Euclidean sphere \(S^n\). Let us remark, that for a compact Riemannian manifold \((M,g)\) with constant negative or zero curvature it holds \(t_1(M)=k_1(M)\), because in this case any conformal Killing vector field is a Killing field (see [9]).

Remark 4.2

Conformal Killing and closed conformal Killing vector field is dual to conformal Killing and planar \(1\)-form, respectively. At the end of the past and beginning of this century, these vector fields have been objects of an intensive interest in connection with the study of groups of infinitesimal conformal transformations of search criteria for conformity and isometry Riemannian manifold to the Euclidean sphere (see [1, 6, 23, 28] and others.) Therefore, there exists a great number of propositions, which are analogical to our proved corollary. Let us mention one of them (see [23]). It says that any compact Riemannian manifold having finite fundamental group \(\pi _1(M)\) and admitting a closed conformal Killing vector field, which is not an infinitesimal isometry, is diffeomorphic to the Euclidean sphere. Adding the condition of constancy of the scalar curvature we obtain that such manifold must be isometric to the Euclidean sphere (see [28]). Le us add, the finiteness of a fundamental group \(\pi _1(M)\) implies automatically that first Betti number \(b_1(M)\) is equal to zero and closed conformal Killing vector field is the gradient at the same time.