MATHEMATICS

I. V. SKRIPNIK

PERIODS OF \(\mathfrak A\)-CLOSED FORMS

(Presented by Academician I. N. Vekua on 25 VII 1964)

In this paper the notion of periods of \(\mathfrak A\)-closed forms with respect to cycles of a manifold is introduced. Results generalizing the well-known theorems of de Rham and Hodge (see \((^{3-6})\)) are established in terms of periods. The concept of a period is of important significance in a number of questions connected with \(A\)-harmonic fields and forms, in particular in the consideration of \(A\)-harmonic fields with singularities.

1. Let \(\mathfrak M\) be a real analytic compact Riemannian space of dimension \(n\) (without boundary), on which analytic tensors \(a_j^{i_1,\ldots,i_q}\) \((q=0,\ldots,m)\), contravariant in the upper indices and covariant in the lower one, are given. We define operators \(\mathfrak A^p\) \((p=0,\ldots,n-1)\), taking forms of degree \(p\) into forms of degree \(p+1\), by the formula

\[ (\mathfrak A^p\alpha)_{k_1,\ldots,k_{p+1}} = \sum_{\nu=1}^{p+1}(-1)^{\nu-1}A_{k_\nu}\alpha_{k_1,\ldots,\hat k_\nu,\ldots,k_{p+1}}; \qquad A_j=\sum_{q=0}^{m}a_j^{i_1,\ldots,i_q}\nabla_{i_1}\cdots\nabla_{i_q}. \]

Here \(\nabla_i\) is the symbol of covariant differentiation. The indices on the forms \(\alpha,\mathfrak A^p_\alpha\) indicate that the corresponding components of the forms are meant; \(k_1,\ldots,\hat k_\nu,\ldots,k_{p+1}\) is obtained from \(k_1,\ldots,k_{p+1}\) by deleting \(k_\nu\). In \(A_j\) summation over repeated indices is carried out. We also define

\[ A^p=(\mathfrak A^p)'\mathfrak A^p+\mathfrak A^{p-1}(\mathfrak A^{p-1})', \]

where \((\mathfrak A^p)'\) is metrically adjoint to \(\mathfrak A^p\) (see \((^4)\)). It is assumed that \(A^0\) is an elliptic operator and the \(A_j\) commute.

A form \(\alpha\) is called \(\mathfrak A\)-closed if \(\mathfrak A\alpha=0\); \(\beta\) is called \(\mathfrak A\)-homologous to zero if there exists on \(\mathfrak M\) such a form \(\gamma\) that \(\mathfrak A\gamma=\beta\). A form \(\varphi\) is called \(A\)-harmonic if \(A\varphi=0\); \(\psi\) is called an \(A\)-harmonic field if \(\mathfrak A\psi=0,\ \mathfrak A'\psi=0\).

The following generalized de Rham theorem holds.

Theorem 1. The quotient group \(\mathcal H_{\mathfrak A}^{p}(\mathfrak M)\) of the group of \(\mathfrak A\)-closed forms of degree \(p\) by the subgroup of \(\mathfrak A\)-homologous-to-zero forms of degree \(p\) is isomorphic to \([H^p(\mathfrak M,R)]^M\), where \(H^p(\mathfrak M,R)\) is the \(p\)-dimensional cohomology group of the space \(\mathfrak M\) with real coefficients. \(M\) is the number of linearly independent \(\mathfrak A\)-closed forms of degree zero.

2. Here forms \(\beta_p(u,\tilde v_j)\), fundamental for what follows, will be introduced and an assumption concerning them will be formulated.

Let

\[ \mathfrak A^{*p}=*(\mathfrak A^{\,n-p})' *^{-1} \]

(with respect to the operator \(*\), see \((^4)\)). On \(\mathfrak M\) there exist \(M\) linearly independent solutions of the equation \(\mathfrak A^{*0}v=0\), and also of the equation \(\mathfrak A^0u=0\). We denote these solutions respectively by

\[ \tilde v_1,\ldots,\tilde v_M;\quad \tilde u_1,\ldots,\tilde u_M. \]

Lemma 1. There exist forms \(\beta_p(u,\tilde v_j)\) \((1\le j\le M)\) of degree \(p\) \((0\le p\le n)\) such that for all \(u\) of degree \(p\) the relations hold

\[ \beta_n(u,\tilde v_j)=u\cdot\tilde v_j;\qquad \beta_{p+1}(\mathfrak A^p u,\tilde v_j)=d\beta_p(u,\tilde v_j)\quad (p<n). \]

Here the coefficients \(\beta_p(\tilde u, v_j)\) are bilinear differential expressions with respect to \(u,\ \tilde v_j\) (\(d\) is the exterior differential).

Let \(S_{\varepsilon_1,\varepsilon_2}:\ \varepsilon_1<r_{P,Q}<\varepsilon_2\) be a geodesic ring with center at an arbitrary point \(P\in \mathfrak M\). By Theorem 1 the group \(H_{\mathfrak A}^{\,n-1}(S_{\varepsilon_1,\varepsilon_2})\) has \(M\) generators \(\hat u_1,\ldots,\hat u_M\). We shall assume that

\[ \det \|B_{ij}\|\ne 0,\qquad (\beta_{i,j})_{j=1}^M\ne (0,\ldots,0), \]

where \(B_{i,j}\) are given by the formula

\[ \int_{r_{P,Q}=\varepsilon}\beta_{n-1}(u_i,\tilde v_j)\qquad (\varepsilon_1<\varepsilon<\varepsilon_2) \]

and do not depend on \(\varepsilon\), while \(\beta_{i,j}\) are numbers on \(\mathfrak M\) equal to \(\beta_0(\tilde u_i,\tilde v_j)\).

1. Let \(K\) be some simplicial decomposition (see \((^3)\)) of the manifold \(\mathfrak M\). Introduce the mapping \(B_i\) of the set of all forms of degree \(p\) on \(\mathfrak M\) into the set of chains of the complex \(K\):

\[ B_i(\varphi^p)=\sum_{C^p}\{\varphi^p,C^p\}_i\,C^p;\qquad \{\varphi^p,C^p\}_j=\int_{C^p}\beta_p(\varphi^p,\tilde v_j). \]

The following theorem is of fundamental importance:

Theorem 2. The operator \(B\varphi=(B_i\varphi)_{i=1}^M\) establishes the isomorphism formulated in Theorem 1.

It is now possible to introduce

Definition. The periods of an \(\mathfrak A\)-closed form \(\varphi\) with respect to a cycle \(Z\) of the manifold \(\mathfrak M\) are the numbers \(\{\varphi,Z\}_i,\ 1\le i\le M\).

1. On the basis of the results of item 3, the following theorems are established:

Theorem 3. An \(\mathfrak A\)-closed form with zero periods with respect to all cycles of the manifold is \(\mathfrak A\)-homologous to zero.

Let \(Z_j^p\ (j=1,\ldots,s)\) be \(p\)-dimensional cycles no linear combination of which is homologous to zero. Then:

Theorem 4. There exists an \(\mathfrak A\)-closed form having arbitrary prescribed periods with respect to the cycles \(Z_j^p\).

Theorem 5. There exists an \(A\)-harmonic form having arbitrary prescribed periods with respect to the cycles \(Z_j^p\). The latter is uniquely determined if \(s\) is equal to the \(p\)-th Betti number.

In the case of the operator \(\mathfrak A\) coinciding with the operator of exterior differentiation, Theorems 3 and 4 were obtained by de Rham \((^5)\), and Theorem 5 is known as Hodge’s theorem \((^{4,6})\).

1. In conclusion we indicate analogues of Theorems 3 and 4 for the case of differential forms with fixed carrier. These results are needed in the consideration of \(A\)-harmonic fields and forms on a manifold with boundary (cf. \((^{1,2})\)).

We shall denote the carrier of the form \(\varphi\) by \(\underline{\varphi}\). Let \(\mathfrak R\) be an \(n\)-dimensional subspace of \(\mathfrak M\) with boundary \(\mathfrak B\), and let \(R_i^p\) be a basis of the relative \(p\)-dimensional cycles of \(\mathfrak R\ (\operatorname{mod}\ \mathfrak B)\). The relative periods of the form \(\varphi\) \((\underline{\varphi}\subseteq \mathfrak R)\) are the numbers \(\{\varphi,R_i^p\}_j\).

Theorem \(3'\). An \(\mathfrak A\)-closed form \(\varphi\) \((\underline{\varphi}\subseteq \mathfrak R)\) with zero relative periods is representable in the form \(\varphi=\mathfrak A\psi\) \((\underline{\psi}\subseteq \mathfrak R)\).

Theorem \(4'\). There exists an \(\mathfrak A\)-closed form \(\theta\) \((\underline{\theta}\subseteq \mathfrak R)\) having arbitrary prescribed relative periods.

Lviv State
University

Received
23 VII 1964

References

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