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V. V. Vishnevskii
On the Complex Structures of One Class of Kähler–Rashevskii Spaces
(Presented by Academician A. N. Kolmogorov on 17 VIII 1962)
We shall call a Riemannian space \(V_n\) with metric tensor \(g_{ij}\) a \(B\)-space if it admits a covariantly constant affinor \(\gamma_i^j\) such that the tensor \(\gamma_{ij}=\gamma_i^k g_{kj}\) is symmetric. Spaces of this type were studied in papers \((^{1-3})\). Since the geometric properties of a \(B\)-space depend essentially on the algebraic structure of the affinor \(\gamma_i^j\), we shall call the characteristic \(((^{4}),\) p. 182) of its matrix the characteristic of the \(B\)-space.
Using A. P. Shirokov’s theorem \((^5)\) on the existence of a holonomic coordinate system in which the matrix of a covariantly constant affinor of a torsion-free affine-connection space has constant components, G. I. Kruchkovich and A. S. Solodovnikov \((^3)\) showed that only \(B\)-spaces of characteristics \([(m_1,m_2,m_3,\ldots,\ldots,m_r)]\)—with one real eigenvalue—and \([(n_1,n_2,n_3,\ldots,\ldots,n_r),(n_1,n_2,n_3,\ldots,n_r)]\)—with two complex conjugate eigenvalues—are irreducible. The present paper aims to study one important class of irreducible \(B\)-spaces with a real eigenvalue \(\lambda\), which may be taken to be zero, by considering the new covariantly constant affinor \(\gamma_i^j-\lambda\delta_i^j\), as will be assumed in what follows.
Since the tensors
\[
\overset{s}{\gamma}{}_i^j=\overset{s\ \text{times}}{\overbrace{\gamma_i^p\gamma_p^q\cdots\gamma_q^j}}
\]
are covariantly constant, in the coordinate system of A. P. Shirokov we have
\[
\Gamma_{km}^j \overset{s}{\gamma}{}_i^m=\Gamma_{ki}^l \overset{s}{\gamma}{}_l^j,
\tag{1}
\]
whence it follows that the components of the symmetric tensors \(\overset{p}{\gamma}_{ij}=g_{ik}\overset{p}{\gamma}{}_j^k\) satisfy the system of differential equations
\[
\overset{q}{\gamma}{}_k^m\,\partial_m \overset{p}{\gamma}_{ij}
=
\partial_k \overset{p+q}{\gamma}_{ij},
\tag{2}
\]
which, in the four-dimensional case, was first obtained by A. P. Norden \((^1)\). This system has the simplest geometric meaning when in the characteristic of the \(B\)-space \(m_1=m_2=\cdots=m_r=m\), and consequently, in a certain holonomic coordinate system, which we shall call canonical, the matrix of the affinor \(\gamma_i^j\) will have on its main diagonal \(r\) Jordan blocks of dimension \(m\). In consequence of the relations \(g_{ik}\gamma_j^k=g_{jl}\gamma_i^l\), the matrix of the metric tensor in this coordinate system decomposes into blocks of the form
\[
B_{\alpha\beta}=
\begin{pmatrix}
b_1 & b_2 & b_3 & \cdots & b_m\\
b_2 & b_3 & \cdots & b_m & 0\\
b_3 & \cdots & b_m & 0 & 0\\
\cdot & \cdot & \cdot & \cdot & \cdot\\
\cdot & \cdot & \cdot & \cdot & \cdot\\
b_m & 0 & \cdots & \cdots & 0
\end{pmatrix},
\qquad
\alpha,\beta=1,2,\ldots,r.
\tag{3}
\]
The matrices \(\overset{1}{\gamma}_{ij}, \overset{2}{\gamma}_{ij}, \ldots, \overset{m-1}{\gamma}_{ij}\), having an analogous structure, are obtained from (3) by raising (when contracting with \(\overset{s}{\gamma}_{i}\)) each element in all cells by \(s\) places upward and replacing the vacated places by zeros.
Let us establish the meaning of the differential equations (2). For \(k=1,2,\ldots,m\) we arrive at the system
\[ \begin{aligned} &\partial_2 g_{ij}=\partial_1 \overset{1}{\gamma}_{ij},\quad \partial_3 g_{ij}=\partial_2 \overset{1}{\gamma}_{ij}=\partial_1 \overset{2}{\gamma}_{ij},\quad \partial_4 g_{ij}=\partial_3 \overset{1}{\gamma}_{ij}=\partial_2 \overset{2}{\gamma}_{ij}=\partial_1 \overset{3}{\gamma}_{ij},\ \ldots\\ &\ldots,\partial_m g_{ij}=\partial_{m-1}\overset{1}{\gamma}_{ij} =\partial_{m-2}\overset{2}{\gamma}_{ij}=\ldots=\partial_1 \overset{m-1}{\gamma}_{ij},\quad \partial_m \overset{1}{\gamma}_{ij}=\partial_{m-1}\overset{2}{\gamma}_{ij} =\partial_{m-2}\overset{3}{\gamma}_{ij}=\ldots\\ &\ldots=\partial_2 \overset{m-1}{\gamma}_{ij}=0,\quad \partial_m \overset{2}{\gamma}_{ij}=\partial_{m-1}\overset{3}{\gamma}_{ij} =\ldots=\partial_3 \overset{m-1}{\gamma}_{ij}=0,\ \ldots \tag{4}\\ &\ldots,\quad \partial_m \overset{m-2}{\gamma}_{ij}=\partial_{m-1}\overset{m-1}{\gamma}_{ij}=0,\quad \partial_m \overset{m-1}{\gamma}_{ij}=0 . \end{aligned} \]
Replacing here the differentiation indices \(1,2,3,\ldots,m\) by \(ms+1\), \(ms+2\), \(ms+3,\ldots,m(s+1)\), respectively, where \(s=1,2,\ldots,r-1\), we obtain another \(r-1\) systems analogous to (4). Introducing the imaginary unit \(\varepsilon\), satisfying the condition \(\varepsilon^m=0\), we assert that conditions (4) are necessary and sufficient for the components of the tensor
\[ G_{ij}=\overset{m-1}{\gamma}_{ij} +\varepsilon \overset{m-2}{\gamma}_{ij} +\varepsilon^2 \overset{m-3}{\gamma}_{ij} +\ldots+\varepsilon^{m-2}\overset{1}{\gamma}_{ij} +\varepsilon^{m-1}g_{ij} \tag{5} \]
to be analytic functions of \(r\) complex coordinates
\[ U^s=u^{m(s-1)+1}+\varepsilon u^{m(s-1)+2} +\varepsilon^2u^{m(s-1)+3}+\ldots+\varepsilon^{m-1}u^{ms},\quad s=1,2,\ldots,r. \]
The matrix of the tensor \(G_{ij}\) has a cellular structure; moreover, in consequence of the described structure of the matrices \(g_{ij}, \overset{1}{\gamma}_{ij},\ldots,\overset{m-1}{\gamma}_{ij}\), the cells \(G_{\alpha\beta}\) have the form
\[ \left( \begin{array}{cccccc} G_{\alpha\beta} & \varepsilon G_{\alpha\beta} & \varepsilon^2G_{\alpha\beta} & \cdots & \cdots & \varepsilon^{m-1}G_{\alpha\beta}\\ \varepsilon G_{\alpha\beta} & \varepsilon^2G_{\alpha\beta} & \cdots & \cdots & \varepsilon^{m-1}G_{\alpha\beta} & 0\\ \varepsilon^2G_{\alpha\beta} & \cdots & \cdots & \varepsilon^{m-1}G_{\alpha\beta} & 0 & 0\\ \cdot & \cdots & \cdot & \cdot & \cdot & \cdot\\ \cdot & \cdots & \cdot & \cdot & \cdot & \cdot\\ \varepsilon^{m-1}G_{\alpha\beta} & 0 & \cdots & \cdots & \cdots & 0 \end{array} \right). \]
Observing that each of them is determined by its left upper corner element, we come to the conclusion that the \(B\)-space of characteristic
\[ \left[\left(\overbrace{m,m,m,\ldots,m}^{r}\right)\right] \]
admits a mapping onto an \(r\)-dimensional complex analytic Riemannian space with metric tensor \(G_{\alpha\beta}\) \((\alpha,\beta=1,2,\ldots,r)\). For \(m=3\) these results were first obtained by A. P. Shirokov (6).
In what follows we shall dwell in detail on the \(B\)-space of characteristic \([(m,m)]\), since it is mapped onto the two-dimensional complex space \(V_2\), and the properties of the latter determine a number of interesting properties of this \(B\)-space.* In particular, in \(V_2\) one can introduce an isothermal coordinate system in which the complex metric has the form
\[ (G_{\alpha\beta})= \begin{pmatrix} B & 0\\ 0 & -\sigma B \end{pmatrix} \quad(\alpha,\beta=1,2), \]
where \(\sigma=\pm1\) determines the signature of this complex \(V_2\). Then for the metric of the original \(B\)-space we obtain a matrix of the same structure, where \(B\) is an \(m\)-dimensional cell of the form (3).
\[ \rule{6em}{0.4pt} \]
* The study of the \(B\)-space of characteristic \([m]\) is of no interest, since it is Euclidean ((3), note on p. 155).
Consider the discriminant bivector \(E_{\alpha\beta}\) of the complex \(V_2\) and define its essential component by means of the equality
\[ E_{12}^2=-\sigma\left(G_{11}G_{22}-G_{12}^2\right). \]
Then in isothermal coordinates \(E_{12}=B\), or, in more detail,
\[ e_{12}^{m-1}+\varepsilon e_{12}^{m-2}+\varepsilon^2 e_{12}^{m-3}+\cdots+\varepsilon^{m-1}e_{12}^{0} = \gamma_{11}^{0}+\varepsilon\gamma_{11}^{m-1}+\varepsilon^2\gamma_{11}^{m-2}+\cdots+\varepsilon^{m-1}\gamma_{11}^{m-3}, \]
whence, by comparison, we obtain \(e_{12}^{p}=\gamma_{11}^{p}\) \((p=0,1,2,\ldots,m-1;\ \gamma_{ij}^{0}=g_{ij})\).
Returning to the \(B\)-space, we shall have \(m\) bivectors \(e_{ij}^{p}\), whose matrices in the isothermal coordinate system have the form
\[ \left(e_{ij}^{p}\right)= \begin{pmatrix} 0 & B_p\\ -B_p & 0 \end{pmatrix}, \tag{6} \]
where \(B_p\) is an entry of the matrix \(\gamma_{ij}^{p}\), and which, as a consequence, satisfy the relations
\[ \gamma_i^{pk} e_{kj}^{q}=e_{ij}^{p+q}. \tag{7} \]
Let us now transform the isothermal coordinates \(u^i\) according to the formulas
\[ x^\alpha=\frac{1}{\sqrt{2}}\left(u^\alpha+e u^{\alpha+m}\right),\qquad \bar{x}^{\alpha}=\frac{1}{\sqrt{2}}\left(u^\alpha-e u^{\alpha+m}\right),\qquad \alpha=1,2,\ldots,m, \tag{8} \]
where \(e\) is the imaginary unit, defined by the condition \(e^2=\sigma\). In the coordinate system \((x^\alpha,\bar{x}^\alpha)\), the matrix \(g_{ij}\) takes the form
\[ (g_{ij})= \begin{pmatrix} 0 & B\\ B & 0 \end{pmatrix}. \tag{9} \]
The structure of the matrices \(\gamma_{ij}^{p}\) will be analogous, and, consequently, the transformation (8) is equivalent to passing to isotropic coordinates of the complex \(V_2\).
Writing the systems of equations (4) in the new variables, we obtain that, by virtue of them, the Kähler–Rashevskii conditions are satisfied:
\[ \frac{\partial g_{\alpha\bar{\beta}}}{\partial \bar{x}^{\gamma}} = \frac{\partial g_{\alpha\bar{\gamma}}}{\partial \bar{x}^{\beta}}, \qquad \frac{\partial g_{\bar{\alpha}\beta}}{\partial x^\gamma} = \frac{\partial g_{\bar{\alpha}\gamma}}{\partial x^\beta}, \tag{10} \]
as a result of which our space, for \(\sigma=-1\), will be a Kähler space \((^7)\), and for \(\sigma=1\), the so-called stratified space of P. K. Rashevskii \((^8)\), whose complex interpretation by means of double numbers was given by B. A. Rozenfeld \((^9)\). The connection of \(B\)-spaces with Kähler–Rashevskii manifolds was first pointed out by A. P. Norden \((^1)\).
Since the remaining tensors \(\gamma_{ij}^{p}\) have a structure similar to that of the tensor \(g_{ij}\), and the conditions (10) are satisfied for them as well, they are also Kähler–Rashevskii metrics, which, however, will be degenerate; moreover, the degree of degeneracy increases together with \(p\).
Transforming the bivectors (6) by means of (8), we conclude that
\[ \left(e_{ij}^{p}\right)= \begin{pmatrix} 0 & -\sigma e\,\gamma_{\alpha\bar{\beta}}^{p}\\ \sigma e\,\gamma_{\alpha\bar{\beta}}^{p} & 0 \end{pmatrix}, \]
whence it follows that each of the bivectors \(e_{ij}^{p}\) is covariantly constant and corresponds to “its own” Kähler metric (cf. \((^7)\), pp. 119–120).
As a consequence of (10), there exist real functions \(\stackrel{p}{\Gamma}(x^\alpha,\bar x^\alpha)\), which we shall call Kern functions, such that
\[ \stackrel{p}{\gamma}_{\alpha\bar\beta} = \partial^2 \stackrel{p}{\Gamma}/\partial x^\alpha \partial \bar x^\beta, \qquad p=0,1,2,\ldots,m-1. \tag{11} \]
Comparing (9) and the analogous matrices of the remaining metrics, we write down the complete set of relations between their components; whence, by virtue of (11), we obtain that the Kern functions satisfy the system of differential equations
\[ \frac{\partial \stackrel{r-s}{\Gamma}}{\partial x^{m-r}} - \frac{\partial \stackrel{r-s-1}{\Gamma}}{\partial x^{m-r+1}} = f_{m-r}^{\,r-s}(x^\alpha), \qquad r=s+1,\ldots,m-1;\quad s=0,1,2,\ldots,m-2; \]
\[ \frac{\partial \stackrel{q}{\Gamma}}{\partial x^{m-t}} = \varphi_t^q(x^\alpha), \qquad t\le q-1;\quad q=1,2,\ldots,m-1. \tag{12} \]
As a consequence of (11), the metrics \(\stackrel{p}{\gamma}_{ij}\) do not change if the Kern functions are replaced by new ones according to the formulas
\[ \stackrel{p}{\Gamma}'(x^\alpha,\bar x^\alpha) = \stackrel{p}{\Gamma}(x^\alpha,\bar x^\alpha) - \stackrel{p}{\Gamma}(x^\alpha,\bar c^\alpha) - \stackrel{p}{\Gamma}(\bar x^\alpha,c^\alpha), \tag{13} \]
where \(c^\alpha=\mathrm{const}\). Solving (13) with respect to \(\stackrel{p}{\Gamma}(x^\alpha,\bar x^\alpha)\), and then substituting them into equations (12), which are satisfied for arbitrary \(\bar x^\alpha\), we obtain that the new Kern functions \(\stackrel{p}{\Gamma}'\) satisfy the system of equations (12) without right-hand sides, and, owing to the reality of \(\stackrel{p}{\Gamma}'\), also the system obtained from it by overlining all \(x^\alpha\). After passing, by means of (8), to the coordinates \(u^i\), we arrive at systems analogous to (4), which express the fact that the combination
\[ \mathcal{G} = \stackrel{m-1}{\Gamma} + \varepsilon \stackrel{m-2}{\Gamma} + \varepsilon^2 \stackrel{m-3}{\Gamma} +\cdots+ \varepsilon^{m-2}\stackrel{1}{\Gamma} + \varepsilon^{m-1}G \qquad \left(G=\stackrel{0}{\Gamma}\right) \tag{14} \]
is an analytic function of the complex coordinates
\(U^1=u^1+\varepsilon u^2+\varepsilon^2u^3+\cdots+\varepsilon^{m-1}u^m\),
\(U^2=u^{m+1}+\varepsilon u^{m+2}+\varepsilon^2u^{m+3}+\cdots+\varepsilon^{m-1}u^{2m}\).
Conversely, if such a function is given and the algebra of complex or dual numbers over which the Kähler space is constructed is specified, then, passing to the variables \(x^\alpha,\bar x^\alpha\), one can construct on the functions \(\stackrel{p}{\Gamma}\) Kähler metrics that will have a special structure of type (9). Referring this space to the coordinates \(u^i\), we arrive at a \(B\)-space of characteristic \([(m,m)]\).
Summarizing what has been said, one may assert that the specification of a \(B\)-space of characteristic \([(m,m)]\) is equivalent to the specification of a function (14) of two complex variables \(U^1,U^2\) and of an algebra of complex or dual numbers.
The present work generalizes results obtained by the author for the four-dimensional case \({}^{10}\).
Kazan State University
named after V. I. Ulyanov-Lenin
Received
17 VIII 1962
REFERENCES
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