A. S. SOLODOVNIKOV

GEOMETRIC DESCRIPTION OF ALL POSSIBLE REPRESENTATIONS OF A RIEMANNIAN METRIC IN LEVI-CIVITA FORM

(Presented by Academician I. G. Petrovskii, 9 VI 1961)

1°. As is known, the problem of the geodesic mapping of Riemannian spaces leads to metrics of the form (see (¹))

\[ ds^{*}= \Pi_{\alpha}' |f_{\alpha}-f_1|\,ds_1^2 + \Pi_{\alpha}' |f_{\alpha}-f_2|\,ds_2^2 +\ldots+ \Pi_{\alpha}' |f_{\alpha}-f_p|\,ds_p^2, \tag{1} \]

which are called Levi-Civita metrics. Here an important role is played by the associated metric

\[ ds^2=\Pi_{\alpha}' |f_{\alpha}-f_1|\,dy_1^2 +\Pi_{\alpha}' |f_{\alpha}-f_2|\,dy_2^2 +\ldots+ \Pi_{\alpha}' |f_{\alpha}-f_p|\,dy_p^2. \]

If \(ds^{*2}\) does not have constant curvature, then the representation of \(ds^2\) in Levi-Civita form is possible in only one way. If \(ds^{*2}\) has constant curvature \(K\), the metric (1) is called exceptional; for exceptional metrics, representation in Levi-Civita form is possible, generally speaking, in an infinite set of ways. The purpose of the present article is to give an effective description of all representations of an exceptional metric in Levi-Civita form. For lack of space we shall confine ourselves to the most interesting case \(K<0\).

2°. Suppose, for definiteness, that

\[ f_1(x_1),\ldots,f_r(x_r)\ne \mathrm{const},\qquad f_{r+1},\ldots,f_p=\mathrm{const}\quad (0<r<p). \tag{2} \]

Writing \(ds^2\) in the form

\[ ds^2=ds_0^2+\varphi_{r+1}ds_{r+1}^2+\ldots+\varphi_p ds_p^2 \quad (\dim ds_0^2=r) \tag{3} \]

(where \(ds_0^2\) denotes the sum of the first \(r\) terms in (1)), we see that \(ds_0^2\) defines a space \(V(K)\), and (3) is a \(K\)-decomposition of the metric \(ds^2\).

Thus the problem posed splits into two:

I. To describe all \(K\)-decompositions in the given space \(V(K)\).

II. For each of such \(K\)-decompositions (3), to indicate all ways of representing it in the form (1); in fact, what is required here is to find certain special \(r\)-orthogonal coordinate systems in the space of constant curvature \(ds_0^2\).

3°. We shall describe certain special coordinate systems in the Lobachevskii space \(L_r\) of curvature \(-1\).

A. Fix a certain plane \(L_{r_0-1}\). Through an arbitrary point \(M\in L_r\) draw the plane \(L'_{r_0}\) containing \(L_{r_0-1}\). As coordinates of the point \(M\) we take: a) the coordinates of the point \(M_0\), the projection of \(M\) onto \(L_{r_0-1}\), and also the number \(\tau=MM_0\); b) the angular coordinates of the plane \(L'_{r_0}\) itself. This is a generalized polar coordinate system with axis (more precisely, “pole”) \(L_{r_0-1}\). The metric of \(L_r\) in such a system has the form \(ds_0^2+\varphi\,d\lambda_0^2\), where \(ds_0^2\) is the metric in \(L'_{r_0}\).

The following items B, C, D exhaust all coordinate systems in which the metric of \(L_r\) is reduced to the form

\[ dS_0^2=ds_0^2+\varphi\,d\lambda_0^2+\psi\,d\mu_0^2, \tag{4} \]

where the functions \(\varphi,\psi\) depend only on the variables \(x^s\).*

B. Two planes \(\widetilde L\), \(\widetilde{\widetilde L}\) of dimensions \(r_0+r_1-1\) and \(r_0+r_2-1\), respectively, are fixed, intersecting orthogonally in some plane \(L^*\). Through an arbitrary point \(M\) pass the plane \(L'_{r_0+r_1}\supset \widetilde L\) and the plane \(L'_{r_0+r_2}\supset \widetilde{\widetilde L}\). As the coordinates of the point \(M\) one takes: a) the coordinates of the point \(M^*\)—the projection of \(M\) onto \(L^*\)—in \(L^*\), and also the numbers \(\tau_1=M\widetilde M\), \(\tau_2=M\widetilde{\widetilde M}\); b) the angular coordinates of the plane \(L'_{r_0+r_1}\) \((x^{\mu_0})\); c) the angular coordinates of the plane \(L'_{r_0+r_2}\) \((x^{\lambda_0})\). In this case \(\varphi=\operatorname{sh}^2\tau_1\), \(\psi=\operatorname{sh}^2\tau_2\).

C. A plane \(\widetilde L\) of dimension \(r_2\) is fixed. At each point \(\widetilde M\in \widetilde L\) a plane \(L'\) is constructed—the complete normal to \(\widetilde L\). In each plane \(L'\) a generalized polar coordinate system is introduced with axis \(L'_{r_0-1}\) passing through \(\widetilde M\); all such coordinate systems (in the various \(L'\)) are coordinated in a natural way. As the coordinates of a point \(M\in L'\) one takes: a) the coordinates of the point \(\widetilde M\) in \(\widetilde L\) \((x^{\mu_0})\); b) the polar coordinates of the point \(M\) in \(L'\). Instead of specifying at each point \(\widetilde M\) the axis \(L'_{r_0-1}\), one may specify one plane \(\widetilde{\widetilde L}=\widetilde L\dot{+}L'_{r_0-1}\). In this case \(\varphi=\operatorname{sh}^2\tau_1\) (\(\tau_1\) is the distance to the axis), \(\psi=\operatorname{ch}^2\tau_2\), where \(\tau_2=M\widetilde M\).

D. An orisphere \(O\) of dimension \(r_2\) is fixed. At each point \(\widetilde M\in O\) a plane \(L'\) is constructed—the complete normal to \(O\). Obviously, \(L'\) contains the line \(l'\)—the axis of the orisphere at the point \(\widetilde M\). In \(L'\) a polar coordinate system is introduced with axis \(L'_{r_0-1}\supset l'\); all such systems are coordinated in a natural way. As the coordinates of a point \(M\in L'\) one takes: a) the coordinates of the point \(\widetilde M\) in \(O\) \((x^{\mu_0})\); b) the polar coordinates of the point \(M\) in \(L'\). In this case \(\varphi=\operatorname{sh}^2\tau_1\), \(\psi=e^{2\tau_2}\), where \(\tau_2\) is the distance to the hyperorisphere containing \(O\). In general the coordinate system is determined by a pair of planes \(\widetilde L\), \(\widetilde{\widetilde L}\) \((\widetilde L=L_{r_2+1}\supset O,\ \widetilde{\widetilde L}=O\dot{+}L'_{r_0-1})\) and a line \(l\) belonging to both planes.

\(4^\circ\). Solution of Problem I. We describe here all \(K\)-decompositions of the form

\[ ds^2=ds_0^2+\varphi\,dS_1^2+\psi\,dS_2^2 \tag{5} \]

(the number of forms \(dS_\alpha^2\) is equal to two). Let

\[ ds^2=dS_0^2+F_1ds_1^2+\ldots+F_qds_q^2 \tag{6} \]

be a maximal \(K\)-decomposition. The problem reduces to finding, in the space \(dS_0^2\) of constant curvature \(K\), all coordinate systems of the form (4) which in a certain way “split” the functions \(F_1,\ldots,F_q\); namely, some of them lead to the form \(\Phi_\alpha=\varphi\rho_\alpha\), and the other part to the form \(\Psi_\beta=\psi\sigma_\beta\), where the functions \(\rho_\alpha\) depend only on the coordinates \(x^{\lambda_0}\), and the functions \(\sigma_\beta\) on \(x^{\mu_0}\). Indeed, then we shall have (5), where

\[ dS_1^2=d\lambda_0^2+\sum_\alpha \rho_\alpha\,d\lambda_\alpha^2, \qquad dS_2^2=d\mu_0^2+\sum_\beta \sigma_\beta\,d\mu_\beta^2 \]

(the forms \(d\lambda_\alpha^2,d\mu_\beta^2\) coincide, in some order, with \(ds_1^2,\ldots,ds_q^2\)).

We regard (6) as given. For \(K=-1\) this decomposition has one of the forms:

\[ ds^2=dS_0^2+y_{r+1}^2ds_1^2+y_1^2ds_2^2+\ldots+y_{q-1}^2ds_q^2 \quad (q\leq r+1); \tag{7} \]

\[ ds^2=dS_0^2+(y_{r+1}-y_r)^2ds_1^2+y_1^2ds_2^2+\ldots+y_{q-1}^2ds_q^2 \quad (q\leq r); \tag{8} \]

\[ ds^2=dS_0^2+y_1^2ds_1^2+y_2^2ds_2^2+\ldots+y_q^2ds_q^2 \quad (q\leq r), \tag{9} \]

where \(dS_0^2\) is the metric of the imaginary unit sphere \(y_1^2+\ldots+y_r^2-y_{r+1}^2=-1\) in the pseudo-Euclidean space \(E_{r+1}^{(1)}\): \(dl^2=dy_1^2+\ldots+dy_r^2-dy_{r+1}^2\).

\[ {}^* x^{s_0}\text{ are coordinates in }ds_0^2,\quad x^{\lambda_0}\text{ in }d\lambda_0^2,\quad x^{\mu_0}\text{ in }d\mu_0^2. \]
We also put \(r_0=\dim ds_0^2,\ r_1=\dim d\lambda_0^2,\ r_2=\dim d\mu_0^2\).

We identify the sphere indicated above with Lobachevsky space \(L_r\). All possible planes in \(L_r\) are sections of the sphere by subspaces of the space \(E_{r+1}^{(1)}{}^*\).

In case (7), all coordinate systems (4) of interest to us are obtained as follows: we divide the coordinates \(y_{r+1}, y_1,\ldots,y_{q-1}\) into two groups, for example, \(y_{r+1}, y_1,\ldots,y_k\) and \(y_{k+1},\ldots,y_{q-1}\); the coordinate system (4) is determined by scheme C, where \(\widetilde{\widetilde L}\) is the section of \(L_r\) by the subspace contained in \(y_{k+1}=\cdots=y_{q-1}=0\), and \(\widetilde L\) is the section of \(L_r\) by the subspace containing the axes \(Oy_{r+1}, Oy_1,\ldots,Oy_k\); moreover, it is required that \(\widetilde L\subset\widetilde{\widetilde L}\).

In case (8), all coordinate systems (4) of interest to us are obtained as follows: we divide the coordinates \(y_1,\ldots,y_{q-1}\) into two groups, for example \(y_1,\ldots,y_k\) and \(y_{k+1},\ldots,y_{q-1}\); the coordinate system (4) is determined by scheme C, where \(\widetilde{\widetilde L}\) is determined as above, and \(\widetilde L\) is the section of \(L_r\) by the subspace containing the axes \(Oy_1,\ldots,Oy_k\) and the line \(y_1=\cdots=y_{r-1}=0,\ y_r=y_{r+1}\); moreover, it is required that \(\widetilde L\subset\widetilde{\widetilde L}\).

In case (9), coordinate systems of all types B, C, D are possible. As above, we divide the coordinates \(y_1,\ldots,y_q\) into two groups, for example \(y_1,\ldots,y_k\) and \(y_{k+1},\ldots,y_q\). For scheme B one takes the planes \(\widetilde L,\widetilde{\widetilde L}\) contained in the subspaces \(y_1=\cdots=y_k=0\) and \(y_{k+1}=\cdots=y_q=0\), respectively; moreover, it is required that \(\widetilde L\) and \(\widetilde{\widetilde L}\) intersect (generally speaking, in some plane \(L^s\)) orthogonally. For scheme C one takes the plane \(\widetilde{\widetilde L}\) lying in the subspace \(y_{k+1}=\cdots=y_q=0\), and \(\widetilde L\) in the subspace containing the axes \(Oy_1,\ldots,Oy_k\); moreover, \(\widetilde L\subset\widetilde{\widetilde L}\). For scheme D one takes a line \(l\) lying in the subspace \(y_1=\cdots=y_q=0\), the plane \(\widetilde{\widetilde L}\) in the subspace \(y_{k+1}=\cdots=y_q=0\), and the plane \(\widetilde L\) in the subspace containing the axes \(Oy_1,\ldots,Oy_k\).

5°. Solution of Problem II. Consider the exceptional metric (1). Without loss of generality, to conditions (2) one may add \(f_1<f_2<\cdots<f_r\). Passing to new coordinates \(\rho_1=f_1,\ldots,\rho_r=f_r\) brings \(ds_0^2\) to the form

\[ ds_0^2=\sum_{i=1}^{r}\frac{1}{P(\rho_i)}\Pi_i'(\rho_1-\rho_i)\,d\rho_i^2, \tag{10} \]

where \(P(\rho)\) is a polynomial among whose roots all the numbers \(f_{r+1},\ldots,f_p\) are contained. For \(K=-1\) the polynomial \(P(\rho)\) has the form
\((-1)^r(\rho-\alpha_1)\cdots(\rho-\alpha_{r-1})(\rho-\beta)(\rho-\gamma)\), \(\alpha_1<\alpha_2<\cdots<\alpha_{r-1}\); for \(\beta\) and \(\gamma\) there are the various possibilities listed in items 1)—6). Thus, starting from a given \(K\)-decomposition (3), we must solve two problems:

1. Find all coordinate systems in \(ds_0^2\) in which (10) occurs. For this purpose we shall use [2], where the required coordinate systems were found for \(r=3\); for \(r>3\) everything is analogous.

2. Among these coordinate systems select those for which

\[ \varphi_{r+1}=A_{r+1}\prod_{i=1}^{r}(\rho_i-f_{r+1}),\ldots,\varphi_p=A_p\prod_{i=1}^{r}(\rho_i-f_p), \]

where \(A_{r+1},\ldots,A_p\) are constants, and \(f_{r+1},\ldots,f_p\) are part of the roots of \(P(\rho)\).

* Containing at least one vector with negative scalar square.

Let us list all the solutions of problem 1. Recall that \(ds_0^2\) is the metric on the imaginary unit sphere in \(E_{r+1}^{(1)}\).

1) \(\alpha_{r-1}<\beta<\gamma\). Let \(z_1,\ldots,z_{r+1}\) be any Cartesian coordinates* in \(E_{r+1}^{(1)}\). Consider the equation with respect to \(\rho\):

\[ \frac{z_1^2}{\rho-\alpha_1}+\cdots+\frac{z_{r-1}^2}{\rho-\alpha_{r-1}}+ \frac{z_r^2}{\rho-\beta}-\frac{z_{r+1}^2}{\rho-\gamma}=0. \tag{11} \]

Its roots satisfy the condition
\(\rho_1<\alpha_1<\rho_2<\cdots<\alpha_{r-1}<\rho_r<\beta<\gamma\).
Taking \(\rho_1,\ldots,\rho_r\) as new coordinates in \(ds_0^2\), we obtain (10).

2) \(\alpha_{k-1}<\beta<\gamma<\alpha_k\) \((1<k\le r-1)\). Equation (11).

3) \(\beta=a+bi,\quad \gamma=a-bi\). Equation (11) with \(z_r\) replaced by
\(\tilde z_r=\frac{1}{\sqrt2}(z_r-iz_{r+1})\) and \(z_{r+1}\) by
\(\tilde z_{r+1}=\frac{i}{\sqrt2}(z_r+iz_{r+1})\).

4) \(\alpha_{r-1}<\beta=\gamma\). The equation

\[ \frac{z_1^2}{\rho-\alpha_1}+\cdots+\frac{z_{r-1}^2}{\rho-\alpha_{r-1}}+ \frac{z_r^2}{\rho-\beta}-\frac{z_{r+1}^2}{\rho-\beta} -\frac{(z_r-z_{r+1})^2}{(\rho-\beta)^2}=0. \tag{12} \]

5) \(\alpha_{k-1}<\beta=\gamma<\alpha_k\). Equation (12).

6) \(\beta=\gamma=\alpha_k\). The equation

\[ \frac{z_1^2}{\rho-\alpha_1}+\cdots+\frac{z_{r-1}^2}{\rho-\alpha_{r-1}}+ \frac{z_r^2}{\rho-\alpha_k}-\frac{z_{r+1}^2}{\rho-\alpha_k} -\frac{2z_k(z_r-z_{r+1})}{(\rho-\alpha_k)^2} +\frac{(z_r-z_{r+1})^2}{(\rho-\alpha_k)^3}=0. \]

Let us list all the solutions of problem 2. First of all, in \(E_{r+1}^{(1)}\) we fix a Cartesian coordinate system that brings the given \(K\)-decomposition (3) to one of the forms (7), (8), (9). All the desired coordinate systems \(\rho_1,\ldots,\rho_r\) in \(ds_0^2\) are obtained from the admissible Cartesian coordinate systems in \(E_{r+1}^{(1)}\) listed below according to the rules indicated in 1)—6).

Case (9). 1)—2)** We fix an (ordered) group of \(q=r-p\) roots of the polynomial \(P(\rho)\), excluding \(\gamma\): \(f_{r+1}=\alpha_{i_1},\ldots,f_p=\alpha_{i_q}\). We perform the corresponding renumbering of the coordinates \(y\): \(y_1\to y_{i_1},\ldots,\ldots,y_q\to y_{i_q}\). Those coordinate systems \(z_1,\ldots,z_{r+1}\) are regarded as admissible in which the axes \(Oz_{i_1},\ldots,Oz_{i_q}\) coincide with the axes \(Oy_{i_1},\ldots,Oy_{i_q}\) (respectively). If one of the selected roots, for example \(f_{r+1}\), is \(\beta\), then we set \(i_1=r\). 3)—6) We fix a group of \(q\) roots, excluding \(\beta,\gamma\). Thereafter everything is as in 1)—2).

Case (7). 1)—2) We fix a group of \(q\) roots, with the first being \(\gamma\): \(f_{r+1}=\gamma,\ f_{r+2}=\alpha_{i_1},\ldots,f_p=\alpha_{i_{q-1}}\). We perform the renumbering:
\(y_1\to y_{i_1},\ldots,y_{q-1}\to y_{i_{q-1}}\). Those systems are regarded as admissible in which the axes \(Oz_{r+1},Oz_{i_1},\ldots,Oz_{i_{q-1}}\) coincide with the axes \(Oy_{r+1},Oy_{i_1},\ldots,Oy_{i_{q-1}}\) (respectively).

Case (8). 4)—5) Everything is as in the preceding case, but instead of coincidence of the axes \(Oz_{r+1}, Oy_{r+1}\) one requires coincidence of the planes \(Oz_rz_{r+1}\) and \(Oy_ry_{r+1}\). 6) We fix the group \(f_{r+1}=\alpha_k,\ f_{r+2}=\alpha_{i_1},\ldots,f_p=\alpha_{i_{q-1}}\). Thereafter everything is as in the preceding case.

Moscow State
Correspondence Pedagogical Institute

Received
1 VI 1961

References

1. A. S. Solodovnikov, Tr. seminara po tenzorn. analizu, vol. XI (1961).
2. M. N. Olevskii, Matem. sborn., 27, No. 3 (1950).

* We use only such Cartesian coordinate systems in \(E_{r+1}^{(1)}\) in which the metric is \(dz_1^2+\cdots+dz_r^2-dz_{r+1}^2\) (the minus sign stands before the last square).

** Below follows a listing of admissible coordinate systems for items 1)—2). The subsequent numbering has an analogous meaning.