MATHEMATICS

Yu. I. Ermakov

THREE-DIMENSIONAL SPACE WITH A CUBIC SEMIMETRIC

(Presented by Academician I. G. Petrovskii, 13 IX 1957)

Let \(F_3^{(3)}\) be a three-dimensional Finsler space whose metric is given by the cubic differential form

\[ ds^3=a_{\alpha\beta\gamma}\,dx^\alpha dx^\beta dx^\gamma \qquad (\alpha,\ \beta,\ \gamma,\ldots,\omega=1,\ 2,\ 3) \]

with nonzero discriminant \(\bigl((^3),\ \text{p. }313\bigr)\). Recently K. Tonooka \((^2)\), extending to the case \(F_3^{(3)}\) the method indicated by A. E. Liber in \((^1)\), constructed a system of algebraic comitants for \(F_3^{(3)}\) and used them to find a linear affine connection. In the present note, a connection in \(F_3^{(3)}\) invariant with respect to a conformal transformation of the metric is constructed, and some questions of the differential geometry of the space \(X_3\) with a given pseudotensor field whose discriminant is nonzero are also considered. We shall denote such a space by \(\mathfrak F_3^{(3)}\).

1. Since the discriminant \(\mathfrak A\) of the tensor \(a_{\alpha\beta\gamma}\) is nonzero, i.e. \(\mathfrak A\ne0\), the fundamental components
   \(B^{\alpha\beta\gamma}\), \(A^{\alpha\beta\gamma}\), \(L_{\alpha\beta\gamma}\), \(P_\mu^{\lambda\alpha\beta\gamma}\) of the fundamental tensor \(a_{\alpha\beta\gamma}\) are successively determined \((^2)\) from the relations

\[ a_{\alpha\beta\gamma}B^{\alpha\beta\gamma}=0,\qquad h_{\alpha\beta\gamma}B^{\alpha\beta\gamma}=\mathfrak A, \tag{1} \]

where \(h_{\alpha\beta\gamma}\) are the coefficients of the Hessian of the cubic ternary form corresponding to the fundamental tensor;

\[ A^{\alpha\beta\gamma} = \frac{1}{2}\varepsilon^{\alpha\alpha_1\alpha_2} \varepsilon^{\gamma\gamma_1\gamma_2} B^{\beta\beta_1\beta_2} a_{\alpha_1\beta_1\gamma_1} a_{\alpha_2\beta_2\gamma_2}, \tag{2} \]

by \(\varepsilon^{\alpha\beta\gamma}\) is denoted the fundamental contravariant trivector density of weight \(+1\);

\[ A^{\alpha\beta\gamma}L_{\alpha\beta\gamma}=0,\qquad B^{\gamma\beta\gamma}L_{\alpha\beta\lambda}=\mathfrak A\delta_\lambda^\gamma; \tag{3} \]

\[ P_\mu^{\lambda\alpha\beta\gamma} = \delta_\mu^\lambda A^{\alpha\beta\lambda} + \varepsilon^{\gamma\sigma_1\rho_1} \varepsilon^{\lambda\alpha\rho_2} a_{\sigma_1\sigma_2\mu} a_{\rho_1\rho_2\omega} B^{\alpha\beta\omega}. \tag{4} \]

The comitants listed are symmetric in the indices \(\alpha,\beta,\gamma\), except for \(L_{\alpha\beta\gamma}\), which is symmetric only in the indices \(\alpha,\beta\), and they satisfy the relations

\[ \text{a)}\quad A^{\alpha\beta\gamma}a_{\alpha\beta\lambda} =\mathfrak A\delta_\lambda^\gamma, \qquad \text{b)}\quad B^{\gamma\beta\gamma}h_{\lambda\beta\lambda} =\frac{1}{3}\mathfrak A\delta_\lambda^\gamma; \tag{5} \]

\[ \text{a)}\quad P_\mu^{\lambda\alpha\beta\gamma}a_{\nu\beta\gamma} = \mathfrak A\delta_\mu^\delta\delta_\nu^\lambda, \qquad \text{b)}\quad P_\mu^{\lambda\alpha\beta\gamma}L_{\alpha\beta\gamma}=0. \tag{6} \]

The linear connection in \(F_3^{(3)}\) is completely determined \((^2)\) by the requirement

\[ P_\mu^{\lambda\alpha\beta\gamma}\nabla_\nu a_{\alpha\beta\gamma}=0, \]

and the connection coefficients \(\Gamma_{\nu\mu}^{\lambda}\) have the form

\[ \Gamma_{\nu\mu}^{\lambda} = \frac{1}{3\mathfrak A} P_\mu^{\lambda\alpha\beta\gamma} \partial_\nu a_{\alpha\beta\gamma}. \tag{7} \]

Theorem. For the mapping of \(F_3^{(3)}\) into Minkowski space, it is necessary and sufficient that the torsion tensor of the connection and the vector density
\(\chi_\nu = B^{\alpha\beta\gamma}\nabla_\nu a_{\alpha\beta\gamma}\) vanish. (In \((^2)\), in the formulation of the theorem the requirement that the torsion tensor of the connection vanish was not taken into account.)

1. Let us now consider a conformal transformation of the metric in \(F_3^{(3)}\), which is defined by the relation

\[ {}^{*}a_{\alpha\beta\gamma}=\sigma a_{\alpha\beta\gamma}. \tag{8} \]

Then the basic algebraic concomitants are transformed as follows:

\[ {}^{*}\mathfrak A=\sigma^{12}\mathfrak A,\qquad {}^{*}B^{\alpha\beta\gamma}=\sigma^9 B^{\alpha\beta\gamma},\qquad {}^{*}A^{\alpha\beta\gamma}=\sigma^{11}A^{\alpha\beta\gamma}, \tag{9} \]

\[ {}^{*}P_{\mu}^{\lambda\alpha\beta\gamma} =\sigma^{11}P_{\mu}^{\lambda\alpha\beta\gamma}. \]

Let us establish a similar connection between the coefficients of the connection (7). Differentiating (8) with respect to \(\xi^\nu\), we have

\[ \partial_\nu{}^{*}a_{\alpha\beta\gamma} =\sigma_\nu a_{\alpha\beta\gamma} +\sigma\cdot \partial_\nu a_{\alpha\beta\gamma} \qquad \left(\sigma_\nu=\frac{\partial_\nu\sigma}{\sigma}\right). \]

Then, by virtue of (6a), (8), and (9), we obtain

\[ {}^{*}P_{\mu}^{\lambda\alpha\beta\gamma} \partial_\nu{}^{*}a_{\alpha\beta\gamma} = \sigma^{12}\left( P_{\mu}^{\lambda\alpha\beta\gamma}\partial_\nu a_{\alpha\beta\gamma} +\mathfrak A\delta_\mu^\lambda\sigma_\nu \right) \]

and, consequently,

\[ {}^{*}\Gamma_{\nu\mu}^{\lambda} = \Gamma_{\nu\mu}^{\lambda} +\frac{1}{3}\sigma_\nu\delta_\mu^\lambda . \tag{10} \]

(10) shows that, under a conformal transformation of the metric, the object (7) is transformed projectively. Further, if \(S_{\nu\mu}^{\lambda}\) is the torsion tensor of the connection (7), then under the transformation (8) \(S_{\nu\mu}^{\lambda}\) is transformed as follows:

\[ {}^{*}S_{\nu\mu}^{\lambda} = S_{\nu\mu}^{\lambda} +\frac{1}{3}\sigma_{[\nu}\delta_{\mu]}^\lambda . \]

Contracting the last equality with respect to the indices \(\lambda,\mu\), then multiplying the result of the contraction by \(\delta_\mu^\lambda\) and subtracting what is obtained from (10), we have

\[ {}^{*}\overset{(k)}{\Gamma}{}_{\nu\mu}^{\lambda} = \overset{(k)}{\Gamma}{}_{\nu\mu}^{\lambda}, \quad\text{where} \]

\[ \overset{(k)}{\Gamma}{}_{\nu\mu}^{\lambda} = \Gamma_{\nu\mu}^{\lambda} - S_\nu\delta_\mu^\lambda \qquad \left(S_\nu=S_{\nu\omega}^{\omega}\right). \tag{11} \]

The constructed connection coefficients (11) are invariant with respect to a conformal transformation of the metric and have the form

\[ \overset{(k)}{\Gamma}{}_{\nu\mu}^{\lambda} = \frac{1}{3\mathfrak A} \left( P_{\mu}^{\lambda\alpha\beta\gamma}\partial_\nu a_{\alpha\beta\gamma} - P_{[\omega}^{\omega\lambda\beta\gamma}\partial_{\nu]}a_{\alpha\beta\gamma}\delta_\mu^\lambda \right). \]

From the results obtained, the following theorem easily follows:

Theorem 1. In order that \(F_3^{(3)}\) be conformally flat, it is necessary and sufficient that the following conditions be fulfilled: the vanishing of the vector density
\(\chi_\nu = B^{\alpha\beta\gamma}\nabla_\nu a_{\alpha\beta\gamma}\), the semisymmetry \((^4)\) of the connection (7), and the vector \(S_\nu{}^{\omega}\) must be a gradient vector.

1. Let a space \(X_3\) be given, in each local \(E_3\) of which there is given a symmetric covariant pseudotensor \(A_{\alpha\beta\gamma}\) of third valence, whose discriminant is distinct from zero. Since the discriminant of the pseudotensor \(A_{\alpha\beta\gamma}\) is not equal to zero, we can construct a tensor density, determined up to sign, of weight \(-1\), with discriminant equal to \(\pm 1\), in the following way:

\[ \mathfrak A_{\alpha\beta\gamma} = \frac{A_{\alpha\beta\gamma}} {\left|\operatorname{Dis}(A_{\alpha\beta\gamma})\right|^{1/12}} \qquad \left(\widetilde{\mathfrak A}=\operatorname{Dis}(\mathfrak A_{\alpha\beta\gamma})=\pm1\right). \tag{12} \]

Now we can consider \(X_3\) with a prescribed field of tensor density, determined up to sign, and whose discriminant satisfies condition (12). We shall denote such a space by \(\mathfrak{F}_3^{(3)}\), and we shall call the prescribed tensor density the fundamental tensor density.

Substituting into relations (1)—(4), instead of the tensor \(a_{\alpha\beta\gamma}\), the fundamental tensor density \(\mathfrak{A}_{\alpha\beta\gamma}\), we construct the tensor densities \(\widetilde{B}^{\alpha\beta\gamma}\), \(\widetilde{A}^{\lambda\alpha\beta\gamma}\), \(\widetilde{L}_{\alpha\beta\gamma}\), \(\widetilde{P}_{\mu}^{\lambda\alpha\beta}\) of weights \(+1, +1, -1, +1\), respectively, which are the fundamental algebraic concomitants of the space \(\mathfrak{F}_3^{(3)}\). These concomitants satisfy relations analogous to (5), (6). A linear symmetric connection in \(\mathfrak{F}_3^{(3)}\) can now be defined by the conditions

\[ \text{a) }\ \widetilde{P}_{(\mu}^{\lambda\alpha\beta}\nabla_{\nu)}\mathfrak{A}_{\alpha\beta\gamma}=0,\qquad \text{b) }\ S_{\nu\mu}^{\lambda}=0. \tag{13} \]

Expanding the covariant differentiation in (13a) and using a relation analogous to (6a) for the tensor densities \(\mathfrak{A}_{\alpha\beta\gamma}\), \(\widetilde{P}_{\mu}^{\lambda\alpha\beta}\), we have

\[ \widetilde{P}_{(\mu}^{\lambda\alpha\beta}\partial_{\nu)}\mathfrak{A}_{\alpha\beta\gamma} -3\mathfrak{A}\Gamma_{\nu\mu}^{\lambda} +\mathfrak{A}\Gamma_{(\nu}\delta_{\mu)}^{\lambda}=0. \]

Contracting with respect to the indices \(\lambda,\mu\), in view of (13b), we obtain

\[ \Gamma_{\mu}=\frac{1}{\mathfrak{A}}\, \widetilde{P}_{(\mu}^{\omega\alpha\beta\gamma}\partial_{\omega)}\mathfrak{A}_{\alpha\beta\gamma}, \]

and, consequently,

\[ \Gamma_{\nu\mu}^{\lambda} = \frac{1}{3\mathfrak{A}} \left( \widetilde{P}_{\mu}^{\lambda\alpha\beta\gamma}\partial_{\nu}\mathfrak{A}_{\alpha\beta\gamma} + \widetilde{P}_{(\nu}^{\omega\alpha\beta\gamma}\delta_{\mu)}^{\lambda} \mathfrak{A}_{\alpha\beta\gamma}\gamma_{\mu} \right). \tag{14} \]

Putting

\[ \varphi_{\nu\mu}^{\lambda} = \frac{1}{\mathfrak{A}}\, \widetilde{P}_{\mu}^{\lambda\alpha\beta\gamma}\nabla_{\nu}\mathfrak{A}_{\alpha\beta\gamma}, \qquad \psi_{\nu} = \frac{1}{3\mathfrak{A}}\, \widetilde{B}^{\alpha\beta\gamma}\nabla_{\nu}\mathfrak{A}_{\alpha\beta\gamma}, \]

it is not difficult to show that, for the covariant derivative of the fundamental tensor density \(\mathfrak{A}_{\alpha\beta\gamma}\), the following equality holds:

\[ \nabla_{\nu}\mathfrak{A}_{\alpha\beta\gamma} = \varphi_{\nu(\alpha}^{\omega}\mathfrak{A}_{\beta\gamma)\omega} + \psi_{\nu}\widetilde{L}_{(\alpha\beta\gamma)}. \]

The tensor \(\varphi_{\nu\mu}^{\lambda}\), as is easy to see, is skew-symmetric in the indices \(\nu,\mu\).

Let us apply the Ricci identity to the fundamental tensor density \(\mathfrak{A}_{\alpha\beta\gamma}\). We have

\[ 2\nabla_{[\lambda}\nabla_{\nu]}\mathfrak{A}_{\alpha\beta\gamma} = -3R_{\lambda\nu(\alpha}^{\omega}\mathfrak{A}_{\beta\gamma)\omega} + R_{\lambda\nu\omega}^{\omega}\mathfrak{A}_{\alpha\beta\gamma}. \]

Let now \(\varphi_{\nu\mu}^{\lambda}=0\), \(\psi_{\nu}=0\); then, multiplying the Ricci identity by \(\widetilde{P}_{\mu}^{\tau\alpha\beta\gamma}\), we obtain

\[ 3R_{\lambda\nu\mu}^{\tau}-R_{\lambda\nu\omega}^{\omega}\delta_{\mu}^{\tau}=0. \tag{15} \]

It is well known \({}^{(4)}\) that in a space of symmetric affine connection the curvature tensor satisfies the identity \(R_{[\lambda\nu\mu]}^{\tau}=0\); then from (15) we have \(R_{[\lambda\nu|\omega|}^{\omega}\delta_{\mu]}^{\tau}=0\), which gives \(R_{\lambda\nu\omega}^{\omega}=0\) and, consequently, \(R_{\lambda\nu\mu}^{\tau}=0\).

Theorem 2. A necessary and sufficient condition for there to exist in \(\mathfrak{F}_3^{(3)}\) a coordinate system in which the components of the fundamental tensor density \(\mathfrak{A}_{\alpha\beta\gamma}\) do not depend on the coordinates of a point of \(\mathfrak{F}_3^{(3)}\), is the vanishing of the tensor \(\varphi_{\nu\mu}^{\lambda}\) and the vector \(\psi_{\nu}\).

Proof. Let \(\partial_{\nu}\mathfrak{A}_{\alpha\beta\gamma}=0\); then, as is seen from (14), \(\Gamma_{\nu\mu}^{\lambda}=0\), and, consequently, \(\varphi_{\nu\mu}^{\lambda}=0\), \(\psi_{\nu}=0\). Conversely, let \(\varphi_{\nu\mu}^{\lambda}=0\), \(\psi_{\nu}=0\); then \(R_{\lambda\nu\mu}^{\tau}=0\), consequently, there exists a coordinate system in which \(\Gamma_{\nu\mu}^{\lambda}=0\), and hence in this coordinate system \(\partial_{\nu}\mathfrak{A}_{\alpha\beta\gamma}=0\).

It is also possible to prove the following theorem.

Theorem 3. Every differential concomitant of the fundamental tensor density \(\mathfrak{A}_{\alpha\beta\gamma}\) is an algebraic concomitant of \(\mathfrak{A}_{\alpha\beta\gamma}\), \(\varphi^\lambda_{\mu\nu}\), \(\psi_\nu\), and their covariant derivatives computed with respect to the object (14).

Saratov State University
named after N. G. Chernyshevsky

Received
28 XII 1956

CITED LITERATURE

\(^{1}\) A. E. Liber, Tr. seminara po vektorn. i tenzorn. analizu, 9, 319 (1952).
\(^{2}\) K. Tonooka, Tensor, 6, No. 1, 60 (1956).
\(^{3}\) G. B. Gurevich, Foundations of the Theory of Algebraic Invariants, Moscow—Leningrad, 1948.
\(^{4}\) I. A. Schouten, D. J. Struik, Introduction to New Methods of Differential Geometry, 1, Moscow, 1939.