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}{},}^{\lambda
\quad\text{where}
]

[
\overset{(k)}{\Gamma}{}{\nu\mu}^{\lambda}
=
\Gamma}^{\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}^{\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}) can now be defined by the conditions}), (\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)

[
\text{a) }\ \widetilde{P}{(\mu}^{\lambda\alpha\beta}\nabla}\mathfrak{A{\alpha\beta\gamma}=0,\qquad
\text{b) }\ S=0.}^{\lambda
\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}), we have}^{\lambda\alpha\beta

[
\widetilde{P}{(\mu}^{\lambda\alpha\beta}\partial}\mathfrak{A{\alpha\beta\gamma}
-3\mathfrak{A}\Gamma}^{\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,}\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}\mathfrak{A{\alpha\beta\gamma}
+
\widetilde{P}}^{\omega\alpha\beta\gamma}\delta_{\mu)}^{\lambda
\mathfrak{A}{\alpha\beta\gamma}\gamma
\right).
\tag{14}
]

Putting

[
\varphi_{\nu\mu}^{\lambda}
=
\frac{1}{\mathfrak{A}}\,
\widetilde{P}{\mu}^{\lambda\alpha\beta\gamma}\nabla}\mathfrak{A{\alpha\beta\gamma},
\qquad
\psi
=
\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}^{\omega}\mathfrak{A{\beta\gamma)\omega}
+
\psi.}\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}^{\omega}\mathfrak{A{\beta\gamma)\omega}
+
R.}^{\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}}) do not depend on the coordinates of a point of (\mathfrak{F3^{(3)}), is the vanishing of the tensor (\varphi).}^{\lambda}) and the vector (\psi_{\nu

Proof. Let (\partial_{\nu}\mathfrak{A}{\alpha\beta\gamma}=0); then, as is seen from (14), (\Gamma=0).}^{\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

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}), (\psi_\nu), and their covariant derivatives computed with respect to the object (14).}), (\varphi^\lambda_{\mu\nu

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.