MATHEMATICS

V. G. LEMLEIN

ON SPACES OF SYMMETRIC ALMOST SYMPLECTIC CONNECTION

(Presented by Academician P. S. Aleksandrov, March 6, 1957)

In the present work, spaces of a symmetric affine connection \(\Gamma^l_{jk}\), invariantly associated with a nondegenerate skew-symmetric tensor \(a_{ij}\), are considered. The basic property defining these spaces is the requirement that the covariant derivative of the fundamental tensor \(a_{ij}\) and its exterior derivative be proportional:

\[ \frac{\partial a_{ij}}{\partial x^k}-a_{il}\Gamma^l_{jk}-a_{lj}\Gamma^l_{ik} =\mu(x)\left(\frac{\partial a_{ij}}{\partial x^k} +\frac{\partial a_{ja}}{\partial x^i} +\frac{\partial a_{ki}}{\partial x^j}\right). \tag{1} \]

Cyclic permutation of this relation with respect to \(i,j,k\) leads to \(\mu(x)=1/3\). Since (1) can be rewritten in the form

\[ \frac{1}{3}\left(\frac{\partial a_{ij}}{\partial x^k} -\frac{\partial a_{ki}}{\partial x^j}\right)-a_{il}\Gamma^l_{jk} = \frac{1}{3}\left(\frac{\partial a_{ji}}{\partial x^k} -\frac{\partial a_{kj}}{\partial x^i}\right)-a_{jl}\Gamma^l_{ik}, \tag{2} \]

then, denoting

\[ \frac{1}{3}\left(\frac{\partial a_{ij}}{\partial x^k} -\frac{\partial a_{ki}}{\partial x^j}\right)-a_{il}\Gamma^l_{jk} =-\frac{1}{3}\gamma_{ijk}(x), \tag{3} \]

we have

\[ \Gamma^l_{jk} = \frac{1}{3}a^{li}\left(\frac{\partial a_{ij}}{\partial x^k} -\frac{\partial a_{ki}}{\partial x^j} +\gamma_{ijk}(x)\right); \tag{4} \]

here \(a^{li}a_{ik}=\delta^l_k\); \(\gamma_{ijk}(x)\) are arbitrary functions, symmetric in any pair of indices.

If we require that, upon passage to the coordinate system \(\{x'\}\), the \(\gamma_{ijk}\) transform according to the law\(^*\)

\[ \gamma_{i'j'k'} = \frac{\partial x^i}{\partial x^{i'}} \frac{\partial x^j}{\partial x^{j'}} \frac{\partial x^k}{\partial x^{k'}} \gamma_{ijk} + a_{pq}\frac{\partial x^p}{\partial x^{(i''}} \frac{\partial^2 x^q}{\partial x^{j''}\partial x^{k'')}}, \tag{5} \]

then, by virtue of the transitivity of the transformation formulas (5), the \(\gamma_{ijk}\) will define a field of geometric objects invariantly associated with the tensor \(a_{ij}\).

The transitivity of (5) follows from the identity

\[ a_{pq}\frac{\partial x^p}{\partial x^{(i''}} \frac{\partial^2 x^q}{\partial x^{j''}\partial x^{k'')}} \equiv \frac{\partial x^{i'}}{\partial x^{i''}} \frac{\partial x^{j'}}{\partial x^{j''}} \frac{\partial x^{k'}}{\partial x^{k''}} a_{pq}\frac{\partial x^p}{\partial x^{(i'}} \frac{\partial^2 x^q}{\partial x^{j'}\partial x^{k')}} + a_{p'q'}\frac{\partial x^{p'}}{\partial x^{(i''}} \frac{\partial^2 x^{q'}}{\partial x^{j''}\partial x^{k'')}}, \tag{6} \]

\(^*\) Parentheses everywhere denote cyclic permutation.

which, in turn, follows from the identity

\[ \frac{\partial^2 x^q}{\partial x^{j''}\partial x^{k''}} \equiv \frac{\partial x^{j'}}{\partial x^{j''}} \frac{\partial x^{k'}}{\partial x^{k''}} \frac{\partial^2 x^q}{\partial x^{j'}\partial x^{k'}} + \frac{\partial x^q}{\partial x^{q'}} \frac{\partial^2 x^{q'}}{\partial x^{j''}\partial x^{k''}} . \tag{7} \]

If now one subjects \(\Gamma^l_{jk}\) to the transformation law of an affine connection

\[ \Gamma^{l'}_{j'k'}= \frac{\partial x^{l'}}{\partial x^l} \frac{\partial^2 x^l}{\partial x^{j'}\partial x^{k'}} + \frac{\partial x^{l'}}{\partial x^l} \frac{\partial x^j}{\partial x^{j'}} \frac{\partial x^k}{\partial x^{k'}} \Gamma^l_{jk}, \tag{8} \]

then, taking relation (5) into account, we shall have

\[ \Gamma^{l'}_{j'k'}= \frac{1}{3}a^{l'i'} \left( \frac{\partial a_{i'j'}}{\partial x^{k'}} - \frac{\partial a_{k'i'}}{\partial x^{j'}} + \gamma_{i'j'k'} \right). \tag{9} \]

Indeed, we have

\[ a_{i'k'}= \frac{\partial x^i}{\partial x^{i'}} \frac{\partial x^k}{\partial x^{k'}}a_{ik}, \tag{10} \]

\[ a^{l'i'}= \frac{\partial x^{l'}}{\partial x^l} \frac{\partial x^{i'}}{\partial x^p}a^{lp}. \tag{11} \]

Differentiating (10) and multiplying by (11), we obtain

\[ a^{l'i'}\frac{\partial a_{i'k'}}{\partial x^{j'}} = a^{l'i'}a_{pq}\frac{\partial x^q}{\partial x^{k'}} \frac{\partial^2 x^p}{\partial x^{i'}\partial x^{j'}} + \frac{\partial x^{l'}}{\partial x^l} \frac{\partial^2 x^l}{\partial x^{j'}\partial x^{k'}} + \frac{\partial x^{l'}}{\partial x^l} \frac{\partial x^k}{\partial x^{k'}} \frac{\partial x^j}{\partial x^{j'}} a^{li}\frac{\partial a_{ik}}{\partial x^j}. \tag{12} \]

and analogously

\[ a^{l'i'}\frac{\partial a_{j'i'}}{\partial x^{k'}} = - \frac{\partial x^{l'}}{\partial x^l} \frac{\partial^2 x^l}{\partial x^{j'}\partial x^{k'}} + a^{l'i'}a_{qp}\frac{\partial x^q}{\partial x^{j'}} \frac{\partial^2 x^p}{\partial x^{i'}\partial x^{k'}} + \frac{\partial x^{l'}}{\partial x^l} \frac{\partial x^j}{\partial x^{j'}} \frac{\partial x^k}{\partial x^{k'}} a^{li}\frac{\partial a_{ji}}{\partial x^k}. \tag{13} \]

Subtracting (13) from (12) and dividing by 3, we obtain:

\[ \begin{aligned} \Gamma^{l'}_{j'k'}-\frac{1}{3}a^{l'i'}\gamma_{i'j'k'} ={}& \frac{\partial x^{l'}}{\partial x^l} \frac{\partial^2 x^l}{\partial x^{j'}\partial x^{k'}} + \frac{\partial x^{l'}}{\partial x^l} \frac{\partial x^j}{\partial x^{j'}} \frac{\partial x^k}{\partial x^{k'}} \left( \Gamma^l_{jk}-\frac{1}{3}a^{li}\gamma_{ijk} \right) \\ &- \frac{1}{3} \frac{\partial x^{l'}}{\partial x^l} \frac{\partial^2 x^l}{\partial x^{j'}\partial x^{k'}} + \frac{1}{3}a^{l'i'}a_{pq} \left( \frac{\partial x^q}{\partial x^{j'}} \frac{\partial^2 x^p}{\partial x^{i'}\partial x^{k'}} + \frac{\partial x^q}{\partial x^{k'}} \frac{\partial^2 x^p}{\partial x^{i'}\partial x^{j'}} \right). \end{aligned} \tag{14} \]

The last two terms of the right-hand side, since

\[ - \frac{\partial^2 x^l}{\partial x^{j'}\partial x^{k'}} \equiv \frac{\partial x^l}{\partial x^m} a^{mi'}a_{pq} \frac{\partial x^q}{\partial x^{i'}} \frac{\partial^2 x^p}{\partial x^{k'}\partial x^{j'}}, \tag{15} \]

take the form

\[ -\frac{1}{3}a^{l'i'}a_{pq} \frac{\partial x^p}{\partial x^{(i'}} \frac{\partial^2 x^q}{\partial x^{j'}\partial x^{k')}} . \tag{16} \]

In order to obtain (8) and (9), it is now sufficient to multiply (5) by \(\frac{1}{3}a^{l'i'}\) and add it to (14).

Thus, \(\Gamma^l_{jk}\) are a symmetric affine connection, and the \(\gamma_{ijk}\) entering them determine a framing of the manifold on which the tensor \(a_{ij}\) is given.

Definition. A differentiable manifold of \(2n\) dimensions with a nondegenerate skew-symmetric tensor \(a_{ij}\) given on it and with an object \(\gamma_{ijk}\) determining a framing is called a space of symmetric almost symplectic connection.

In the case

\[ T_{ijk}\equiv \frac{1}{3}\left(\frac{\partial a_{ij}}{\partial x^k}+\frac{\partial a_{jk}}{\partial x^i}+\frac{\partial a_{ki}}{\partial x^l}\right)=0 \tag{17} \]

the space defined by us coincides with the space of a symmetric symplectic connection.

Let us also indicate the important relation

\[ -a_{l(i}\Gamma^l_{jk)}=\gamma_{ijk}. \tag{18} \]

We now consider the curvature tensor

\[ R_{ijk}^{\cdot\ \cdot\ \alpha}= \frac{\partial \Gamma^\alpha_{ik}}{\partial x^j} - \frac{\partial \Gamma^\alpha_{jk}}{\partial x^i} + \Gamma^\beta_{ik}\Gamma^\alpha_{j\beta} - \Gamma^\beta_{jk}\Gamma^\alpha_{i\beta}. \tag{19} \]

If we denote \(B_{ij,kl}=a_{l\alpha}R_{ij,k}^{\cdot\ \cdot\ \alpha}\), then, taking (4) into account and

\[ \nabla_l a_{ij}=T_{ijl}, \tag{20} \]

we can put \(B_{ij,kl}\) in the form

\[ B_{ij,kl} = \frac{1}{3}\left( \frac{\partial^2 a_{jl}}{\partial x^k\partial x^i} - \frac{\partial^2 a_{il}}{\partial x^k\partial x^j} \right) + \frac{1}{3}\left( \frac{\partial \gamma_{lik}}{\partial x^j} - \frac{\partial \gamma_{ljk}}{\partial x^i} \right) + \]

\[ + a_{\alpha\beta}\left(\Gamma^\alpha_{ki}\Gamma^\beta_{lj} - \Gamma^\alpha_{kj}\Gamma^\beta_{li}\right) - T_{l\alpha j}\Gamma^\alpha_{ik} + T_{l\alpha i}\Gamma^\alpha_{jk}. \tag{21} \]

In the case \(T_{ijk}=0\), (21) takes the form

\[ B_{ij,kl} = -\frac{1}{3}\frac{\partial^2 a_{jl}}{\partial x^k\partial x^i} + \frac{1}{3}\left( \frac{\partial \gamma_{lik}}{\partial x^j} - \frac{\partial \gamma_{ljk}}{\partial x^i} \right) + a_{\alpha\beta}\left(\Gamma^\alpha_{ki}\Gamma^\beta_{lj} - \Gamma^\alpha_{kj}\Gamma^\beta_{li}\right). \tag{22} \]

Consider the case

\[ B_{ij,kl}=0. \tag{23} \]

When (23) holds, we can pass to a Cartesian coordinate system characterized by the equality

\[ \Gamma^l_{jk}=0, \tag{24} \]

which, by virtue of (18), leads to

\[ \gamma_{ijk}=0. \tag{25} \]

From (4), (24), and (25) we obtain

\[ \frac{\partial a_{ij}}{\partial x^l} = \frac{\partial a_{jl}}{\partial x^i} = \frac{\partial a_{li}}{\partial x^j}. \tag{26} \]

But, since from (21)

\[ \frac{\partial^2 a_{jl}}{\partial x^k\partial x^i} = \frac{\partial^2 a_{il}}{\partial x^k\partial x^j}, \]

and from (26)

\[ \frac{\partial^2 a_{jl}}{\partial x^i\partial x^k} = \frac{\partial^2 a_{li}}{\partial x^j\partial x^k}, \]

it follows that

\[ \frac{\partial^2 a_{ij}}{\partial x^k\partial x^l}=0, \]

and, consequently,

\[ a_{ij}=\alpha_{ijk}x^k+\beta_{ij}, \tag{27} \]

where \(\alpha_{ijk}\) and \(\beta_{ij}\) are constants, skew-symmetric in any pair of indices.

From (20), (26), and (27) we find \(\alpha_{ijk}=T_{ijk}\), and consequently, \(\beta_{ij}=a_{ij}-T_{ijk}x^k\).

Let us note that in the case \(T_{ijk}=0\) it is necessary that \(a_{ij}=\beta_{ij}\). Spaces characterized by condition (23) should naturally be called flat spaces of symmetric almost symplectic connection.

Considering a coordinate system geodesic at the given point, we arrive at the conclusion that to each point of the general manifold one can attach a flat space of the indicated type.

Let us now consider the covariant vector

\[ b_i=\frac{1}{2}a^{pq}T_{pqi}. \tag{28} \]

It is easy to see that

\[ b_i=-\frac{1}{3}\frac{1}{\sqrt{a}}\frac{\partial a}{\partial x^i} +\frac{1}{3}a^{jq}\frac{\partial a_{qi}}{\partial x^j}, \tag{29} \]

where \(a=|a_{ij}|\). But the covariant derivative of \(\sqrt{a}\) has the form

\[ \nabla_i\sqrt{a}=\frac{\partial\sqrt{a}}{\partial x^i}-\sqrt{a}\,\Gamma^j_{ij}, \tag{30} \]

or, according to (4),

\[ \nabla_i\sqrt{a} = \frac{1}{3}\frac{\partial\sqrt{a}}{\partial x^i} + \frac{1}{3}\sqrt{a}\,a^{iq}\frac{\partial a_{iq}}{\partial x^j}. \tag{31} \]

Taking (29) into account, we have

\[ \nabla_i\sqrt{a}=-\sqrt{a}\,b_i . \tag{32} \]

But in a coordinate system geodesic at the given point, we have

\[ \nabla_i\sqrt{a}=\frac{\partial\sqrt{a}}{\partial x^i}. \tag{33} \]

Comparing (32) and (33), we obtain
\(\dfrac{\partial\sqrt{a}}{\partial x^i}=-\sqrt{a}b_i\), or
\(\dfrac{\partial\ln\sqrt{a}}{\partial x^i}=-b_i\).

Hence, in particular, we obtain: in order that a space of symmetric almost symplectic connection be a space with invariant volume \(a\det\|u^i_k\|\), it is necessary and sufficient that the covariant vector \(b_i=0\).

In conclusion I express my gratitude to Prof. S. P. Finikov and Prof. G. F. Laptev for their comments on the work.

Moscow City Pedagogical Institute
named after V. P. Potemkin

Received
5 III 1957