Reports of the Academy of Sciences of the USSR
1964. Volume 159, No. 2

MATHEMATICS

B. P. GEIDMAN

THE OBJECT OF A CENTRO-PROJECTIVE CONNECTION ON A MANIFOLD WITH AN ALMOST COMPLEX STRUCTURE

(Presented by Academician P. S. Novikov on 22 V 1964)

1. Let \(\mathfrak M_{2n}\) be a \(2n\)-dimensional differentiable manifold with an almost complex structure, i.e., on \(\mathfrak M_{2n}\) there is defined a tensor \(F^i_j\) \((i,j,k,l=1,2,\ldots,2n)\) such that \(F^i_jF^k_i=-\delta^k_j\). Suppose that on \(\mathfrak M_{2n}\) an object of centro-projective connection of a differential neighborhood of the third order \((\Gamma^i_{jk},\Gamma_{jk})\) and a copuncture \(a_i\) are given \((^1)\).

The object with components

\[ \begin{gathered} \widetilde{\Gamma}^i_{jk}=\Gamma^i_{jk}+F^s_jF^i_{s,k},\\ \widetilde{\Gamma}_{jk}=\Gamma_{jk}-a_iF^s_jF^i_{s,k}+(\delta^s_j-F^s_j)D_ka_s, \end{gathered} \tag{1} \]

where \(D_ka_s=\partial a_s/\partial x^k-a_p\Gamma^p_{sk}-\Gamma_{sk}\), and \(F^i_{s,k}\) denotes covariant differentiation in the connection \(\Gamma^i_{jk}\), defines on \(\mathfrak M_{2n}\) a centro-projective connection, the transport of a puncture in which along a curve \(L\) from a point \(M\) to a point \(M_1\) is carried out as follows: a puncture \(u^i\) at the point \(M\) is transformed into the puncture

\[ v^i=\frac{F^i_ku^k}{-a_pF^p_ju^j+a_pu^p+1}, \tag{2} \]

then \(v^i\) is transported along the curve \(L\) from the point \(M\) to the point \(M_1\) in the centro-projective connection \((\Gamma^i_{jk},\Gamma_{jk})\) and at the point \(M_1\) is transformed into the puncture

\[ \widetilde{u}^i=\frac{-F^i_j\widetilde{v}^j}{a_pF^p_j\widetilde{v}^j+a_p\widetilde{v}^p+1} \]

by the transformation inverse to (2).

The object \((\widetilde{\Gamma}^i_{jk},\widetilde{\Gamma}_{jk})\) is called the centro-projective connection conjugate to the connection \((\Gamma^i_{jk},\Gamma_{jk})\) with respect to the copuncture \(a_i\). From (1) it is seen that the subobject \(\widetilde{\Gamma}^i_{jk}\) of the object of the conjugate centro-projective connection \((\widetilde{\Gamma}^i_{jk},\widetilde{\Gamma}_{jk})\) is an affine connection conjugate to the affine connection \(\Gamma^i_{jk}\) \((^2)\), which forms the subobject of the original object of centro-projective connection \((\Gamma^i_{jk},\Gamma_{jk})\).

Theorem 1. If the centro-projective connection \((\Gamma^i_{jk},\Gamma_{jk})\) is attached to an affine connection \(\Gamma^i_{jk}\) and \(a_i\) is a corresponding copuncture covariantly constant in this connection, then the conjugate centro-projective connection \((\widetilde{\Gamma}^i_{jk},\widetilde{\Gamma}_{jk})\) is also attached to the connection \(\widetilde{\Gamma}^i_{jk}\) conjugate to the affine connection \(\Gamma^i_{jk}\).

Proof. Since \(D_ja_k=\partial a_k/\partial x^j-a_p\Gamma^p_{kj}-\Gamma_{kj}=0\), we have

\[ \widetilde{\Gamma}_{jk} = \Gamma_{jk}-a_iF^s_jF^i_{s,k} = \Gamma_{jk}-a_i(\widetilde{\Gamma}^i_{jk}-\Gamma^i_{jk}) = \frac{\partial a_j}{\partial x^k}-a_p\widetilde{\Gamma}^p_{jk}. \]

The connection \((\widetilde{\Gamma}^i_{jk},\widetilde{\Gamma}_{jk})\) is attached to the affine connection \(\widetilde{\Gamma}^i_{jk}\), and the copuncture \(a_k\) is covariantly constant in this connection.

Let us note that, for a covariant tensor of degree one, the analogous property does not hold: a tensor, being covariantly constant in an affine connection—

of \(\Gamma^i_{jk}\), is not at all obliged to be covariantly constant in the conjugate connection \(\widetilde{\Gamma}^i_{jk}\).

1. The object of a centro-projective connection
   \[ \left\{\Gamma^i_{jk},\quad \Gamma_{jk}=-\frac{1}{2n+1}\left(\frac{\partial \Gamma^m_{mj}}{\partial x^k}-\Gamma^m_{ml}\Gamma^l_{jk}\right)\right\}, \]
   defined by the copunctor \(-\dfrac{1}{2n+1}\Gamma^m_{ml}\), will be called left-invariantly associated with the affine connection \(\Gamma^i_{jk}\), and the object of a centro-projective connection defined by the copunctor \(-\dfrac{1}{2n+1}\Gamma^m_{lm}\), right-invariantly associated with the affine connection \(\Gamma^i_{jk}\). If the left- and right-invariantly associated objects of a centro-projective connection coincide, then we shall say that the object of a centro-projective connection \((\Gamma^i_{jk},\Gamma_{jk})\) is invariantly associated with \(\Gamma^i_{jk}\). It is clear that both objects of a centro-projective connection associated with a symmetric affine connection \(\Gamma^i_{jk}\) coincide.

The connection \(\widetilde{\Gamma}^i_{jk}\) conjugate to the symmetric affine connection \(\Gamma^i_{jk}\) is, generally speaking, nonsymmetric, and therefore the objects of a centro-projective connection left- and right-invariantly associated with it need not coincide.

Theorem 2. If the connection \((\Gamma^i_{jk},\Gamma_{jk})\), invariantly associated with a symmetric affine connection \(\Gamma^i_{jk}\), then the conjugate centro-projective connection \((\widetilde{\Gamma}^i_{jk},\widetilde{\Gamma}_{jk})\) is left-invariantly associated with \(\widetilde{\Gamma}^i_{jk}\).

Proof. From Theorem 1 it follows that the copunctor
\(-\dfrac{1}{2n+1}\Gamma^k_{kj}\) is covariantly constant in the connection \((\widetilde{\Gamma}^i_{jk},\widetilde{\Gamma}_{jk})\), and since \(\widetilde{\Gamma}^k_{kj}=\Gamma^k_{kj}\), everything is proved.

Consider the tensor \(B_j=F^l_s F^s_{j,l}\). A symmetric affine connection \(\Gamma^i_{jk}\) for which \(B_j=0\) is called a normal affine connection. The tensor \(B_j\) itself we shall call the normal tensor of the connection \(\Gamma^i_{jk}\).

Theorem 3. Let \((\Gamma^i_{jk},\Gamma_{jk})\) be a centro-projective connection invariantly associated with a symmetric affine connection \(\Gamma^i_{jk}\). In order that the conjugate centro-projective connection \((\widetilde{\Gamma}^i_{jk},\widetilde{\Gamma}_{jk})\) be invariantly associated with \(\widetilde{\Gamma}^i_{jk}\), it is necessary and sufficient that the normal tensor \(B_j\) of the connection \(\Gamma^i_{jk}\) be covariantly constant in the conjugate connection \(\widetilde{\Gamma}^i_{jk}\).

1. The connection
   \[ \gamma^i_{jk}=\frac{1}{2}\bigl(\widetilde{\Gamma}^i_{jk}+\widetilde{\Gamma}^i_{kj}\bigr),\qquad \gamma_{jk}=\frac{1}{2}\bigl(\widetilde{\Gamma}_{jk}+\widetilde{\Gamma}_{kj}\bigr) \tag{3} \]
   is called the symmetrically conjugate centro-projective connection for the connection \((\Gamma^i_{jk},\Gamma_{jk})\).

Theorem 4. Let \((\Gamma^i_{jk},\Gamma_{jk})\) be associated with a symmetric affine connection \(\Gamma^i_{jk}\), i.e. \(\Gamma^i_{kj}=\Gamma^i_{jk}\), and suppose there exists a copunctor \(a_i\) such that
\(\Gamma_{jk}=\partial a_j/\partial x^k-a_p\Gamma^p_{jk}\). In order that the complete curvature object of the connection \((\Gamma^i_{jk},\Gamma_{jk})\) be equal to zero, it is necessary and sufficient that the symmetrically conjugate centro-projective connection \((\gamma^i_{jk},\gamma_{jk})\) be associated with the affine connection \(\gamma^i_{jk}\) by means of the copunctor \(a_i\).

Proof. The necessity is obvious.

Sufficiency. Let \(\gamma_{jk}=\partial a_j/\partial x^k-a_p\gamma^p_{jk}\). Taking into account relation (3), we obtain
\[ \frac{1}{2}\widetilde{\Gamma}_{jk}+\frac{1}{2}\widetilde{\Gamma}_{kj} = \frac{\partial a_j}{\partial x^k} -\frac{1}{2}a_p\widetilde{\Gamma}^p_{jk} -\frac{1}{2}a_p\widetilde{\Gamma}^p_{kj}, \]
or \(\partial a_k/\partial x^j=\partial a_j/\partial x^k\).

Theorem 5. Let \(a_j\) be a concircular and \((\Gamma^i_{jk}=\Gamma^i_{kj}, \Gamma^i_{jk})\) a centro-projective connection in which this concircular is covariantly constant. If \((\gamma^i_{jk}, \gamma^i_{jk})\) is the symmetrically conjugate connection for \((\Gamma^i_{jk}, \Gamma^i_{jk})\), and \((\widetilde{\gamma}^i_{jk}, \widetilde{\gamma}^i_{jk})\) is the conjugate centro-projective connection for the connection \((\gamma^i_{jk}, \gamma^i_{jk})\), then the complete torsion object \((\widetilde{S}^i_{jk}, \widetilde{S}_{jk})\) of the connection \((\widetilde{\gamma}^i_{jk}, \widetilde{\gamma}_{jk})\) does not depend on the choice of the original connection \((\Gamma^i_{jk}, \Gamma_{jk})\) and is equal to

\[ \widetilde{S}^i_{jk}=2t^i_{jk}; \]

\[ \widetilde{S}_{jk}=2a_i t^i_{kj} +\frac{1}{2}\left(\delta^s_j-F^s_j\right) \left(\frac{\partial a_s}{\partial x^k}-\frac{\partial a_k}{\partial x^s}\right) -\frac{1}{2}\left(\delta^s_k-F^s_k\right) \left(\frac{\partial a_s}{\partial x^j}-\frac{\partial a_j}{\partial x^s}\right), \]

where \(t^i_{jk}\) is the torsion tensor of the almost complex structure.

1. A normal centro-projective connection is defined as a symmetric centro-projective connection invariantly attached to a normal affine connection.

Theorem 6. If \(\Gamma^i_{jk}\) is an equiaffine connection and \(\gamma^i_{jk}\) is an affine connection symmetrically conjugate to it, then the centro-projective connection invariantly attached to \(\gamma^i_{jk}\) will be normal if and only if the normal tensor \(B_m\) of the equiaffine connection \(\Gamma^i_{jk}\) is a gradient.

Proof. Necessity. \(\gamma^m_{mj}=\Gamma^m_{mj}-{}^{1}/_{2}B_j\) and \(\partial\gamma^m_{mj}/\partial x^k=\partial\gamma^m_{mk}/\partial x^j\); moreover, \(\partial\Gamma^m_{mj}/\partial x^k=\partial\Gamma^m_{mk}/\partial x^j\), and therefore \(\partial B_j/\partial x^k=\partial B_k/\partial x^j\) and \(B_j=\partial\varphi/\partial x^j\), where \(\varphi\) is an arbitrary scalar function.

Sufficiency. a) We shall prove that the normal tensor \(\widetilde{B}_l\) of the connection \(\gamma^i_{jk}\) is equal to zero.

\[ \widetilde{B}_l = F^k_m\left( \frac{\partial F^m_l}{\partial x^k} +F^p_l\gamma^m_{pk} -F^m_p\gamma^p_{lk} \right) = \]

\[ = \frac{1}{2}F^k_m\left( \frac{\partial F^m_l}{\partial x^k} +F^p_l\widetilde{\Gamma}^m_{pk} -F^m_p\widetilde{\Gamma}^p_{lk} \right) + \frac{1}{2}F^k_m\left( \frac{\partial F^m_l}{\partial x^k} +F^p_l\widetilde{\Gamma}^m_{kp} -F^m_p\widetilde{\Gamma}^p_{kl} \right) = \]

\[ = \frac{1}{2}B_l +\frac{1}{2}F^k_mF^p_lF^s_pF^m_{s,k} -\frac{1}{2}F^k_mF^m_pF^s_lF^p_{s,k} + \]

\[ + \frac{1}{2}B_l +\frac{1}{2}F^k_mF^p_lF^s_kF^m_{s,p} -\frac{1}{2}F^k_mF^m_pF^s_kF^p_{s,l} = \]

\[ = B_l-\frac{1}{2}B_l-\frac{1}{2}B_l-\frac{1}{2}F^p_lF^s_{s,p} +\frac{1}{2}F^s_kF^k_{s,l}=0. \]

b) We shall prove that \(\gamma_{jk}=\gamma_{kj}\):

\[ \frac{\partial\gamma^m_{mk}}{\partial x^j} - \frac{\partial\gamma^m_{mj}}{\partial x^k} = \frac{\partial\Gamma^m_{mk}}{\partial x^j} - \frac{\partial\Gamma^m_{mj}}{\partial x^k} - \frac{1}{2}\left( \frac{\partial B_k}{\partial x^j} - \frac{\partial B_j}{\partial x^k} \right) =0. \]

Theorem 7. If \((\Gamma^i_{jk}, \Gamma_{jk})\) is a normal centro-projective connection, then the centro-projective connection \((\gamma^i_{jk}, \gamma_{jk})\) symmetrically conjugate to it is also normal.

The proof follows from Theorem 4, taking into account that \(\gamma^m_{mk}=\Gamma^m_{mk}\). The author expresses gratitude to V. G. Lemlein for valuable suggestions.

Moscow State Pedagogical Institute
named after V. I. Lenin

Received
15 V 1964

CITED LITERATURE

1. V. G. Lemlein, Litovsk. matem. sborn., vol. 4, 1, 41 (1964).
2. V. A. Gaukhman, DAN, 142, No. 4 (1962).