Abstract Generated abstract
The paper studies centro-projective connections on differentiable manifolds with an almost complex structure, focusing on connections conjugate with respect to a copuncture and their relation to associated affine connections. It defines conjugate and symmetrically conjugate centro-projective connections, then establishes conditions under which attachment, invariant association, vanishing curvature, and torsion properties are preserved or characterized. The results include criteria involving covariantly constant copunctures, the normal tensor of an affine connection, and gradient conditions, culminating in preservation theorems for normal centro-projective connections under symmetric conjugation.
Full Text
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 \(\widetilde{\Gamma}^i_{jk}\).
- 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 \(\widetilde{\Gamma}^i_{jk}\).
- 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.
- 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
- V. G. Lemlein, Litovsk. matem. sborn., vol. 4, 1, 41 (1964).
- V. A. Gaukhman, DAN, 142, No. 4 (1962).