V. A. Gaukman

Symmetric Almost Symplectic Affine Connections on Hermitian Manifolds

(Presented by Academician P. S. Novikov, 14 IV 1962)

Let \(\mathfrak M\) be a \(2n\)-dimensional Hermitian manifold \((^1)\), i.e., on \(\mathfrak M\) there are given a tensor \(F^i_j\) and a positive definite symmetric tensor \(g_{ij}\), satisfying the conditions:
\[ F^i_k F^k_j=-\delta^i_j;\quad g_{lm}F^l_iF^m_j=g_{ij};\quad t^i_{jk}=\frac14\left( F^l_j\frac{\partial F^i_k}{\partial x^l} - F^l_j\frac{\partial F^i_l}{\partial x^k} - F^l_k\frac{\partial F^i_j}{\partial x^l} + F^l_k\frac{\partial F^i_l}{\partial x^j} \right)=0. \]
The tensor \(a_{ij}=-F^l_i g_{lj}\) is skew-symmetric and
\[ a_{lm}F^l_iF^m_j=a_{ij}. \tag{1} \]

The quantities
\[ T_{ijk}=\frac13\left(\partial a_{ij}/\partial x^k+\partial a_{jk}/\partial x^i+\partial a_{ki}/\partial x^j\right) =\frac13 a_{(ijk)}, \]
where parentheses denote cyclic summation and \(a_{ijk}=\partial a_{ij}/\partial x^k\), define a tensor whose vanishing means that the space is Kähler.

In the present note we shall find a symmetric affine connection invariantly associated with the tensors \(F^i_j, g_{ij}\), satisfying the conditions indicated above.

We note that on Hermitian manifolds various canonical affine connections have already been constructed (see, for example, \((^{1,2})\)); however, all of them except the Frölicher connection \((^2)\) are not symmetric. The Frölicher connection is almost complex, i.e.,
\(F^i_{j,k}=0\) (the comma here and below denotes covariant differentiation). We set ourselves the goal of finding a symmetric affine connection that is not almost complex, but instead belongs to the class of so-called almost symplectic connections \((^3)\).

It is not difficult to show that the condition \(a_{ij,k}=0\) implies \(T_{ij,k}=0\). However, V. G. Lemlein \((^3)\) showed that there exists a pencil of symmetric affine connections satisfying the condition \(a_{ij,k}=T_{ijk}\). The components of these connections have the form:
\[ \Gamma^i_{jk}=\frac13 a^{il}\left(a_{ljk}-a_{klj}+\gamma_{ljk}\right), \tag{2} \]
where \(a^{il}a_{lj}=\delta^i_j\), and \(\gamma_{ijk}\) is an arbitrary geometric object symmetric in each pair of indices, whose transformation law for its components under a change of local coordinates can be obtained from the equality \(\gamma_{ijk}=-a_{l(i}\Gamma^l_{jk)}\). Lemlein called these connections almost symplectic.

Let us require that the desired connection be almost symplectic. Our problem is now reduced to the choice of the components of the object \(\gamma_{ijk}\).

To facilitate the further calculations it is useful to note that on a Hermitian manifold there exist so-called adapted systems of coordinates, in which the components of the tensor \(F^i_j\) are constant \((^1)\). Passing to an adapted coordinate system, differentiation of equality (1) now gives
\[ F^l_iF^m_j a_{lmk}=a_{ijk}. \tag{3} \]

Consider the tensor \(F^i_{j,k}\). It has two pure \((1)\)-components (see (4)):

\[ A^i_{jk}=-(F^l_jF^i_{l,k}+F^l_kF^i_{j,l}),\qquad B^i_{jk}=F^l_j(F^i_{k,l}-F^i_{l,k}). \]

It can be verified that the identity

\[ T_{ijk}=\frac12 F^l_{(i}F^n_jB^m_{k)m}a_{tl} \]

holds, from which it follows that the vanishing of the tensor \(B^i_{jk}\) entails the vanishing of the tensor \(T_{ijk}\).

We now require that, for the desired connection, the equality

\[ A^i_{jk}=0. \tag{4} \]

hold. This gives:

\[ \gamma_{ijk}-F^l_{(i}F^m_j\gamma_{k)lm}=0. \tag{5} \]

For what follows we shall need the notion of a conjugate affine connection \((4)\). Let \(\Gamma^i_{jk}\) be a symmetric affine connection. By the conjugate affine connection \(\widetilde{\Gamma}^i_{jk}\) we shall mean the connection in which parallel transport from a point \(x_1\) to a point \(x_2\) along a curve \(L\) is carried out in the following way: the vector is first subjected to the automorphism \(F(x_1)=(F^i_j(x_1))\), then transported from \(x_1\) to \(x_2\) along \(L\) parallel with respect to the connection \(\Gamma^i_{jk}\), and finally subjected to the inverse automorphism \(F^{-1}(x_2)=-F(x_2)\). The components of this connection have the form \(\widetilde{\Gamma}^i_{jk}=\Gamma^i_{jk}+F^l_jF^i_{l,k}\).

We now require that the equality

\[ g_{ij,k}=g_{ij|k}, \tag{6} \]

hold, where \(g_{ij|k}\) is the covariant derivative of the tensor \(g_{ij}\) in the connection \(\widetilde{\Gamma}^i_{jk}\). We obtain

\[ \gamma_{ijk}-F^l_iF^m_j\gamma_{klm} =\frac12(a_{kji}+a_{kij}-F^l_iF^m_ja_{klm}-F^l_iF^m_ja_{kml}), \]

whence it follows:

\[ 3\gamma_{ijk}-F^l_{(i}F^m_j\gamma_{k)lm} =-\frac12F^l_{(i}F^m_ja_{k)lm} -\frac12F^l_{(i}F^m_ja_{k)ml}. \tag{7} \]

From (5) and (7) we have

\[ \gamma_{ijk}=-\frac12F^l_{(i}F^m_ja_{k)lm}. \tag{8} \]

We must now verify that the connection (8) indeed satisfies conditions (4) and (6). First note that the identity

\[ T_{ijk}=F^l_{(i}F^m_jT_{k)lm}, \tag{9} \]

holds; for its proof it is sufficient to use equality (3). By direct computation we obtain

\[ 6a_{il}F^l_{j,k}=F^l_ja_{ikl}-F^l_ia_{jkl}+F^l_ia_{lkj}-F^l_ja_{lki}. \]

Hence

\[ a_{il}F^l_{j,k}+a_{jl}F^l_{i,k}=0. \tag{10} \]

Further:

\[ g_{ij,k}=F^l_{i,k}a_{lj}+F^l_ia_{lj,k}. \tag{11} \]

Interchanging \(i\) and \(j\) and adding, we obtain

\[ g_{ij,k}=\frac12(F^l_iT_{jkl}+F^l_jT_{ikl}). \tag{12} \]

From (11) we have

\[ F^i_{j,k}=a^{li}g_{jl,k}-F^i_ja^{mt}T_{lmk}, \]

whence

\[ F^i_{j,k}=\frac12a^{li}(F^m_lT_{jkm}-F^m_jT_{lkm}). \]

Now a computation gives

\[ a_{il}A^l_{jk}=\frac12T_{ijk}-\frac12F^l_{(i}F^m_jT_{k)lm}, \]

which, by virtue of (9), proves (4). To prove (6), note that

\[ g_{ij,k}-g_{ij|k}=-(a_{il}F^l_{j,k}+a_{jl}F^l_{i,k})=0 \]

by virtue of (10). Thus, it has been proved that the connection (8) satisfies conditions (4) and (6).

Using the method of prolongation and envelopments of geometric objects of G. F. Laptev \({}^{(5)}\), extended by A. M. Vasil'ev \({}^{(6)}\) to the case of infinite transformation groups, one can show that the expressions (2), in which

\[ \gamma_{ijk}=-\frac12 F^l_{(i}F^m_j a_{k)lm} -\frac12 F^m_{(i}a_{j|l}F^l_{k)m} +\frac12 F^m_{(j}a_{i|l}F^l_{mk)} +\frac12 F^m_{(i}a_{j|l}F^l_{mk)}, \tag{13} \]

where \(F^i_{jk}=\partial F^i_j/\partial x^k\), and there is no cyclic summation over the underlined indices, define the components of an object of affine connection. For an adapted coordinate system \(F^i_{jk}=0\), and therefore expression (13) coincides with (8). Thus formula (13) makes it possible to compute the components of our connection for an arbitrary coordinate system. Hence the following has been proved.

Theorem 1. On a Hermitian manifold there exists, and moreover uniquely, a symmetric affine connection satisfying the conditions:
1) \(a_{ij,k}=T_{ijk}\); 2) \(A^i_{jk}=0\); 3) \(g_{ij,k}=g_{ij|k}\). The components of this connection are determined by formulas (2) and (13).

In \({}^{(4)}\) the so-called normal connections were considered, i.e. connections for which \(B_i=B^l_{il}=0\). Such connections play a role in the study of families of geodesic lines. For our connection \(B_i=-\frac12 a^{lm}T_{ilm}\). The vanishing of this vector means \({}^{(3)}\) that the volume

\[ \sqrt{\det\|a_{ij}\|\,\det\|X^k_{(l)}\|} \]

is invariant. But

\[ \det\|a_{ij}\|=\det\|g_{il}F^l_j\|=\det\|g_{ij}\|\times\det\|F^i_j\|=\pm\det\|g_{ij}\|. \]

Consequently, we have proved:

Theorem 2. In order that the almost symplectic connection (13) be normal, it is necessary and sufficient that the space have the invariant volume \(\sqrt{\det\|g_{ij}\|\,\det\|X^k_{(l)}\|}\).

Denote \(B_{ijkl}=a_{im}R^m_{jkl}\), where

\[ R^i_{jkl}=\partial\Gamma^i_{jl}/\partial x^k-\partial\Gamma^i_{jk}/\partial x^l +\Gamma^m_{jl}\Gamma^i_{mk}-\Gamma^m_{jk}\Gamma^i_{ml} \]

is the curvature tensor of the connection (13). From the general properties of the curvature tensor there follow the relations

\[ B_{ijkl}=-B_{ijlk}; \tag{I} \]

\[ B_{i(jkl)}=0. \tag{II} \]

For almost symplectic connections one additionally has \({}^{(3)}\)

\[ B_{ijkl}+B_{jilk}+B_{klij}+B_{lkji}=0. \tag{III} \]

For our connection (13) we have, in addition,

\[ \begin{aligned} &B_{ijkl}-F^r_jF^s_kB_{irsl}-F^r_jF^s_lB_{irks} -F^r_kF^s_lB_{ijrs}-F^r_iF^s_jB_{rskl}\\ &\qquad -F^r_iF^s_kB_{rjsl}-F^r_iF^s_lB_{rjks} +F^r_iF^s_jF^p_kF^q_lB_{rspq}=0; \end{aligned} \tag{IV} \]

\[ \begin{aligned} &B_{(i|jk)}-F^r_iF^s_jB_{(k|rs)} -F^r_kF^s_iB_{(j|rs)} -F^r_jF^s_kB_{(i|rs)}\\ &\qquad=-2F^r_{(i,l}F^s_jT_{k)rs} -2F^r_{(i}F^s_{j,l}T_{k)rs}; \end{aligned} \tag{V} \]

\[ \begin{aligned} &4F^m_jB_{mikl}+4F^m_iB_{mjl k} +F^m_iB_{(jklm)}+F^m_jB_{(iklm)} -F^m_iB_{(jlkm)}-F^m_jB_{(ilkm)}\\ &\qquad=2F^m_{i,l}T_{jkm}+2F^m_{j,l}T_{ikm} -2F^m_{i,k}T_{jlm}-2F^m_{j,k}T_{ilm}. \end{aligned} \tag{VI} \]

(IV) follows from the relation \(A^i_{jk}=0\). (V) is obtained by covariant differentiation of the identity \(T_{ijk}=F^l_{(i}F^m_jT_{k)lm}\) with the use of the relation (see \({}^{(3)}\)) \(T_{ijk,l}=-\frac12 B_{(i|jk)l}\). To obtain (VI), one must, with the aid of (12), compute \(\nabla_k\nabla_l g_{ij}-\nabla_l\nabla_k g_{ij}\) and use the Ricci identity.

If one symmetrizes the connections of the works \((^{1,2})\), then one obtains symmetric affine connections defined on a Hermitian manifold. The connection (2), (13) can coincide with one of these connections only in the Kähler case. However, there is a deep connection between the connection constructed by us and the connections of the works \((^{1,2})\). Let us introduce the notion of an associated almost symplectic connection: let \(\Gamma^i_{jk}\) be an arbitrary affine connection. Then the quantities \(\gamma_{ijk}=-\tfrac12 a_{l(i}\Gamma^l_{jk)}-\tfrac12 a_{l(i}\Gamma^l_{kj)}\) define an object symmetric in any pair of indices, whose substitution into (2) determines some symmetric almost symplectic affine connection. It is called the connection associated with the connection \(\Gamma^i_{jk}\). It turns out that the following holds:

Theorem 3. The connection (2), (13) is an almost symplectic affine connection associated with the Riemannian connection \(\left\{\begin{matrix} i\\ jk \end{matrix}\right\}\) corresponding to the tensor \(g_{ij}\), as well as with any of the connections of the works \((^{1,2})\).

Let us also note that our connection is expressed in the following way through the components of the Riemannian connection \(\left\{\begin{matrix} i\\ jk \end{matrix}\right\}\):

\[ \Gamma^i_{jk}=\left\{\begin{matrix} i\\ jk \end{matrix}\right\} +\frac13 a^{il}a_{lj/k} +\frac13 a^{il}a_{lk/j}, \]

where \(a_{ij/k}\) is the covariant derivative of the tensor \(a_{ij}\) in the Riemannian connection \(\left\{\begin{matrix} i\\ jk \end{matrix}\right\}\).

In conclusion, let us note that if the manifold \(\mathfrak M\) is regarded as almost Hermitian, i.e., if one does not assume that the tensor

\[ t^i_{jk}=\frac14\left(F^l_j\partial F^i_k/\partial x^l -F^l_j\partial F^i_l/\partial x^k -F^l_k\partial F^i_j/\partial x^l +F^l_k\partial F^i_l/\partial x^j\right), \]

is zero, then the quantities (2), (13) still define an affine connection; however, relations (4) and (6) now take the form:

\[ A^i_{jk}=2t^i_{jk} +\frac23 a^{im}a_{j l}t^l_{mk} +\frac23 a^{lm}a_{kl}t^l_{mj}; \]

\[ g_{ij,k}-g_{ij/k} =-\frac43\left(g_{il}t^l_{jk}+g_{jl}t^l_{ik}\right). \]

The vector \(B_i=B^l_{il}\) has the same form as before: \(B_i=-\tfrac12 a^{lm}T_{ilm}\), and therefore Theorem 2 holds also for almost Hermitian manifolds.

Moscow State University
named after M. V. Lomonosov

Received
9 IV 1962

CITED LITERATURE

\(^{1}\) A. Lichnerowicz, Theory of Connections in the Large and Holonomy Groups, Moscow, 1960.
\(^{2}\) T. Uesugi, Mem. Fac. Sci. Kyusyu. Univ., A, 10, No. 2 (1956).
\(^{3}\) V. G. Lemlein, DAN, 115, No. 4 (1957).
\(^{4}\) V. A. Gaukman, DAN, 142, No. 4 (1962).
\(^{5}\) G. F. Laptev, Tr. Moscow Math. Soc., 2, 275 (1953).
\(^{6}\) A. M. Vasil’ev, DAN, 79, No. 1 (1951).