Mathematics

V. A. Gaukhman

On Geodesic Surfaces in Manifolds of Affine Connection with a Complex Structure

(Presented by Academician P. S. Aleksandrov, 25 V 1961)

In the present work the investigation is local in character. Let \(\mathfrak M_{2n}\) be a \(2n\)-dimensional manifold with a complex analytic structure, on which an almost complex affine connection without torsion is given \((^{1})\).

For frames adapted to the complex structure, the equations of infinitesimal displacements have the form:

\[ \begin{aligned} dM &= \omega^i e_i+\widetilde{\omega}^{\,i} e_{n+i},\\ de_i &= \omega_i^{\,j} e_j+\widetilde{\omega}_i^{\,j} e_{n+j},\qquad (i,j,k=1,\ldots,n).\\ de_{n+i} &= -\,\widetilde{\omega}_i^{\,j} e_j+\omega_i^{\,j} e_{n+j}. \end{aligned} \]

The structure equations of an almost complex connection are written in the form:

\[ \begin{aligned} D\omega^i &= [\omega^j\omega_j^{\,i}]-[\widetilde{\omega}^{\,j}\widetilde{\omega}_j^{\,i}],\\ D\widetilde{\omega}^{\,i} &= [\widetilde{\omega}^{\,j}\omega_j^{\,i}]+[\omega^j\widetilde{\omega}_j^{\,i}],\\ D\omega_i^{\,j} &= [\omega_i^{\,k}\omega_k^{\,j}] -[\widetilde{\omega}_i^{\,k}\widetilde{\omega}_k^{\,j}] +[\omega^k, R_{jkl}^{\,i}\omega^l+\widetilde{R}_{jkl}^{\,i}\widetilde{\omega}^{\,l}] -[\widetilde{\omega}^{\,k}, S_{jkl}^{\,i}\omega^l+\widetilde{S}_{jkl}^{\,i}\widetilde{\omega}^{\,l}],\\ D\widetilde{\omega}_i^{\,j} &= [\omega_i^{\,k}\widetilde{\omega}_k^{\,j}] +[\widetilde{\omega}_i^{\,k}\omega_k^{\,j}] +[\widetilde{\omega}^{\,k}, R_{jkl}^{\,i}\omega^l+\widetilde{R}_{jkl}^{\,i}\widetilde{\omega}^{\,l}] +[\omega^k, S_{jkl}^{\,i}\omega^l+\widetilde{S}_{jkl}^{\,i}\widetilde{\omega}^{\,l}]. \end{aligned} \]

Denoting
\(\Omega^i=\omega^i+\sqrt{-1}\,\widetilde{\omega}^{\,i}\);
\(\Omega_i^{\,j}=\omega_i^{\,j}+\sqrt{-1}\,\widetilde{\omega}_i^{\,j}\), we obtain:

\[ D\Omega^i=[\Omega^k\Omega_k^{\,i}], \tag{1} \]

\[ D\Omega_i^{\,j}=[\Omega_i^{\,k}\Omega_k^{\,j}] +K_{jkl}^{\,i}[\Omega^k\Omega^l] +L_{jkl}^{\,i}[\Omega^k\overline{\Omega}^{\,l}], \tag{2} \]

where

\[ K_{jkl}^{\,i} =\frac12\bigl[(R_{j[kl]}^{\,i}+\widetilde{S}_{j[kl]}^{\,i}) +\sqrt{-1}\,(S_{j[kl]}^{\,i}-\widetilde{R}_{j[kl]}^{\,i})\bigr], \]

\[ L_{jkl}^{\,i} =\frac12\bigl[(R_{jkl}^{\,i}-\widetilde{S}_{jkl}^{\,i}) +\sqrt{-1}\,(S_{jkl}^{\,i}+\widetilde{R}_{jkl}^{\,i})\bigr]. \]

Exterior differentiation of equations (1) gives:

\[ K_{(jkl)}^{\,i}=0;\qquad L_{[jk]l}^{\,i}=0. \tag{3} \]

Let \(C_n\) be a complex analytic manifold of complex dimension \(n\) (see \((^{2})\)), whose equations of infinitesimal displacements have the form:

\[ dM=\Omega^i E_i, \]

\[ dE_i=\Omega_i^{\,j}E_j. \]

Then (1), (2) are the structure equations of this complex manifold.

Every \(2r\)-dimensional analytic surface\(^*\) in \(\mathfrak M_{2n}\) is given by equations of the form:
\[ \omega^\lambda=a_\tau^\lambda\omega^\tau-\widetilde a_\tau^\lambda\widetilde\omega^\tau, \qquad \widetilde\omega^\lambda=\widetilde a_\tau^\lambda\omega^\tau+a_\tau^\lambda\widetilde\omega^\tau, \tag{4} \]
\[ (\tau=1,\ldots,r;\ \lambda=r+1,\ldots,n), \]
or, what is the same, by equations of the form
\[ \Omega^\lambda=A_\tau^\lambda\Omega^\tau, \]
where
\[ A_\tau^\lambda=a_\tau^\lambda+\sqrt{-1}\,\widetilde a_\tau^\lambda . \]
It follows from this that a \(2r\)-dimensional analytic surface in \(\mathfrak M_{2n}\) is the image of an arbitrary complex analytic surface of complex dimension \(r\) in \(C_n\).

Let \(\mathfrak N_2\subset \mathfrak M_{2n}\) be a two-dimensional analytic and totally geodesic (see (3)) surface. \(\mathfrak N_2\) can be given by equations (4), in which \(r=1\). The condition that \(\mathfrak N_2\) be totally geodesic has the form:
\[ da_1^\lambda+a_1^\mu\omega_\mu^\lambda-\widetilde a_1^\mu\widetilde\omega_\mu^\lambda -a_1^\lambda\omega_1^1+\widetilde a_1^\lambda\widetilde\omega_1^1+\omega_1^\lambda -a_1^\lambda(a_1^\mu\omega_\mu^1-\widetilde a_1^\mu\widetilde\omega_\mu^1)+ \]
\[ +\widetilde a_1^\lambda(\widetilde a_1^\mu\omega_\mu^1+a_1^\mu\widetilde\omega_\mu^1)=0, \]
\[ d\widetilde a_1^\lambda+\widetilde a_1^\mu\omega_\mu^\lambda+a_1^\mu\widetilde\omega_\mu^\lambda -\widetilde a_1^\lambda\omega_1^1-a_1^\lambda\widetilde\omega_1^1-\widetilde\omega_1^\lambda -\widetilde a_1^\lambda(a_1^\mu\omega_\mu^1-\widetilde a_1^\mu\widetilde\omega_\mu^1)- \]
\[ -a_1^\lambda(\widetilde a_1^\mu\omega_\mu^1+a_1^\mu\widetilde\omega_\mu^1)=0. \tag{5} \]

Equations (5) can be written more compactly in the form
\[ dA_1^\lambda+A_1^\mu\Omega_\mu^\lambda-A_1^\lambda\Omega_1^1+\Omega_1^\lambda -A_1^\lambda A_1^\mu\Omega_\mu^1=0. \tag{6} \]

But (6) means that the line \(\Omega^\lambda=A_1^\lambda\Omega'\) is geodesic in \(C_n\), i.e. that along this line \(d^2M=\Lambda\cdot dM\). Consequently, a two-dimensional analytic totally geodesic surface in \(\mathfrak M_{2n}\) is the image of a geodesic line in \(C_n\).

Lemma. In the space \(C_n\), through every point and in every complex direction there passes, moreover uniquely, a geodesic line.

Proof. We exteriorly differentiate the system (6). In view of (2), (3), (6), we obtain an identity. Hence the system (6) is completely integrable, which proves the lemma.

It follows immediately from the lemma that in \(\mathfrak M_{2n}\), through any point and in any two-dimensional analytic direction, there passes a unique analytic totally geodesic surface.

Now let \(x_0\) be a point of the manifold \(\mathfrak M_{2n}\) and let \(\pi_{2r}\) be some analytic \(2r\)-dimensional direction at \(x_0\). In \(C_n\) this direction corresponds to some complex direction \(\Pi_r\) of complex dimension \(r\), issuing from a certain point \(X_0\in C_n\). Consider in \(C_n\) the surface formed by all complex geodesic lines passing through the point \(X_0\) and tangent at this point to the plane \(\Pi_r\). In \(\mathfrak M_{2n}\) there corresponds to it an analytic surface. Obviously, this surface is formed by geodesic lines tangent at the point \(x_0\) to the plane \(\pi_{2r}\), i.e. this surface is geodesic at the point \(x_0\) (see (3)). We have proved:

Theorem. Let \(\mathfrak M_{2n}\) be a \(2n\)-dimensional manifold with a complex structure and with an almost complex torsion-free affine connection. Let

\(^*\) A subspace of the tangent space to \(\mathfrak M_{2n}\) at the point \(x\) is called analytic if it is invariant under the automorphism
\(I:e_i\to e_{n+i},\ e_{n+i}\to -e_i\)
(see (2)). A surface is called analytic at a point \(x\in\mathfrak M_{2n}\) if its tangent plane at this point is analytic. A surface is called analytic if all its tangent planes are analytic.

\(x_0\) is an arbitrary point of \(\mathfrak M_{2n}\); \(\pi_{2r}\) is an arbitrary \(2r\)-dimensional analytic subspace of the tangent space at the point \(x_0\). Then the \(2r\)-dimensional surface formed by all geodesic lines passing through the point \(x_0\) and tangent at this point to the plane \(\pi_{2r}\) is analytic. If \(r=1\), then the corresponding two-dimensional surface is also totally geodesic.

Corollary. If an analytic two-dimensional surface \(\mathfrak M_2 \subset \mathfrak M_{2n}\) contains one geodesic line, then it is totally geodesic.

The corollary becomes obvious if one observes that through every line in \(\mathfrak M_{2n}\) there passes a unique two-dimensional analytic surface.

In conclusion we note that the following assertion can be proved: if, on a manifold \(\mathfrak M_{2n}\) with complex structure, a torsion-free affine connection is given such that all the surfaces discussed in the theorem are analytic, then this connection is projectively equivalent to an almost complex torsion-free connection, i.e., on \(\mathfrak M_{2n}\) there exists an almost complex torsion-free connection having the same geodesic lines as the original connection.

The author expresses gratitude to A. M. Vasiliev for valuable suggestions.

Moscow State University
named after M. V. Lomonosov

Received
23 V 1961

REFERENCES

\(^{1}\) A. Lichnerowicz, Theory of connections as a whole and holonomy groups, Moscow, 1960.
\(^{2}\) K. Yano, S. Bochner, Curvature and Betti numbers, Moscow, 1957.
\(^{3}\) Riemannian geometry in an orthogonal frame, from lectures by É. Cartan, Moscow State University Press, 1960.