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Integrability of generalized pluriharmonic maps. (English) Zbl 1323.53067

The author studies generalized pluriharmonic maps \(\phi:M\to N\) where \((M,J)\) is an almost complex manifold and \((N,h)\) is a pseudo-Riemannian manifold. A complex connection \(D\) (i.e., \(DJ=0\)) on an almost complex manifold \((M,J)\) is called of Nijenhuis type if \(T^{1,1}=0\), where \(T\) is the torsion of \(D\). A map \(\phi:(M,J)\to (N,h)\) is called generalized pluriharmonic if it satisfies the equation \((\nabla d\phi)^{1,1}=0\) where \(\nabla\) is the connection on the bundle \(T^*M\otimes\phi^*TN\) induced by a Nijenhuis-type complex connection \(D\) on \((M,J)\) and the Levi-Civita connection \(\nabla^h\) of \((N,h)\). If \((M,g)\) and \((N,h)\) are pseudo-Riemannian manifolds then a smooth map \(\phi:M\to N\) is called harmonic if \(\mathrm{tr}_g\nabla d\phi=0\) where \(\nabla\) is the connection on \(T^*M\otimes\phi^*TN\) induced by the Levi-Civita connections \(\nabla^g,\nabla^h\). The author proves that if \((M,J,g)\) is nearly Kähler then a generalized pluriharmonic map \(\phi:(M,J)\to (N,h)\) is a harmonic map \(\phi:(M,g)\to (N,h)\). An associated family for \(\phi:(M,J)\to (N,h)\) is a family of maps \(\phi :M\to N,\in \mathbb R\) such that \(\Psi\circ d\phi=d\phi\circ R\) where \(R=\exp(J)\) and \(\Psi\) is some bundle isomorphism \(\Psi:\phi^*TN\to\phi^*TN\) which is parallel with respect to \(\nabla^h\), i.e., \(\Psi\circ \phi^*\nabla^h=\phi^*\circ\Psi\). Using the twistor theory the author proves that if \((N^{2n},h)\) is a Riemannian manifold and \(\mathcal Z\) is its twistor space then \(p:(\mathcal Z,J_2)\to N\) is pluriharmonic, where \(J_2\) is the standard almost complex structure on \(\mathcal Z\) and gives examples of generalized pluriharmonic maps from the twistor space \(\phi:(\mathcal Z(S^{2n}),J_2)\to S^{2n}\) and \(\phi:(\mathcal Z_r(\mathbb CP^{n}),J_2)\to \mathbb CP^{n}\) which do not admit an associated family. The author also proves that if \((\mathcal Z,g,J)\) is a twistor space of a positive quaternionic Kähler manifold \(N\) and \(J\) is a canonical nearly Kähler structure then the projection \(P:\mathcal Z\to N\) is a generalized pluriharmonic map which does not admit an associated family.

MSC:

53C43 Differential geometric aspects of harmonic maps
53C28 Twistor methods in differential geometry
53C15 General geometric structures on manifolds (almost complex, almost product structures, etc.)
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