Horizontal lift of affinor structures and its applications (Q1888562)

In this paper, the authors study the horizontal lifts of tensor fields of type (1,1) to tensor bundles and the integrability conditions for the horizontal lifts of special types of complex and tangent structures. It is shown that if \(\varphi\) is an almost complex structure on a manifold \(M\) with a symmetric connection \(\nabla\), then the horizontal lift \(^ H\!\varphi\) of \(H\) is an almost complex structure in the tensor bundle \(T^ p_ q(M)\). If \(F\) is a Kählerian structure on \(M\), then the horizontal lift \(^ H\!F\) of \(F\) is an almost complex structure in \(T^ p_ q(M)\). Finally, the authors show that if \(F\) is a tangent structure on \(M\), then the horizontal lift \(^ H\!F\) of \(F\) is a tangent structure in \(T^ p_ q(M)\).