Almost Complex and Hypercomplex Norden Structures Induced by Natural Riemann Extensions

The Riemann extension, introduced by E. K. Patterson and A. G. Walker, is a semi-Riemannian metric with a neutral signature on the cotangent bundle T ∗ M of a smooth manifold M, induced by a symmetric linear connection ∇ on M . In this paper we deal with a natural Riemann extension g ¯ , which is a generalization (due to M. Sekizawa and O. Kowalski) of the Riemann extension. We construct an almost complex structure J ¯ on the cotangent bundle T ∗ M of an almost complex manifold ( M , J , ∇ ) with a symmetric linear connection ∇ such that ( T ∗ M , J ¯ , g ¯ ) is an almost complex manifold, where the natural Riemann extension g ¯ is a Norden metric. We obtain necessary and sufficient conditions for ( T ∗ M , J ¯ , g ¯ ) to belong to the main classes of the Ganchev–Borisov classification of the almost complex manifolds with Norden metric. We also examine the cases when the base manifold is an almost complex manifold with Norden metric or it is a complex manifold ( M , J , ∇ ′ ) endowed with an almost complex connection ∇ ′ ( ∇ ′ J = 0 ). We investigate the harmonicity with respect to g ¯ of the almost complex structure J ¯ , according to the type of the base manifold. Moreover, we define an almost hypercomplex structure ( J ¯ 1 , J ¯ 2 , J ¯ 3 ) on the cotangent bundle T ∗ M 4 n of an almost hypercomplex manifold ( M 4 n , J 1 , J 2 , J 3 , ∇ ) with a symmetric linear connection ∇. The natural Riemann extension g ¯ is a Hermitian metric with respect to J ¯ 1 and a Norden metric with respect to J ¯ 2 and J ¯ 3 .