On the Hermitian structures of the sequence of tangent bundles of an affine manifold endowed with a Riemannian metric

Abstract Let ( M , ∇, 〈 〉 ) be a manifold endowed with flat torsionless connection r and Riemannian metric T k ≥1 the sequence of tangent bundles given by = −1 1 TM . We show that, for any ≥ 1, carries Hermitian structure J g ∇ when is Lie group (∇, are left invariant there on each such that invariant. It well-known Kähler if only Hessian, i.e, in system affine coordinates x ., n ), 〈 ∂ x i , j 〉 = 2 φ \left\langle {{\partial _x}_{_i},{\partial _{{x_j}}}} \right\rangle {{{\partial ^2}\phi } \over _x}_{_i}{\partial _x}_j}} Having mind many generalizations condition introduced recently, we give conditions so balanced, locally conformally Kähler, pluriclosed, Gauduchon, Vaisman or Calabi-Yau torsion. Moreover, can control at level insuring some all them satisfy generalized condition. For instance, classes balanced non-Kähler By carefully studying geometry develop powerful machinery to build large manifolds.