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

Let (M, ∇, 〈, 〉) be a manifold endowed with a flat torsionless connection r and a Riemannian metric 〈, 〉 and (TkM)k≥1 the sequence of tangent bundles given by TkM = T(Tk−1M) and T1M = TM. We show that, for any k ≥ 1, TkM carries a Hermitian structure (Jk, gk) and a flat torsionless connection ∇k and when M is a Lie group and (∇, 〈, 〉) are left invariant there is a Lie group structure on each TkM such that (Jk, gk, ∇k) are left invariant. It is well-known that (TM, J1, g1) is Kähler if and only if 〈, 〉 is Hessian, i.e, in each system of affine coordinates (x1, . . ., xn), 〈∂xi,∂xj〉=∂2φ∂xi∂xj\left\langle {{\partial _x}_{_i},{\partial _{{x_j}}}} \right\rangle = {{{\partial ^2}\phi } \over {{\partial _x}_{_i}{\partial _x}_j}}. Having in mind many generalizations of the Kähler condition introduced recently, we give the conditions on (∇, 〈, 〉) so that (TM, J1, g1) is balanced, locally conformally balanced, locally conformally Kähler, pluriclosed, Gauduchon, Vaisman or Calabi-Yau with torsion. Moreover, we can control at the level of (∇, 〈, 〉) the conditions insuring that some (TkM, Jk, gk) or all of them satisfy a generalized Kähler condition. For instance, we show that there are some classes of (M, ∇, 〈, 〉) such that, for any k ≥ 1, (TkM, Jk, gk) is balanced non-Kähler and Calabi-Yau with torsion. By carefully studying the geometry of (M, ∇, 〈, 〉), we develop a powerful machinery to build a large classes of generalized Kähler manifolds.