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Let ( M , ∇,  〈 ,  〉 ) be a manifold endowed with a flat torsionless connection  r  and a Riemannian metric  〈 ,  〉  and ( T k M )  k   ≥1  the sequence of tangent bundles given by  T k M  =  T ( T k   −1  M ) and  T  1  M  =  TM . We show that, for any  k  ≥ 1,  T k M  carries a Hermitian structure ( J k  ,  g k  ) 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  T k M  such that ( J k  ,  g k  ,  ∇ k  ) are left invariant. It is well-known that ( TM ,  J  1 ,  g  1 ) is Kähler if and only if  〈 ,  〉  is Hessian, i.e, in each system of affine coordinates ( x  1 , . . .,  x n  ),〈∂   xi,    ∂x   j〉   =∂   2   φ∂   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 ,  J  1 ,  g  1 ) 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 ( T k M ,  J k  ,  g k  ) 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, ( T k M ,  J k  ,  g k  ) 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.

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