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

Let $(M,\nabla,\langle\;,\;\rangle)$ be a manifold endowed with a flat torsionless connection $\nabla$ and a Riemannian metric $\langle\;,\;\rangle$ and $(T^kM)_{k\geq1}$ the sequence of tangent bundles given by $T^kM=T(T^{k-1}M)$ and $T^1M=TM$. We show that, for any $k\geq1$, $T^kM$ carries a Hermitian structure $(J_k,g_k)$ and a flat torsionless connection $\nabla^k$ and when $M$ is a Lie group and $(\nabla,\langle\;,\;\rangle)$ are left invariant there is a Lie group structure on each $T^kM$ such that $(J_k,g_k,\nabla^k)$ are left invariant. It is well-known that $(TM,J_1,g_1)$ is K\"ahler if and only if $\langle\;,\;\rangle$ is Hessian, i.e, in each system of affine coordinates $(x_1,\ldots,x_n)$, $\langle\partial_{x_i},\partial_{x_j}\rangle=\frac{\partial^2\phi}{\partial_{x_i}\partial_{x_j}}$. Having in mind many generalizations of the K\"ahler condition introduced recently, we give the conditions on $(\nabla,\langle\;,\;\rangle)$ so that $(TM,J_1,g_1)$ is balanced, locally conformally balanced, locally conformally K\"ahler, pluriclosed, Gauduchon, Vaismann or Calabi-Yau with torsion. Moreover, we can control at the level of $(\nabla,\langle\;,\;\rangle)$ the conditions insuring that some $(T^kM,J_k,g_k)$ or all of them satisfy a generalized K\"ahler condition. For instance, we show that there are some classes of $(M,\nabla,\langle\;,\;\rangle)$ such that, for any $k\geq1$, $(T^kM,J_k,g_k)$ is balanced non-K\"ahler and Calabi-Yau with torsion. By carefully studying the geometry of $(M,\nabla,\langle\;,\;\rangle)$, we develop a powerful machinery to build a large classes of generalized K\"ahler manifolds.