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

Abstract

Let $(M,\nabla ,⟨,⟩)$ be a manifold endowed with a flat torsionless connection $\nabla$ and a Riemannian metric $⟨,⟩$ and $(T^{k}M{)}_{k\ge 1}$ 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 $k\ge 1$, $T^{k}M$ carries a Hermitian structure $(J_k,g_k)$ and a flat torsionless connection ${\nabla }^k$ and when $M$ is a Lie group and $(\nabla ,⟨,⟩)$ are left invariant there is a Lie group structure on each $T^{k}M$ such that $(J_k,g_k,{\nabla }^k)$ 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 $(x_1,\dots ,x_n)$, $⟨{\partial }_{x_i},{\partial }_{x_j}⟩=\frac{{\partial }^2\varphi }{{\partial }_{x_i}{\partial }_{x_j}}$. Having in mind many generalizations of the Kähler condition introduced recently, we give the conditions on $(\nabla ,⟨,⟩)$ so that $(TM,J_1,g_1)$ is balanced, locally conformally balanced, locally conformally Kähler, pluriclosed, Gauduchon, Vaismann or Calabi-Yau with torsion. Moreover, we can control at the level of $(\nabla ,⟨,⟩)$ 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 $(M,\nabla ,⟨,⟩)$ such that, for any $k\ge 1$, $(T^{k}M,J_k,g_k)$ is balanced non-Kähler and Calabi-Yau with torsion. By carefully studying the geometry of $(M,\nabla ,⟨,⟩)$, we develop a powerful machinery to build a large classes of generalized Kähler manifolds.

arXiv comment40 pages, 8 Tables