2021/06/23 by Boucetta, Mohamed
#Differential Geometry (math.DG) #FOS: Mathematics #Symplectic Geometry (math.SG)
paper · doi:10.48550/arxiv.2106.12246
Let (M,∇,⟨ , ⟩) be a manifold endowed with a flat torsionless connection ∇ 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⟩=\frac∂2ϕ∂xi∂xj. 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, Vaismann 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.