Narrow Results

Contrary to complex structures, a generic almost complex structure J has no local symmetries. We give a criterion for finite-dimensionality of the pseudogroup G of local symmetries for a given almost complex structure J. It will be indicated that a large symmetry pseudogroup (infinite-dimensional) is a signature of some integrable structure, like a pseudoholomorphic foliation. We classify the sub-maximal (from the viewpoint of the size of G) symmetric structures J. Almost complex structures in dimensions 4 and 6 are discussed in greater details in this paper.

We study almost complex structures on parallelizable manifolds via the rank of their Nijenhuis tensor. First, we show how the computations of such rank can be reduced to finding smooth functions on the underlying manifold solving a system of first order PDEs. On specific manifolds, we find an explicit solution. Then we compute the Nijenhuis tensor on curves of almost complex structures, showing that there is no constraint (except for lower semi-continuity) to the possible jumps of its rank. Finally, we focus on $6$-nilmanifolds and the associated Lie algebras. We classify which $6$-dimensional, nilpotent, real Lie algebras admit almost complex structures whose Nijenhuis tensor has a given rank, deducing the corresponding classification for invariant structures on $6$-nilmanifolds. We also find a topological upper-bound for the rank of the Nijenhuis tensor for invariant almost complex structures on solvmanifolds of any dimension, obtained as a quotient of a completely solvable Lie group. Our results are complemented by a large number of examples.

We study intrinsic deformations of Levi flat structures on a smooth manifold. A Levi flat structure on a smooth manifold L is a couple (ξ, J) where ξ ⊂ T(L) is an integrable distribution of codimension 1 and J : ξ → ξ is a bundle automorphism which defines a complex integrable structure on each leaf. A deformation of a Levi flat structure (ξ, J) is a smooth family {(ξ_t, J_t)}_t∈]-ε,ε[ of Levi flat structures on L such that (ξ₀, J₀) = (ξ, J). We define a complex whose cohomology group of order 1 contains the infinitesimal deformations of a Levi flat structure. In the case of real analytic Levi flat structures, this cohomology group is where (𝒵*(L), δ, {⋅,⋅}) is the differential graded Lie algebra associated to ξ.

We show that every Kaehler algebraic curvature tensor is geometrically realizable by a Kaehler manifold of constant scalar curvature. We also show that every para-Kaehler algebraic curvature tensor is geometrically realizable by a para-Kaehler manifold of constant scalar curvature.

We review origins and main properties of the most important bracket operations appearing canonically in differential geometry and mathematical physics in the classical, as well as in the supergeometric setting. The review is supplemented by some new concepts and examples.

We give the strong form and the weak form of the square of Nijenhuis tensor, and some vanish results of the square.

It is well known that the tensor field $J$ of type $(1,1)$ on the manifold $M$ is an almost complex structure if $J^2=-I,I$ is an identity tensor field and the manifold $M$ is called the complex manifold. Let ^kM be the $k$ order extended complex manifold of the manifold $M$. A tensor field $J_k$ on ^kM is called extended almost complex structure if ${(J_k)}^2=-I$. This paper aims to study the higher order complete and vertical lifts of the extended almost complex structures on an extended complex manifold ^kM. The proposed theorems on the Nijenhuis tensor of an extended almost complex structure $J_k$ on the extended complex manifold ^kM are proved. Also, a tensor field $\widetilde{J_k}$ of type $(1,1)$ is introduced and shows that it is an extended almost complex structure. Furthermore, the Lie derivative concerning higher-order lifts is studied and basic results on the almost analytic complex vector concerning an extended almost complex structure on ^kM are investigated. Finally, for more detailed explanation and better understanding a tensor field $J_k^{\ast }$ of type $(1,1)$ is introduced on ^kM, proving that it is a metallic structure on ^kM. A study of a golden structure, which is a type of metallic structure, is also carried out.