Almost Complex Research Articles

Novel theorems for the cotangent bundle endowed with metallic structures on a differentiable manifold

The present paper aims to study the complete, horizontal and diagonal lifts of metallic structures in the cotangent bundle. Furthermore, the Nijenhuis tensor of a metallic structure is calculated and its integrability conditions by means of partial differential equations are established.

Rank of the Nijenhuis tensor on parallelizable almost complex manifolds

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 [Formula: see text]-nilmanifolds and the associated Lie algebras. We classify which [Formula: see text]-dimensional, nilpotent, real Lie algebras admit almost complex structures whose Nijenhuis tensor has a given rank, deducing the corresponding classification for invariant structures on [Formula: see text]-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.

Characterizations of the Frame Bundle Admitting Metallic Structures on Almost Quadratic ϕ-Manifolds

In this work, we have characterized the frame bundle FM admitting metallic structures on almost quadratic ϕ-manifolds ϕ2=pϕ+qI−qη⊗ζ, where p is an arbitrary constant and q is a nonzero constant. The complete lifts of an almost quadratic ϕ-structure to the metallic structure on FM are constructed. We also prove the existence of a metallic structure on FM with the aid of the J˜ tensor field, which we define. Results for the 2-Form and its derivative are then obtained. Additionally, we derive the expressions of the Nijenhuis tensor of a tensor field J˜ on FM. Finally, we construct an example of it to finish.

Gabor frames from contact geometry in models of the primary visual cortex

We analyze the interplay between contact geometry and Gabor filters signal analysis in geometric models of the primary visual cortex. We show in particular that a specific framed lattice and an associated Gabor system is determined by the Legendrian circle bundle structure of the $3$-manifold of contact elements on a surface (which models the V1-cortex), together with the presence of an almost-complex structure on the tangent bundle of the surface (which models the retinal surface). We identify a scaling of the lattice, also dictated by the manifold geometry, that ensures the frame condition is satisfied. We then consider a $5$-dimensional model where receptor profiles also involve a dependence on frequency and scale variables, in addition to the dependence on position and orientation. In this case we show that a proposed profile window function does not give rise to frames (even in a distributional sense), while a natural modification of the same generates Gabor frames with respect to the appropriate lattice determined by the contact geometry.

On Para-Sasakian Manifold with Respect to the Schouten-van Kampen Connection

In the present paper, we have studied the curvature properties of the Schouten-van Kampen connection on the n-dimensional Para Sasakian manifold and obtained some new results. Also, we studied projective curvature tensor, concircular curvature tensor, and Nijenhuis tensor for the Para-Sasakian manifold with respect to the Schouten-van Kampen connection.

Kerr–Schild Tetrads and the Nijenhuis Tensor

We write the Kerr–Schild tetrads in terms of the flat space–time tetrads and of a (1, 1) tensor Sμλ. This tensor can be considered as a projection operator, since it transforms (i) flat space–time tetrads into non-flat tetrads, and vice-versa, and (ii) the Minkowski space–time metric tensor into a non-flat metric tensor, and vice-versa. The Sμλ tensor and its inverse are constructed in terms of the standard null vector field lμ that defines the Kerr–Schild form of the metric tensor in general relativity, and that yields black holes and non-linear gravitational waves as solutions of the vacuum Einstein’s field equations. We demonstrate that the condition for the vanishing of the Ricci tensor obtained by Kerr and Schild, in empty space–time, is also a condition for the vanishing of the Nijenhuis tensor constructed out of Sμλ. Thus, a theory based on the Nijenhuis tensor yields an important class of solutions of the Einstein’s field equations, namely, black holes and non-linear gravitational waves. We also demonstrate that the present mathematical framework can easily admit modifications of the Newtonian potential that may explain the long range gravitational effects related to galaxy rotation curves.

Combinatorics of Vogan diagrams for almost-Kähler manifolds

Let [Formula: see text] be a non-compact classical semisimple Lie group and let [Formula: see text] be the adjoint orbit with respect to a fixed element in [Formula: see text]. These manifolds can be equipped with an almost-Kähler structure and we provide explicit formulae for the existence of special almost-complex structures on [Formula: see text] purely in terms of the combinatorics of the associated Vogan diagram. The formulae are given separately for Lie groups whose Lie algebras are of type [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text], where [Formula: see text] denotes the rank of the Lie algebra.

A study of different class of almost Para-Sasakian Riemannian manifold

In this work, the author thought about and looked into the Nijenhuis tensor, the Affine Connection, and the conformal curve tensor on Nearly Para Sasakian different class-Riemannian differential Manifold. We developed and proved the different results for above mentioned class. We also established new condition of Geodesic for different class-Riemannian differential Manifold.

Biharmonic almost complex structures

Abstract This project uses methods in geometric analysis to study almost complex manifolds. We introduce the notion of biharmonic almost complex structure on a compact almost Hermitian manifold and study its regularity and existence in dimension four. First, we show that there always exists smooth energy-minimizing biharmonic almost-complex structures for any almost Hermitian four manifold. Then, we study the existence problem where the homotopy class is specified. Given a homotopy class $[\tau ]$ of an almost complex structure, using the fact $\pi _4(S^2)=\mathbb {Z}_2$ , there exists a canonical operation p on the homotopy classes satisfying $p^2=\text {id}$ such that $p([\tau ])$ and $[\tau ]$ have the same first Chern class. We prove that there exists an energy-minimizing biharmonic almost complex structure in the companion homotopy classes $[\tau ]$ and $p([\tau ])$ . Our results show that, When M is simply connected, there exists an energy-minimizing biharmonic almost complex structure in the homotopy classes with the given first Chern class.

Some pseudo-Kähler Einstein 4-symmetric spaces with a “twin” special almost complex structure

On 4-symmetric symplectic spaces, invariant almost complex structures -up to sign- arise in pairs. We exhibit some 4-symmetric symplectic spaces, with a pair of “natural” compatible (usually not positive) invariant almost complex structures, one of them being integrable and the other one being maximally non-integrable (i.e. the image of its Nijenhuis tensor at any point is the whole tangent space at that point). The integrable one defines a pseudo-Kähler Einstein metric on the manifold, and the non-integrable one is Ricci Hermitian (in the sense that the almost complex structure preserves the Ricci tensor of the associated Levi Civita connection) and special in the sense that the associated Chern Ricci form is proportional to the symplectic form.

Nearly para-Kähler geometry on Lie groups

In this paper, we introduce left-invariant (strict) nearly para-Kaähler structures on Lie groups (nearly para-Kähler Lie algebras) and present some properties of them. We prove the existence of a unique connection in terms of the characteristic connection on these Lie groups with totally skew-symmetric torsion tensor and we show that the Nijenhuis tensor is parallel with respect to the characteristic connection. We determine conditions that allow the curvature tensor of the characteristic connection satisfies the first Bianchi identity. Finally, we focus on anti-abelian almost para-complex structures on Lie groups and we study nearly para-Kähler conditions for these structures. In this study we encounter bi-invariant metrics as pseudo-Riemannian metrics that accept nearly para-Kähler structures. Moreover, we present some examples of nearly para-Kähler Lie algebras.

On Lie algebras associated with a spray

The Lie algebra of infinitesimal isometries of a Riemannian manifold contains at most two commutative ideals. One coming from the horizontal nullity space of the Nijenhuis tensor of the canonical connection, the other coming from the constant vectors fields independent of the Riemannian metric.

Maximally non-integrable almost complex structures: an h-principle and cohomological properties

We study almost complex structures with lower bounds on the rank of the Nijenhuis tensor. Namely, we show that they satisfy an h-principle. As a consequence, all parallelizable manifolds and all manifolds of dimension 2nge 10 (respectively ge 6) admit a almost complex structure whose Nijenhuis tensor has maximal rank everywhere (resp. is nowhere trivial). For closed 4-manifolds, the existence of such structures is characterized in terms of topological invariants. Moreover, we show that the Dolbeault cohomology of non-integrable almost complex structures is often infinite dimensional (even on compact manifolds).

Slice regular functions and orthogonal complex structures over $\mathbb{R}^8$

This work looks at the theory of octonionic slice regular functions through the lens of differential topology. It proves a full-fledged version of the open mapping theorem for octonionic slice regular functions. Moreover, it opens the path for a possible use of slice regular functions in the study of almost-complex structures in eight dimensions.

Nijenhuis tensor and invariant polynomials

We discuss the diagonalization problem of the Nijenhuis tensor in a class of Poisson-Nijenhuis structures defined on compact hermitian symmetric spaces. We study its action on the ring of invariant polynomials of a Thimm chain of subalgebras. The existence of ϕ-minimal representations defines a suitable basis of invariant polynomials that completely solves the diagonalization problem. We prove that such representations exist in the classical cases AIII, BDI, DIII and CI, and do not exist in the exceptional cases EIII and EVII. We discuss a second general construction that in these two cases computes partially the spectrum and hints at a different behavior with respect to the classical cases.

Interaction of Codazzi Pairs with Almost Para Norden Manifolds

In this paper, we research some properties of Codazzi pairs on almost para Norden manifolds. Let $(M_{2n},\ \varphi ,\ g,G)$ be an almost para Norden manifold. Firstly, $g$-conjugate connection, $G$-conjugate connection and $\varphi $-conjugate connection of a linear connection $\mathrm{\nabla }$ on $M_{2n}$ denoted by ${\mathrm{\nabla }}^{*\
 },\ {\mathrm{\nabla }}^{\dagger \ }$ and ${\mathrm{\nabla }}^{\varphi \ }$ are defined and it is demonstrated that on the spaces of linear connections, $\left(id,\ *,\dagger ,\varphi \right)$ acts as the four-element Klein group. We also searched some properties of these three types conjugate
 connections. Then, Codazzi pairs $\left(\mathrm{\nabla },\varphi \right)\ ,\left(\mathrm{\nabla },g\right)$ and $\left(\mathrm{\nabla },G\right)$ are introduced and some properties of them are given. Let $R\ ,\ R^{*\ }$and $R^{\dagger \ }$are $(0,4)$-curvature tensors of conjugate connections
 $\mathrm{\nabla }\mathrm{\ ,\ }{\mathrm{\nabla }}^{*\ }$and ${\mathrm{\nabla }}^{\dagger \ }$, respectively. The relationship among the curvature tensors is investigated. The condition of $N_{\varphi }=0$ is obtained, where $N_{\varphi }$ is Nijenhuis tensor field on $M_{2n}$ and it is known
 that the condition of integrability of almost para complex structure $\varphi $ is $N_{\varphi }=0$. In addition, Tachibana operator is applied to the pure metric $g$ and a necessary and sufficient condition $\left(M,\varphi ,\ g,G\right)$ being a para Kahler Norden manifold is found. Finally, we examine $\varphi $-invariant linear connections and statistical manifolds.

Proposed theorems for lifts of the extended almost complex structures on the complex manifold

It is well known that the tensor field [Formula: see text] of type [Formula: see text] on the manifold [Formula: see text] is an almost complex structure if [Formula: see text] is an identity tensor field and the manifold [Formula: see text] is called the complex manifold. LetkM be the [Formula: see text] order extended complex manifold of the manifold [Formula: see text]. A tensor field [Formula: see text] onkM is called extended almost complex structure if [Formula: see text]. This paper aims to study the higher order complete and vertical lifts of the extended almost complex structures on an extended complex manifoldkM. The proposed theorems on the Nijenhuis tensor of an extended almost complex structure [Formula: see text] on the extended complex manifoldkM are proved. Also, a tensor field [Formula: see text] of type [Formula: see text] 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 onkM are investigated. Finally, for more detailed explanation and better understanding a tensor field [Formula: see text] of type [Formula: see text] is introduced onkM, proving that it is a metallic structure onkM. A study of a golden structure, which is a type of metallic structure, is also carried out.

Constants of motion associated with canonoid transformations for non-autonomous systems

We consider non-autonomous systems of ordinary differential equations that can be expressed in Hamiltonian form in terms of two different coordinate systems, not related by a canonical transformation. We show that the relationship between these coordinate systems leads to a, possibly time-dependent, tensor field, $S^{\alpha}_{\beta}$, whose eigenvalues are constants of motion. We prove that if the Nijenhuis torsion tensor of $S^{\alpha}_{\beta}$ is equal to zero then the eigenvalues of $S^{\alpha}_{\beta}$ are in involution, and that these eigenvalues may be in involution even if the Nijenhuis tensor is not zero.

Novel theorems for metallic structures on the frame bundle of the second order

It is well known that ?an almost complex structure? J that is J2 = ?I on the manifold M is called ?an almost Hermitian manifold? (M, J,G) if G(JX, JY) = G(X,Y) and proved that (F2M, JD,GD) is ?an almost Hermitian manifold? on the frame bundle of the second order F2M. The term ?an almost complex structure? refers to the general quadratic structure J2 = pJ + qI, where p = 0, q = ?1. However, this paper aims to study the general quadratic equation J2 = pJ + qI, where p, q are positive integers, it is named as a metallic structure. The diagonal lift of the metallic structure J on the frame bundle of the second order F2Mis studied and shows that it is also a metallic structure. The proposed theorem proves that the diagonal lift GD of a Riemannian metric G is a metallic Riemannian metric on F2M. Also, a new tensor field ?J of type (1,1) is defined on F2M and proves that it is a metallic structure. The 2-form and its derivative dF of a tensor field ?J are determined. Furthermore, the Nijenhuis tensor N?J of a metallic structure ?J and the Nijenhuis tensor NJD of a tensor field JD of type (1,1) on the frame bundle of the second order F2M are calculated.

Alternative Equations for Horizontal Lifts of the Metallic Structures from Manifold onto Tangent Bundle

We explore “the horizontal lift” of the structure J satisfying J2 − αJ − βI = 0 and establish that it as a kind of metallic structure. An analysis of Nijenhuis tensor of metallic structure JH is presented, and a new tensor field of (1,1)‐type is introduced and demonstrated to be metallic structure. Some results on the Nijenhuis tensor and the Lie derivative of in TM are proved and explicit examples are given. Moreover, the metallic structure endowed with projection operators and in TM is studied.