Almost Complex Research Articles

Further remarks on the geometry of two-dimensional non-linear σ-models

A class of extended supersymmetric two-dimensional non-linear sigma -models can be formulated in terms of constrained extended superfields. The conditions for these constraints to be linearisable are examined. These conditions involve generalised Nijenhuis tensors and curvature tensors constructed from the complex structures on the target manifold. Off-shell constrained extended superfields are constructed for the Yang-Mills sector of the (2, 0) and (4, 0) models. These superfields will exist provided that certain conditions are satisfied; these conditions are most conveniently expressed in terms of the geometry of fibre bundles. The existence of (4, 0) superfields implies that the general (4, 0) sigma -model with Yang-Mills sector is ultraviolet finite.

The integrability condition for dynamical systems

We study dynamical systems with a dual Poisson bracket structure from the lagrangian point of view. We show that the condition for integrability corresponds to the vanishing of the Nijenhuis tensor. The conserved quantities are obtained in closed form and the Lax equation as well as the higher order equations in the hierarchy are also presented.

Bi-Hamiltonian structure and invariant endomorphism for rigid body dynamics

A mixed tensor field, with vanishing Nijenhuis tensor and doubly degenerate eigenvalues, invariant for Lagrange-Poisson gyroscope dynamics is constructed.

Affine bundles and integrable almost tangent structures

An almost tangent structure on a manifold N is a type (1,1) tensor field S onN with the property that at each point y ∊N the kernel of Sy (regarded as a linear endo-morphism of TyN) coincides with its image. An almost tangent structure is said to be integrable if its Nijenhuis tensor vanishes.

Submanifolds of differentiable manifolds endowed with differential-geometric structures. IV. Submanifolds of codimension 1 of manifolds with an almost-complex structure

We continue the series of surveys published in Problemy Geometrii. We present a theory of multidimensional surfaces in manifolds with an almost-complex structure.

Non-Canonical Symmetries, Bi-Hamiltonian Structures, and Complete Integrability

It is shown that a non-canonical symmetry of a finite-dimensional Hamiltonian system leads to a bi-Hamiltonian structure for the system. If the recursion operator has a vanishing Nijenhuis tensor and minimal degeneracy, it generates a sequence of conserved quantities in involution. The recursion operator is also the Lax matrix of an iso spectral representation, and its eigenvalues are conserved quantities in involution. If these conserved quantities are functionally independent, the system is completely integrable.

Feuilletages transversalement tangents

A foliation ℱ on a differentiable manifold M is said to be transversally tangent if the jacobian matrices of the change of the transverse coordinates are in the tangent group. Such foliation exists if and only if there is an endomorphism J of its normal bundle such that J2=0 and such that the Nijenhuis tensor of J is the zero-tensor. In the special case of the tangent bundle of order 2, T2M, its total space has a natural (1, 1)-tensor F such that F3=0 and an integrable almost-tangent structure. We study several Lie algebras associated to these structures.

A new characterization of completely integrable systems

A characterization of separability, projectability and integrability of dynamical systems in terms of the spectral properties of invariant mixed tensor fields with vanishing Nijenhuis tensor is given. In addition, some preliminary results on the inverse problem (from Liouville integrability to Lax representation) are illustrated.

On the Nijenhuis tensor in relativistic electrodynamics

The Nijenhuis tensorsNab••c corresponding to the different types of relativistic electromagnetic fields in the Ruse-Synge classification are considered. By using the Newman-Penrose spin coefficient formalism, it is shown that for the type-A fieldsNab••c=0 if and only if i) all the optical parameters of the two real null congruencesla andna vanish and ii)la;bnb=na;blb=0. In the case of the type-B fields,Nab••c=0 if and only if i)la is twist free,la;bnb=0, and ii)\(I_1 = \varrho \bar \varphi _2 \). The conditions for the type-C fields are obtainable from the type-B fields by just interchangingla withna. The effect ofNab••c=0 on the 4-currentJa is that i)Ja=0 for the type-A′ field and ii)JaJa<0 for the type-A″ field as well as for the singular fields (typeB and typeC).

Null electromagnetic fields, total gravitational radiation and collineations in general relativity

The null electromagnetic field, or (F, g, r, S)-structure, and the corresponding Nijenhuis tensor have been studied in an invariant index-free manner. It is seen that the null electromagnetic fields are characterized by the relation F3= 0, and the Nijenhuis tensor plays a very natural role in the study of null electromagnetic fields. The Lichnerowicz contions for the total gravitational radiation have been given in the present setting, and the condition F3=0 has been translated into a corresponding condition on the Ricci tensor. Further, different types of collineations, for (F, g, r, S)-structure, along the propagation and polarization vectors S and T, respectively, have been studied. It is also shown that $$\mathop \pounds\limits_S F_{ij} = 0$$ implies $$F_j^k \mathop \pounds\limits_S g_{ik} = 0$$ . Finally, a covariant conservation law generator has been given for the present structure.

Sobre condiciones de integrabilidad de una estructura $J$ satisfaciendo $(J^2-p^2)(J^2+q^2) = 0$

We consider the structures defined by a tensor field J of type (1,1) which satisfy the condition (J2-p2)(J2+q2)=0 characterizing its integrability in term of its Nijenhuis tensor.

Structures induced on submanifolds of almost-complex manifolds

Structures induced on submanifolds of almost-complex manifolds

Obstruction Formulas and Almost-Complex Manifolds

This paper contains three theorems about almost-complex manifolds. The first theorem states that, under certain conditions, the Euler characteristic of an almost-complex manifold ${M^{2n}}$ must be divisible by $(n - 1)!$. This theorem implies that if ${M^{2n}}$ is an almost-complex homology sphere, then $n \leq 3$. The next two theorems concern the maximal number of vector fields of an almost-complex manifold which are linearly independent over the complex numbers.

Obstruction formulas and almost-complex manifolds

This paper contains three theorems about almost-complex manifolds. The first theorem states that, under certain conditions, the Euler characteristic of an almost-complex manifold M2n must be divisible by (n 1)!. This theorem implies that if MV 2n is an almost-complex homology sphere, then n < 3. The next two theorems concern the maximal number of vector fields of an almost-complex manifold which are linearly independent over the complex numbers.

A note on a differential concomitant

If h {\mathbf {h}} and k {\mathbf {k}} are vector 1 1 -fotms, the vanishing of the concomitant [ h , k ] [{\mathbf {h}},\;{\mathbf {k}}] is an integrability condition fot certain problems on manifolds. In the case that h = k {\mathbf {h}} = {\mathbf {k}} the vanishing of the Nijenhuis tensor [ h , h ] [{\mathbf {h}},\;{\mathbf {h}}] implies d ( tr ⁡ h ) d(\operatorname {tr} {\mathbf {h}}) is a conservation law for h {\mathbf {h}} , provided that tr ⁡ h \operatorname {tr} {\mathbf {h}} is not constant. When the trace of h {\mathbf {h}} is constant, a conservation law for h {\mathbf {h}} exists if one can find a vector 1 1 -form k {\mathbf {k}} with nonconstant trace such that [ h , k ] = 0 [{\mathbf {h}},\;{\mathbf {k}}] = 0 .

The Second Fundamental Forms of S 6 and P n (C)

A Riemannian metric G on a 2n-dimensional manifold M with almostcomplex operator J will be called a J-metric if it is almost-hermitian with respect to J: G (JX, JY) = G (X, Y) for any tangent vectors X and Y of M. Associated with any J-metric G is a real-valued 2-form IIG on M, the second fundamental form of G, defined in [2] by means of an isometric imbedding of M in a high-dimensional Euclidean sphere. The form IIG is described briefly in section 1 of this paper. The metric G will be called a distinguished J-metric if there exists an isometric imbedding with the property that IIG (X, JX) > 0 for any tangent vector X of M. The following theorem is then proved in sections 2-4. If {Jt} is a differentiable family of almost-complex structures on M, 0 < t < 1, and if there exists a distinguished Jo-metric, then there exists a distinguished J,-metric. This result has as corollary the proposition that, if there exists a distinguished JO-metric and if J1 is integrable, then there exists a Kiihler J,-metric (section 1). This proposition can be applied to the spaces S6 and Pn(C), because the canonical almost-complex structures of each of these spaces do in fact admit distinguished metrics. We therefore conclude (sections 5 and 6) that

Hypertangent BundlesW(M 2n ) with (H, ξ, η, G)-structure(M 2n ) with (H, ξ, η, G)-structure

LetJ(M2n) be a vector bundle of (2n+1) dimensions over a differentiable manifold M2n. IfJ(M2n) admits an almost contact metric structure,i.e., (Φ, ξ, η, G)-structure, then it is a hypertagent bundle over M2n and the base manifold M2n admits an almost Hermitian structure. If in such a hypertangent bundleW(M2n) with (Φ, ξ, η, G)-structure the structure tensor field coincides with the second fundamental tensor field H ofW(M2n), we call it the hypertangent bundleW(M2n) with (H, ξ, η, G) structure. The base manifold M2n of such a hypertangent bundle is always totally geodesic The Nijenhuis tensor of the induced almost complex structure which is the second fundamental tensor field of M2n can be expressed interms of the covector field η ofW(M2n) and of the Poisson bracket operator [Bc, Bb] made by the tangent frame vectors Bb of M2n. If h is equal to the covariant derivative of a vector field u in M, one calls such aW(M2n) the hypertangent bundle with\((H,\xi ,\eta ,G:h = \mathop \nabla \limits^o u)\)-structure. In this case u is a Killing vector field and is a contravariant almost analytic vector field at the same time. Its base manifold M2n admits an almost Kaehlerian structure.

Note on the Ślebodziński Tensor of an almost-Complex Structure

Note on the Ślebodziński Tensor of an almost-Complex Structure

Integrability conditions for almost-complex manifolds.

Integrability conditions for almost-complex manifolds.

Differential Geometry on Complex and Almost Complex Spaces. By Kentaro Yano. Pp. xii, 323. 90s. 1965. (Pergamon.)

Differential Geometry on Complex and Almost Complex Spaces. By Kentaro Yano. Pp. xii, 323. 90s. 1965. (Pergamon.)