International Journal of Geometric Methods in Modern Physics

In this paper, we investigate the spectrum and spectrality of the one-dimensional Schrödinger operator with a periodic PT-symmetric complex-valued potential.

In this paper, we study structures and properties of Null scrolls. We define the (relative) invariants for Null scrolls by using a kind of standard equation. Using these (relative) invariants of Null scrolls, we give some new characterizations and classifications of Null scrolls and B-scrolls.

Conformal, concircular, quasi-conformal and conharmonic curvature tensors play an important role in Riemannian geometry. In this paper, we study on normal complex contact metric manifolds under flatness conditions of these tensors.

In this paper, we introduce pseudo-perpendicular vectors in Minkowski plane and give geometric interpretations of the oriented pseudo-angles in terms of the hyperbolic arcs of finite hyperbolic lengths.

In this study, a model of accretive growth for arbitrary surfaces in three-dimensional Minkowski space is formulated by evolving a curve. An analytical approach to surfaces is also given in terms of a few parameters which are effective in the accretive growth of surfaces. The proposed method is visualized on some test surfaces and displayed in terms of figures.

In this paper, we study the geometric properties of generators for the Klein–Gordon equation on classes of space-time homogeneous Gödel-type metrics. Our analysis complements the study involving the “Symmetries of geodesic motion in Gödel-type spacetimes” by U. Camci (J. Cosmol. Astropart. Phys., doi:10.1088/1475-7516/2014/07/002). These symmetries or Killing vectors (KVs) are used to construct potential functions admitted by the Klein–Gordon equation. The criteria for the potential function originates from three primary sources, viz. through generators that are identically the Killing algebra, or with the KV fields that are recast into linear combinations and third, real subalgebras within the Killing algebra. This leads to a classification of the $(1+3)$ Klein–Gordon equation according to the catalogue of infinitesimal Lie and Noether point symmetries admitted. A comprehensive list of group invariant functions is provided and their application to analytic solutions is discussed.

We make some observations about Rosenberg’s Levi-Civita connections on noncommutative tori, noting the non-uniqueness of general torsion-free metric-compatible connections without prescribed connection operator for the inner *-derivations, the nontrivial curvature form of the inner *-derivations, and the validity of the Gauss–Bonnet theorem for two classes of nonconformal deformations of the flat metric on the noncommutative two-tori, including the case of noncommuting scalings along the principal directions of a two-torus.

Small nonassociative corrections for the SUSY operators $Q_{a,ȧ}$ are considered. The smallness is controlled by the ratio of the Planck length and a characteristic length ${\ell }_0={\mathrm{\Lambda }}^{-1/2}$. Corresponding corrections of the momentum operator arising from the anticommutator of the SUSY operators are considered. The momentum operator corrections are defined via the anticommutator of the unperturbed SUSY operators $Q_{a,ȧ}$ and nonassociative corrections $Q_{1,a,ȧ}$. Choosing different anticommutators, one can obtain either a modified or $q$-deformed commutator of position $x^{\mu }$ and momentum operators $P_{\nu }$.

Electromagnetic knots are electromagnetic fields in which the magnetic and electric lines are level curves of two complex scalar fields. If these scalar fields are chosen so that they can be interpreted as maps between the three-sphere and the two-sphere every time, then the electromagnetic helicity is proportional to the sum of the Hopf indices of both maps, that are topological invariants. An important example of this kind of electromagnetic fields with topological properties is often called the Hopfion. It is an electromagnetic knot in which the scalar fields are built from Hopf maps. In this work, we study the conditions for these electromagnetic fields to be null, i.e. to have both Lorentz invariant quantities $E\cdot B$ and $E^2-B^2$ equal to zero. We derive from these fields explicit vector potentials $A$ and $C$ satisfying force-free like conditions for every time. In particular, equations $A\cdot B=B^2/2$, $C\cdot E=E^2/2$ are derived for them. As a consequence, the energy, which is discretized, is proportional to the electromagnetic helicity. This relation between Physics and Topology is intriguing and worth of future research.

As a generalization of anti-invariant ${\xi }^{\perp }$-Riemannian submersions, we introduce semi-invariant ${\xi }^{\perp }$-Riemannian submersions from Sasakian manifolds onto Riemannian manifolds. We give examples, investigating the geometry of foliations which arise from the definition of a Riemannian submersion and proving a necessary and sufficient condition for a semi-invariant ${\xi }^{\perp }$-Riemannian submersion to be totally geodesic. Moreover, we study semi-invariant ${\xi }^{\perp }$-Riemannian submersions with totally umbilical fibers.

In the light of the analogy between electromagnetism and fluid dynamics, the Maxwell-type equations of compressible fluids are reformulated on the basis of spacetime algebra. In this paper, it is proved that this algebra provides an efficient mathematical tool for describing fluid fields in a compact and elegant way. Moreover, the fluid wave equation in terms of potentials are derived in a form similar to electromagnetic and gravitational counterparts previously derived using spacetime algebra.

In this paper, we study ruled surface in 3-dimensional almost contact metric manifolds by using surface theory defined by Gök [Surfaces theory in contact geometry, PhD thesis (2010)]. We also studied the theory of curves using cross product defined by Camcı. In this study, we obtain the distribution parameters of the ruled surface and then some results and theorems are presented with special cases. Moreover, some relationships among asymptotic curve and striction line of the base curve of the ruled surface have been found.

Before 1998, it was usually expected that the universe was expanding with a constant rate or the expansion was slowing down. In 1998, the surprising discovery based on type Ia supernovae, that the rate of expansion of the universe is increasing, forced the researchers to reconsider the various cosmological models proposed so far. The current study is also an effort to revisit the LRS Bianchi type-II, dark energy (DE) model by taking time-dependent deceleration parameter (DP) instead of constant DP. We have assumed the variable scale factor $a(t)={[\mathrm{\text{sinh}}(\alpha t)]}^{\frac{1}{n}}$, which gives the variable DP as $q(t)=n{\mathrm{\text{sech}}}^2(\alpha t)-1$, with these considerations, the solutions of field equations are calculated. Various parameters of DE models are also calculated, and it is found that these are consistent with the recent observations.

In the differential geometry of certain $F$-structures, the role of $W$-curvature tensor is very well known. A detailed study of this tensor has been made on the spacetime of general relativity. The spacetimes satisfying Einstein field equations with vanishing $W$-tensor have been considered and the existence of Killing and conformal Killing vector fields has been established. Perfect fluid spacetimes with vanishing $W$-tensor have also been considered. The divergence of $W$-tensor is studied in detail and it is seen, among other results, that a perfect fluid spacetime with conserved $W$-tensor represents either an Einstein space or a Friedmann-Robertson-Walker cosmological model.

Shokri et al. [Approximate bihomomorphisms and biderivations in 3-Lie algebras, Int. J. Geom. Methods Mod. Phys. 10 (2013) 1220020, 13pp.] proved the Hyers–Ulam stability of bihomomorphisms and biderivations on normed 3-Lie algebras. It is easy to see that the definition of bihomomorphism in normed 3-Lie algebras is meaningless and so the results of [J. Shokri, A. Ebadian and R. Aghalari, Approximate bihomomorphisms and biderivations in 3-Lie algebras, Int. J. Geom. Methods Mod. Phys. 10 (2013) 1220020, 13pp., Sec. 3] are meaningless. Moreover, there is a serious problem in the main functional equation (1.2). So, we replace the functional equation (1.2) by a suitable functional equation. In this paper, we correct the definition of bihomomorphism and the statements of the results in [J. Shokri, A. Ebadian and R. Aghalari, Approximate bihomomorphisms and biderivations in 3-Lie algebras, Int. J. Geom. Methods Mod. Phys. 10 (2013) 1220020, 13pp., Sec. 3], and prove the corrected theorems.

Regarding a Dynkin diagram as a specific point-line incidence structure (where each line has just two points), one can associate with it a Veldkamp space. Focusing on extended Dynkin diagrams of type ${\tilde{D}}_n$, $4\le n\le 8$, it is shown that the corresponding Veldkamp space always contains a distinguished copy of the projective space PG$(3,2)$. Proper labeling of the vertices of the diagram (for $4\le n\le 7$) by particular elements of the two-qubit Pauli group establishes a bijection between the 15 elements of the group and the 15 points of the PG$(3,2)$. The bijection is such that the product of three elements lying on the same line is the identity and one also readily singles out that particular copy of the symplectic polar space $W(3,2)$ of the PG$(3,2)$ whose lines correspond to triples of mutually commuting elements of the group; in the latter case, in addition, we arrive at a unique copy of the Mermin–Peres magic square. In the case of $n=8$, a more natural labeling is that in terms of elements of the three-qubit Pauli group, furnishing a bijection between the 63 elements of the group and the 63 points of PG$(5,2)$, the latter being the maximum projective subspace of the corresponding Veldkamp space; here, the points of the distinguished PG$(3,2)$ are in a bijection with the elements of a two-qubit subgroup of the three-qubit Pauli group, yielding a three-qubit version of the Mermin–Peres square. Moreover, save for $n=4$, each Veldkamp space is also endowed with some exceptional point(s). Interestingly, two such points in the $n=8$ case define a unique Fano plane whose inherited three-qubit labels feature solely the Pauli matrix $Y$.

In the present paper, we consider generalized Kropina change of $m$th root Finsler metric and prove that it is locally projectively flat if and only if it is locally Minkowskian. We also establish a necessary and sufficient condition under which the generalized Kropina change of $m$th root metric is locally dually flat. Further it is proved that a generalized Kropina change of $m$th root metric cannot be conformal to an $m$th root Finsler metric.