The Painlevé expansion for the second Painlevé equation (PII) and fourth Painlevé equation (PIV) have two branches. The singular manifold method therefore requires two singular manifolds. The double singular manifold method is used to derive Miura transformations from PII and PIV to modified Painlevé type equations for which auto-Bäcklund transformations are obtained. These auto-Bäcklund transf...

We consider a first order linear partial differential operator of principal type on a closed connected orientable two-dimensional manifold sending sections of one complex line bundle to sections of another. We prove that the assumption of global hypoellipticity of the operator implies a relation between the degrees of the line bundles and the Euler characteristic of the manifold.

In this paper, we prove a global rigidity theorem for negatively curved Finsler metrics on a compact manifold of dimension n ≥ 3. We show that for such a Finsler manifold, if the flag curvature is a scalar function on the tangent bundle, then the Finsler metric is of Randers type. We also study the case when the Finsler metric is locally projectively flat.

We construct the twistor space associated with an HKT manifold, that is, a hyper-Kähler manifold with torsion, a type of geometry that arises as the target space geometry in two-dimensional sigma models with (4,0) supersymmetry. We show that this twistor space has a natural complex structure and is a holomorphic fibre bundle over the complex projective line with fibre the associated HKT manifol...

In this paper we establish a Nadel-type vanishing theorem and Kawamata–Viehweg-type concerning the asymptotic multiplier ideal sheaf on compact Kähler manifold X. After that, provide relative variant.

The vorticity of a vector field on 3-dimensional Euclidean space is usually given by the curl field. In this paper, we extend concept to n-dimensional compact and oriented Riemannian manifold. We analyse many properties operation. prove that manifold admits unique Helmholtz decomposition establish every smooth an open neighbourhood Stokes’ type identity.

We provide necessary and sufficient conditions for some particular couples ( g , ? ) of pseudo-Riemannian metrics affine connections to be statistical structures if we have gradient almost Einstein, Ricci, Yamabe solitons, or a more general type solitons on the manifold. In cases, establish formula volume manifold give lower an upper bound norm Ricci curvature tensor field.

In the Higgs mechanism, mediators of the weak force acquire masses by interacting with the Higgs condensate, leading to a vector boson mass matrix. On the other hand, a rigid body accelerated through an inviscid, incompressible and irrotational fluid feels an opposing force linearly related to its acceleration, via an added-mass tensor. We uncover a striking physical analogy between the two eff...

In this article, we apply the Nehari manifold method to study the Kirchhoff type equation