Characterizing Base in Warped Product Submanifolds of Complex Projective Spaces by Differential Equations

In this study, a link between the squared norm of the second fundamental form and the Laplacian of the warping function for a warped product pointwise semi-slant submanifold Mn in a complex projective space is presented. Some characterizations of the base NT of Mn are offered as applications. We also look at whether the base NT is isometric to the Euclidean space Rp or the Euclidean sphere Sp, subject to some constraints on the second fundamental form and warping function.

A Symplectic Dynamics Proof of the Degree–Genus Formula

We classify global surfaces of section for the Reeb flow of the standard contact form on the 3-sphere (defining the Hopf fibration), with boundaries oriented positively by the flow. As an application, we prove the degree-genus formula for complex projective curves, using an elementary degeneration process inspired by the language of holomorphic buildings in symplectic field theory.

On the Topology of Warped Product Pointwise Semi-Slant Submanifolds with Positive Curvature

In this paper, we obtain some topological characterizations for the warping function of a warped product pointwise semi-slant submanifold of the form Ωn=NTl×fNϕk in a complex projective space CP2m(4). Additionally, we will find certain restrictions on the warping function f, Dirichlet energy function E(f), and first non-zero eigenvalue λ1 to prove that stable l-currents do not exist and also that the homology groups have vanished in Ωn. As an application of the non-existence of the stable currents in Ωn, we show that the fundamental group π1(Ωn) is trivial and Ωn is simply connected under the same extrinsic conditions. Further, some similar conclusions are provided for CR-warped product submanifolds.

Aspherical manifolds, Mellin transformation and a question of Bobadilla–Kollár

In their paper from 2012, Bobadilla and Kollár studied topological conditions which guarantee that a proper map of complex algebraic varieties is a topological or differentiable fibration. They also asked whether a certain finiteness property on the relative covering space can imply that a proper map is a fibration. In this paper, we answer positively the integral homology version of their question in the case of abelian varieties, and the rational homology version in the case of compact ball quotients. We also propose several conjectures in relation to the Singer–Hopf conjecture in the complex projective setting.

Unobstructedness of hyperkähler twistor spaces

A family of irreducible holomorphic symplectic (ihs) manifolds over the complex projective line has unobstructed deformations if its period map is an embedding. This applies in particular to twistor spaces of ihs manifolds. Moreover, a family of ihs manifolds over a subspace of the period domain extends to a universal family over an open neighborhood in the period domain.

THE GENERALIZED GEOMETRIC UNCERTAINTY PRINCIPLE FOR SPIN 1/2 SYSTEM`

Geometric Quantum Mechanics is a version of quantum theory that has been formulated in terms of Hamiltonian phase-space dynamics. The states in this framework belong to points in complex projective Hilbert space, the observables are real valued functions on the space, and the Hamiltonian flow is described by the Schr{\"o}dinger equation. Besides, one has demonstrated that the stronger version of the uncertainty relation, namely the Robertson-Schr{\"o}dinger uncertainty relation, may be stated using symplectic form and Riemannian metric. In this research, the generalized Robertson-Schr{\"o}dinger uncertainty principle for spin $\frac{1}{2}$ system has been constructed by considering the operators corresponding to arbitrary direction.

THE DYNAMICAL EVOLUTION OF GEOMETRIC UNCERTAINTY PRINCIPLE FOR SPIN 1/2 SYSTEM

Geometric Quantum Mechanics is a formulation that demonstrates how quantum theory may be casted in the language of Hamiltonian phase-space dynamics. In this framework, the states are referring to points in complex projective Hilbert space, the observables are real valued functions on the space and the Hamiltonian flow is defined by Schr{\"o}dinger equation. Recently, the effort to cast uncertainty principle in terms of geometrical language appeared to become the subject of intense study in geometric quantum mechanics. One has shown that the stronger version of uncertainty relation i.e. the Robertson-Schr{\"o}dinger uncertainty relation can be expressed in terms of the symplectic form and Riemannian metric. In this paper, we investigate the dynamical behavior of the uncertainty relation for spin $\frac{1}{2}$ system based on this formulation. We show that the Robertson-Schr{\"o}dinger uncertainty principle is not invariant under Hamiltonian flow. This is due to the fact that during evolution process, unlike symplectic area, the Riemannian metric is not invariant under the flow.