A Survey on Geometric Dynamics of 4-Walker Manifold

Abstract

A Walker n-manifold is a semi-Riemannian n-manifold, which admits a field of parallel null r-planes, with r ≤ 2/n . It is well-known that semi-Riemannian geometry has an important tool to describe spacetime events. Therefore, solutions of some structures about 4-Walker manifold can be used to explain spacetime singularities. Then, here we present complex and paracomplex analogues of Lagrangian and Hamiltonian mechanical systems on 4-Walker manifold. Finally, the geometrical-physical results related to complex (paracomplex) mechanical systems are also discussed

Conformal Weyl-Euler-Lagrangian equations on 4-Walker manifolds

“…For the almost complex structure J given by (15), the form on Walker manifold M 4 is the closed 2-form determined by…”

“…In the problem of a mass on the end of a spring, T = mẋ 2 /2 and V = kx 2 /2. We consider the closed 2-form and base space (J) on T M given by (17) is named as Lagrange dynamical equation [15]. [2].…”

“…Davidov showed that any proper almost Hermitian structure on a Walker 4-manifold is isotropic Kähler [14]. Tekkoyun showed that a Walker n-manifold is a semi-Riemannian n-manifold, which admits a field of parallel null r-planes, with r ≤ n/2 [15]. Law examined that a four-dimensional Walker geometry is a four-dimensional manifold M with a neutral metric g and a parallel distribution of totally null two-planes [16].…”

“…Ghanam and Thompson submitted that special interest manifolds are Walker manifolds of even dimensions (n = 2m) admitting a field of null planes of maximum dimensionality (r = m) [17]. [15].…”