• Journal of the Korean Mathematical Society
• /
• v.57 no.1
• /
• pp.195-213
• /
• 2020

We define two types of null hypersurfaces as; isoparametric and quasi isoparametric null hypersurfaces of Lorentzian space forms, based on the two shape operators associated with a null hypersurface. We prove that; on any screen conformal isoparametric null hypersurface, the screen geodesics lie on circles in the ambient space. Furthermore, we prove that the screen distributions of isoparametric (or quasi isoparametric) null hypersurfaces with at most two principal curvatures are generally Riemannian products. Several examples are also given to illustrate the main concepts.

• Communications of the Korean Mathematical Society
• /
• v.6 no.1
• /
• pp.73-87
• /
• 1991

• Journal of The Korean Astronomical Society
• /
• v.46 no.3
• /
• pp.97-102
• /
• 2013

In the present work, we prove the validity of two theorems on null-paths in a version of absolute parallelismgeometry. A version of these theorems has been originally established and proved by Kermak, McCrea and Whittaker (KMW) in the context of Riemannian geometry. The importance of such theorems lies in their applications to derive a general formula for the redshift of spectral lines coming from distant objects. The formula derived in the present work can be applied to both cosmological and astrophysical redshifts. It takes into account the shifts resulting from gravitation, different motions of the source of photons, spin of the moving particle (photons) and the direction of the line of sight. It is shown that this formula cannot be derived in the context of Riemannian geometry, but it can be reduced to a formula given by KMW under certain conditions.

• Bulletin of the Korean Mathematical Society
• /
• v.50 no.4
• /
• pp.1109-1126
• /
• 2013

Let $\mathbb{M}^3_q(c)$ denote the 3-dimensional space form of index $q=0,1$, and constant curvature $c{\neq}0$. A curve ${\alpha}$ immersed in $\mathbb{M}^3_q(c)$ is said to be a Bertrand curve if there exists another curve ${\beta}$ and a one-to-one correspondence between ${\alpha}$ and ${\beta}$ such that both curves have common principal normal geodesics at corresponding points. We obtain characterizations for both the cases of non-null curves and null curves. For non-null curves our theorem formally agrees with the classical one: non-null Bertrand curves in $\mathbb{M}^3_q(c)$ correspond with curves for which there exist two constants ${\lambda}{\neq}0$ and ${\mu}$ such that ${\lambda}{\kappa}+{\mu}{\tau}=1$, where ${\kappa}$ and ${\tau}$ stand for the curvature and torsion of the curve. As a consequence, non-null helices in $\mathbb{M}^3_q(c)$ are the only twisted curves in $\mathbb{M}^3_q(c)$ having infinite non-null Bertrand conjugate curves. In the case of null curves in the 3-dimensional Lorentzian space forms, we show that a null curve is a Bertrand curve if and only if it has non-zero constant second Frenet curvature. In the particular case where null curves are parametrized by the pseudo-arc length parameter, null helices are the only null Bertrand curves.

• Journal of the Korean Mathematical Society
• /
• v.33 no.3
• /
• pp.507-512
• /
• 1996

A totally umbilic submanifold of a pseudo-Riemanian manifold is a submanifold whose first fundamental form and second fundamental form are proportiona. An ordinary hypersphere $S^n(r)$ of an affine (n + 1)-space of the Euclidean space $E^m$ is the best known example of totally umbilic submanifolds of $E^m$.

• The Journal of the Convergence on Culture Technology
• /
• v.5 no.1
• /
• pp.385-388
• /
• 2019

Considering spherically symmetric spacetimes embedded in a 5-dimensional static Lorentzian manifold, within the large distance limit of DGP model, we study null geodesic equations. We discuss possible relations of particles following the geodesics with the gravitational waves detected recently, in comparison with the electromagnetic waves propagaing in these brane spacetimes.