Fully nonlinear curvature flow of axially symmetric hypersurfaces

In this thesis we consider axially symmetric evolving hypersurfaces mostly with boundary conditions between two parallel planes. The speed function is a fully nonlinear function of the principal curvatures of the hypersurface, homogeneous of degree one. We have results for several boundary conditions. Specifically, with a natural class of Neumann boundary conditions we show that evolving hypersurfaces exist for a finite maximal time. The maximal time is characterised by a curvature singularity at either boundary. Generally, the singularities of the flow are classified as Type I in the case of pure Neumann boundary conditions. In addition to the curvature pinching estimate that is obtained, Sturmian theory is applied to show the discreteness of singularities. Furthermore, some results carry over to higher degrees of homogeneity. Finally, we have some additional results including a gradient bound for the hight function in the volume preserve case

Interpolation of surfaces with asymptotic curves in Euclidean 3-space

In this paper, we investigate the interpolation of surfaces which are obtained from an isoasymptotic curve in 3D-Euclidean space. We prove that there exist a unique $C^0$-Hermite surface interpolation related to an isoasymptotic curve under some special conditions on the marching scale functions. Finally, we present some examples and plot their graphs

Bi-slant lightlike submanifolds of golden semi-Riemannian manifolds

In this paper, we introduce the notion of a bi-slant lightlike submanifold of a golden semi-Riemannian manifold. We provide two examples. We give some characterizations about the geometry of such submanifolds

On Riemannian warped-twisted product submersions

In this paper, we introduce the concepts of Riemannian warped-twisted product submersions and examine their fundamental properties, including total geodesicity, total umbilicity and minimality. Additionally, we investigate the Ricci tensor of Riemannian warped-twisted product submersions, specifically about the horizontal and vertical distributions. Finally, we obtain Einstein condition for base manifold if the horizontal and vertical distributions of the ambient manifold is Einstein

On the Oscillation of Solutions of Differential Equations with Neutral Term

In this work, new criteria for the oscillatory behavior of even-order delay differential equations with neutral term are established by comparison technique, Riccati transformation and integral averaging method. The presented results essentially extend and simplify known conditions in the literature. To prove the validity of our results, we give some examples

Fully nonlinear curvature flow of axially symmetric hypersurfaces with boundary conditions

Inspired by earlier results on the quasilinear mean curvature flow, and recent investigations of fully nonlinear curvature flow of closed hypersurfaces which are not convex, we consider contraction of axially symmetric hypersurfaces by convex, degree-one homogeneous fully nonlinear functions of curvature. With a natural class of Neumann boundary conditions, we show that evolving hypersurfaces exist for a finite maximal time. The maximal time is characterised by a curvature singularity at either boundary. Some results continue to hold in the cases of mixed Neumann-Dirichlet boundary conditions and more general curvature-dependent speeds

Geometric Inequalities of Bi-Warped Product Submanifolds of Nearly Kenmotsu Manifolds and Their Applications

The present paper aims to construct an inequality for bi-warped product submanifolds in a special class of almost metric manifolds, namely nearly Kenmotsu manifolds. As geometric applications, some exceptional cases that generalized several other inequalities are discussed. We also deliberate some applications in the context of mathematical physics and derive a new relation between the Dirichlet energy and the second fundamental form. Finally, we present a constructive remark at the end of this paper which shows the motive of the study

A Study on the Bertrand Offsets of Timelike Ruled Surfaces in Minkowski 3-Space

This work extends some classical results of Bertrand curves to timelike ruled and developable surfaces using the E. Study map. This provides support to define two timelike ruled surfaces which are offset in the sense of Bertrand. It is proved that every timelike ruled surface has a Bertrand offset if and only if an equation should be satisfied among their dual invariants. In addition, some new results and theorems concerning the developability of the Bertrand offsets of timelike ruled surfaces are gained

Interpolation of Surfaces with Asymptotic Curves in Euclidean 3-Space

In this paper, we investigate the interpolation of surfaces which are obtained from an isoasymptotic curve in 3D-Euclidean space. We prove that there exists a unique C0-Hermite surface interpolation related to an isoasymptotic curve under some special conditions on the marching scale functions. Finally, we present some examples and plot their graphs