53 research outputs found
Pseudo-spherical submanifolds with 1-type pseudo-spherical Gauss map
In this work, we study the pseudo-Riemannian submanifolds of a pseudo-sphere
with 1-type pseudo-spherical Gauss map. First, we classify the Lorentzian
surfaces in a 4-dimensional pseudo-sphere with index s,
, and having harmonic pseudo-spherical Gauss map. Then we give a
characterization theorem for pseudo-Riemannian submanifolds of a pseudo-sphere
with 1-type pseudo-spherical Gauss
map, and we classify spacelike surfaces and Lorentzian surfaces in the de
Sitter space with 1-type
pseudo-spherical Gauss map. Finally, according to the causal character of the
mean curvature vector we obtain the classification of submanifolds of a
pseudo-sphere having 1-type pseudo-spherical Gauss map with nonzero constant
component in its spectral decomposition
On spherical submanifolds with finite type spherical Gauss map
Chen and Lue (2007) initiated the study of spherical submanifolds with finite type spherical Gauss map. In this paper, we firstly prove that a submanifold Mn of the unit sphere double-struck Sm-1 has non-mass-symmetric 1-type spherical Gauss map if and only if Mn is an open part of a small n-sphere of a totally geodesic (n + 1)-sphere double-struck Sn+1 ⊂ double-struck Sm-1. Then we show that a non-totally umbilical hypersurface M of double-struck Sn+1 with nonzero constant mean curvature in double-struck Sn+1 has mass-symmetric 2-type spherical Gauss map if and only if the scalar curvature curvature of M is constant. Finally, we classify constant mean curvature surfaces in double-struck S3 with mass-symmetric 2-type spherical Gauss map.Publisher's Versio
On Geometric Applications of Quaternions
Quaternions have become a popular and powerful tool in various engineering fields, such as robotics, image
and signal processing, and computer graphics. However, classical quaternions are mostly used as a representation of
rotation of a vector in 3-dimensions, and connection between its geometric interpretation and algebraic structures is
still not well-developed and needs more improvements. In this study, we develop an approach to understand quaternions
multiplication defining subspaces of quaternion H, called as Plane(N) and Line(N), and then, we observe the effects
of sandwiching maps on the elements of these subspaces. Finally, we give representations of some transformations in
geometry using quaternion
Pseudo-Riemannian Submanifolds of Minkowski Space with Generalized 1-Type Gauss Map
Bu makalede, genelleştirilmiş 1-tipinden Gauss tasvirine sahip Minkowski uzayındaki dönel yüzeyler ve regle alt manifoldları üzerine çalışılmıştır. İlk olarak, ikinci çeşit noktasal 1-tipinden Gauss tasviri ile genelleştirilmiş 1-tipinden Gauss tasviri kavramları arasındaki ilişki verilmiştir. Daha sonra, 3-boyutlu Minkowski uzayında sabit ortalama eğriliğe sahip tümden jeodezik olmayan herhangi bir yüzeyin genelleştirilmiş 1-tipinden Gauss tasvirine sahip olamayacağı ispatlanmıştır. Diğer bölümde, 13 uzayındaki bütün dönel yüzeylerin genelleştirilmiş 1-tipinden Gauss tasvirine sahip olduğu gösterilmiştir. Ayrıca, genelleştirilmiş 1-tipinden Gauss tasvirine sahip dönel yüzeylerle ilgili bir örnek verilmiştir. Son bölümde ise, Minkowski uzayındaki regle alt manifoldları üzerine çalışılmıştır ve genelleştirilmiş 1-tipinden Gauss tasvirine sahip silindirik regle alt manifoldları incelenmiştir.In this article, we study on rotational surfaces and regle submanifolds of the Minkowski space with generalized 1-type Gauss map. First of all, we give a relation between notions of pointwise 1-type Gauss map of the second kind and generalized 1-type Gauss map. Then, we prove that any non-totally geodesic surface in 3-dimensional Minkowski space with constant mean curvature does not have a generalized 1-type Gauss map. In other section, we show that all rotational surfaces in 13 have generalized 1-type Gauss map. Furthermore, we give an example for the rotational surface having generalized 1-type Gauss map. In last section, we study the ruled submanifolds in the Minkowski space and we examine the cylindrical ruled submanifolds having generalized 1-type Gauss map
Timelike Loxodromes on Lorentzian Helicoidal Surfaces in Minkowski N–Space
In this paper, we examine timelike loxodromes on three kinds of
Lorentzian helicoidal surfaces in Minkowski n–space. First, we obtain the first
order ordinary differential equations which determine timelike loxodromes on
the Lorentzian helicoidal surfaces in E
n
1
according to the causal characters of
their meridian curves. Then, by finding general solutions, we get the explicit
parametrizations of such timelike loxodromes. In particular, we investigate the
timelike loxodromes on the three kinds of Lorentzian right helicoidal surfaces
in E
n
1
. Finally, we give an example to visualize the result
Characterizations of Loxodromes on Rotational Surfaces in Euclidean 3–Space
In this paper, we study on the characterizations of loxodromes on the rotational surfaces satisfying
some special geometric properties such as having constant Gaussian curvature and a constant
ratio of principal curvatures (CRPC rotational surfaces). First, we give the parametrizations of
loxodromes parametrized by arc-length parameter on any rotational surfaces in E
3 and then, we
calculate the curvature and the torsion of such loxodromes. Then, we give the parametrizations
of loxodromes on rotational surfaces with constant Gaussian curvature. Also, we investigate
the loxodromes on the CRPC rotational surfaces. Moreover, we give the parametrizations of
loxodromes on the minimal rotational surface which is a special case of CRPC rotational surfaces.
Finally, we give some visual examples to strengthen our main results via Wolfram Mathematica
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