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# The singular ruled surface transformed into the singular dual spherical curve

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Singular ruled surface is an interesting research object and is the breakthrough point of exploring new problems. However, because of singularity, it’s difficult to study the properties of singular ruled surfaces. In this paper, we combine singularity theory and Clifford algebra to study singular ruled surfaces. We take advantage of the dual number...

## Citations

... There are some research studies about hypersurfaces immersed in different spaces from the viewpoint of singularity theory [11][12][13][14]. Furthermore, some of the latest research about singularity theory and the submanifold theory can be seen in [3,4,[15][16][17][18][19][20][21][22][23][24][25][26][27]. For instance, J. Sun and D. Pei considered the Lorentzian hypersurface on the pseudo n-sphere and classified the singularities of this hypersurface in [11]. ...
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In this paper, we consider Frenet type framed base curves that may have singular points and define one-parameter developable surfaces associated with such curves. By using the singularity theory, we classify the generic singularities of the developable surfaces, which are cuspidal edges and swallowtails. In order to characterize these singularities, two geometric invariants are discovered. At last, an example is given to demonstrate the main results.
... There are some articles concerning singularity theory and submanifolds for several types of geometry. In the next work, we will combine the main results in this paper with the methods and techniques of singularity theory and submanifolds theory, etc., presented in [13][14][15][16][17][18][19][20][21][22] to explore new results and theorems related with more symmetric properties about this topic. ...
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Singularity theory is a significant field of modern mathematical research. The main goal in most problems of singularity theory is to understand the dependence of some objects in analysis and geometry, or physics; or from some other science on parameters. In this paper, we study the singularities of the spherical indicatrix and evolute of space-like ruled surface with space-like ruling. The main method takes advantage of the classical unfolding theorem in singularity theory, which is a classical method to study singularity problems in Euclidean space and Minkowski space. Finally, we provide an example to illustrate our results.
... For future research, we will engage with the new ideas that Gaussian and mean curvatures of these Bertrand offsets can be calculated, when the Weingarten map for the Bertrand offsets spacelike ruled surfaces is defined. We will also consider integrating the study of singularity theory and submanifolds theory and so forth, presented in [23][24][25][26][27][28][29][30][31][32][33][34][35][36][37][38][39][40][41][42], with the results of this paper to explore new methods to find more results and theorems related to symmetric properties about this topic. ...
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This work examines some classical results of Bertrand curves for timelike ruled and developable surfaces using the E. Study map. This provides the ability to define two timelike ruled surfaces which are offset in the sense of Bertrand. It is shown that every timelike ruled surface has a Bertrand offset if and only if an equation should be satisfied between the dual geodesic curvatures. Some new results and theorems related to the developability of the Bertrand offsets of timelike ruled surfaces are also obtained.
... Clifford algebra only depends on a few simple concepts such as number fields, vector space etc., providing a unified, standard, elegant and open language and tool for numerous complicated mathematical and physical theories [2] [3] [4] [5] [6]. In recent years, Clifford algebra has made brilliant achieve-ments in differential geometry, theoretical physics, classical analysis and other aspects, and has been widely used engineering, such as robotics and computer vision [7] [8] [9] [10]. Clifford algebra also has applications in mathematical mechanization such as machine proof of geometric theorem. ...
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The Clifford algebra is a unification and generalization of real number, complex number, quaternion, and vector algebra, which accurately and faithfully characterizes the intrinsic properties of space-time, providing a unified, standard, elegant, and open language and tools for numerous complicated mathematical and physical theories. So it is worth popularizing in the teaching of undergraduate physics and mathematics. Clifford algebras can be directly generalized to 2 n-ary associative algebras. In this generalization, the matrix representation of the orthonormal basis of the space-time plays an important role. The matrix representation carries more information than the abstract definition, such as determinant and the definition of inverse elements. Without this matrix representation, the discussion of hypercomplex numbers will be difficult. The zero norm set of hypercomplex numbers is a closed set of special geometric meanings, like the light-cone in the realistic space-time, has no substantial effect on the algebraic calculus. The physical equations expressed in Clifford algebra have a simple formalism, symmetrical structure, standard derivation, complete content. Therefore, we can hope that this magical algebra can complete a new large synthesis of modern science.
... A dual ruled surface results from the motion of line in the dual space D 3 , similarly to the way a dual curve represents the motion of a dual point.A dual ruled surface is a surface swept out by a dual straight line L with moving direction w(u 1 ) along a dual curve α. Such a surface always has a parameterization in the ruled form (see [11,13,14]) ...
Article
As a continuation to our results in [1], we study the dual ruled surface defined on the set of dual numbers. The idea of the dual part are defined similar to quaternion space. The dual part of this represents a ruled dual submanifold. The geometric properties of their dual parts are investigated. Some examples are given and plotted.
... When a manifold is endowed with a geometric structure, we have more opportunities to explore its geometric properties. There are different classes of submanifolds such as warped product submanifolds, biharmonic submanifolds and singular submanifolds, etc., which motivates further exploration and attracts many researchers from different research areas [26][27][28][29][30][31][32][33][34][35][36][37][40][41][42][43][44][45][46][47][48][49][50]. After A. Bejancu et al. [7] in 1993, introduced the concept of an idefinite manifold namely -Sasakian manifold, it gained attention of various researchers and it was established by X. Xufeng et al. [53] that the class of -Sasakian manifolds are real hypersurfaces of indefinite Kaehlerian manifolds. ...
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The present paper is to deliberate the class of-Kenmotsu manifolds which admits conformal η-Ricci soliton. Here, we study some special types of Ricci tensor in connection with the conformal η-Ricci soliton of-Kenmotsu manifolds. Moving further, we investigate some curvature conditions admitting conformal η-Ricci solitons on-Kenmotsu manifolds. Next, we consider gradient conformal η-Ricci solitons and we present a characterization of the potential function. Finally, we develop an illustrative example for the existence of conformal η-Ricci soliton on-Kenmotsu manifold.
... Theorem 33) to enable our study to uncover the useful applications of the obtained inequality in physics. The work described in this paper will be combined with the singularity theory techniques presented in [20][21][22][23][24]. ...
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In this study, we develop a general inequality for warped product semi-slant submanifolds of type Mn=NTn1×fNϑn2 in a nearly Kaehler manifold and generalized complex space forms using the Gauss equation instead of the Codazzi equation. There are several applications that can be developed from this. It is also described how to classify warped product semi-slant submanifolds that satisfy the equality cases of inequalities (determined using boundary conditions). Several results for connected, compact warped product semi-slant submanifolds of nearly Kaehler manifolds are obtained, and they are derived in the context of the Hamiltonian, Dirichlet energy function, gradient Ricci curvature, and nonzero eigenvalue of the Laplacian of the warping functions.
... e geometric structure and topological properties of submanifolds in different spaces have been studied on a large scale during the past few years [4][5][6][7][8][9][10][11][12][13][14][15][16][17]. Many results showed that there is a closed relationship between stable currents which are nonexistent and the vanished homology groups of submanifolds in a different class of the ambient manifold obtained by imposing conditions on the second fundamental form (1). ...
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In this paper, we show that if the Laplacian and gradient of the warping function of a compact warped product submanifold Ωp+q in the hyperbolic space ℍm−1 satisfy various extrinsic restrictions, then Ωp+q has no stable integral currents, and its homology groups are trivial. Also, we prove that the fundamental group π1Ωp+q is trivial. The restrictions are also extended to the eigenvalues of the warped function, the integral Ricci curvature, and the Hessian tensor. The results obtained in the present paper can be considered as generalizations of the Fu–Xu theorem in the framework of the compact warped product submanifold which has the minimal base manifold in the corresponding ambient manifolds.
... We proved a compact warped product submanifold M n in a Euclidean space E n+k , that there are no stable p-currents, homology groups vanish, and that M 3 is homotopic to the Euclidean sphere S 3 under various extrinsic restrictions, involving the eigenvalue of the warped function, integral Ricci curvature and the Hessian tensor. In our next work, we will combine the singularity theory presented in [31][32][33][34] to study compact warped product submanifold M n in a Euclidean space E n+k . ...
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In this paper, we prove that, for compact warped product submanifolds Mn in an Euclidean space En+k, there are no stable p-currents, homology groups are vanishing, and M3 is homotopic to the Euclidean sphere S3 under various extrinsic restrictions, involving the eigenvalue of the warped function, integral Ricci curvature, and the Hessian tensor. The results in this paper can be considered an extension of Xin’s work in the framework of a compact warped product submanifold, when the base manifold is minimal in ambient manifolds.
... With the deepening of theoretical research, the application of the singularity theory is more and more extensive [13][14][15][16][17][18][19][20][21][22]. Z. Wang [13,20,21] considered the singularity classifications of ruled lightlike surfaces in S 3 1 . ...
... With the deepening of theoretical research, the application of the singularity theory is more and more extensive [13][14][15][16][17][18][19][20][21][22]. Z. Wang [13,20,21] considered the singularity classifications of ruled lightlike surfaces in S 3 1 . However, very little has been researched about the differential geometric properties of the lightlike Killing magnetic curves. ...
Article
In this article, we mainly discuss the local differential geometrical properties of the lightlike Killing magnetic curve $\mathit{\boldsymbol{\gamma }}(s)$ in $\mathbb{S}^{3}_{1}$ with a magnetic field $\boldsymbol{ V}$. Here, a new Frenet frame $\{\mathit{\boldsymbol{\gamma }}, \boldsymbol{ T}, \boldsymbol{ N}, \boldsymbol{ B}\}$ is established, and we obtain the local structure of $\mathit{\boldsymbol{\gamma }}(s)$. Moreover, the singular properties of the binormal lightlike surface of the $\mathit{\boldsymbol{\gamma }}(s)$ are given. Finally, an example is used to understand the main results of the paper.