Home Singularities of spherical surface in R4
Article Open Access

Singularities of spherical surface in R4

  • Haiming Liu EMAIL logo , Yuefeng Hua and Wanzhen Li
Published/Copyright: August 10, 2024
Download Article (PDF)
Open Mathematics
From the journal Open Mathematics Volume 22 Issue 1

Abstract

In this article, we mainly study the geometric properties of spherical surface of a curve on a hypersurface Σ in four-dimensional Euclidean space. We define a family of tangent height functions of a curve on Σ as the main tool for research and combine the relevant knowledge of singularity theory. It is shown that there are three types of singularities of spherical surface, that is, in the local sense, the spherical surface is respectively diffeomorphic to the cuspidal edge, the swallowtail, and the cuspidal beaks. In addition, we give two examples of the spherical surface.

Keywords: spherical surface; hypersurface; cuspidal beaks
MSC 2010: 58K05; 53D10; 53B20

1 Introduction

Singularity theory is a subject with strong application, which runs through the fields of differential geometry and differential topology, and is also one of the flourishing fields in modern mathematics. The classification of singular points of curves has always been the focus of research in singularity theory. For surfaces, we can also study their differential geometric properties from the viewpoint of singularity theory. In other words, when studying an unknown surface, we hope to make its local diffeomorphic to a familiar surface, thereby obtaining the properties of the unknown surface.

The application of singularity theory has achieved significant results in different spaces. Most of the research focuses on the classification of singular points of sub-manifolds [1–12]. On the other hand, the study of singularities on hypersurface has also received extensive attention from scholars. In [13], Sun and Pei introduced in detail the geometric property of Lorentzian hypersurfaces on pseudo n-spheres and the one parameter Gauss indicatrices on Lorentzian hypersurfaces. Moreover, they used the Legendrian singularity theory to complete the singularity analysis of the one parameter Gauss indicatrices of Lorentzian hypersurfaces on pseudo n-spheres. In [14], Izumiya et al. classified singularities of lightlike hypersurfaces in Minkowski 4-space. As a generalization of the study on lightlike hypersurface in Minkowski space, Pei et al. studied the singularities of lightlike hypersurface and Lorentzian surface in semi-Euclidean 4-space with index 2 in [15]. The aforementioned research makes the obtained results more systematic, which is what scholars are willing to see. In [16], Izumiya et al. defined the hyperbolic surface and de Sitter surface of a curve in a spacelike hypersurface in Minkowski 4-space and techniques from singularity theory were applied to obtain the generic shape of such surface and their singular value sets. There are also many studies on spherical surfaces. Not only in the field of mathematics but also in fields such as chemistry and physics [17–19]. However, the classification of singular points on spherical surfaces has not been resolved yet. This is also our main research motivation.

Inspired by the aforementioned research, we chose the four-dimensional Euclidean space R 4 as the outer space. Then we considered a embedding Π : U → R 4 , from an open subset U ⊂ R 3 and identify hypersurface Σ and U through the embedding Π . For a curve whose curvature does not disappear γ : I → Σ , we defined a spherical surface in S 3 associated with curve γ . We used the classical deformation theory of singularity theory to study the generic differential geometry of spherical surfaces and their singular sets. The conclusion reached is that spherical surface is respectively diffeomorphic to the cuspidal edge, the swallowtail, and the cuspidal beaks.

The article is organized as follows: In Section 2, we introduce the definition of A k -singularities and discriminant sets. Moreover, we build a moving frame along γ and calculate the Frenet-Serret type formulae. In Section 3, we define the tangential height functions that measure the contact of curve t with special hyperplanes and whose differentiation yields invariants related to each surface. In Section 4, the spherical surface of γ is described as the discriminant set of the family of tangential height functions. By using the theory of deformations, we get a classification and a characterisation of the diffeomorphim type of such surfaces. Finally, we provide some examples of spherical surfaces in Section 5.

2 Preliminaries

The four-dimensional Euclidean space is

R 4 = { ( a 0 , a 1 , a 2 , a 3 ) ∣ a i ∈ R ( i = 0 , 1 , 2 , 3 ) }

with scalar product

⟨ a , b ⟩ = a 0 b 0 + a 1 b 1 + a 2 b 2 + a 3 b 3 ,

for any vectors a = ( a 0 , a 1 , a 2 , a 3 ) , and b = ( b 0 , b 1 , b 2 , b 3 ) in R 4 . We define the vector product of a , b and z = ( z 0 , z 1 , z 2 , z 3 ) as follows:

a ∧ b ∧ z = e 0 e 1 e 2 e 3 a 0 a 1 a 2 a 3 b 0 b 1 b 2 b 3 z 0 z 1 z 2 z 3 ,

where a , b , z ∈ R 4 , and { e 0 , e 1 , e 2 , e 3 } is the canonical basis of R 4 , e 0 = ( 1 , 0 , 0 , 0 ) . The norm of a nonzero vector a ∈ R 4 is defined by ‖ a ‖ = ⟨ a , a ⟩ , and when ‖ a ‖ = 1 , we call a a unit vector.

For a non-zero vector v ∈ R 4 and a real number c , we define a hyperplane with pseudo-normal v by

HP ( v , c ) = { a ∈ R 4 ∣ ⟨ a , v ⟩ = c } .

The sphere in R 4 is defined by

S 3 = { a ∈ R 4 ∣ ⟨ a , a ⟩ = 1 } .

We consider an embedding Π : U → R 4 , where U is an open subset in R 3 . We write Σ = Π ( U ) and identify Σ and U through the embedding Π . Let γ ¯ : I → U be a regular curve. Then we have a curve γ : I → Σ ⊂ R 4 defined by γ ( s ) = Π ( γ ¯ ( s ) ) . We say that γ is a curve in the hypersurface Σ . To facilitate calculation, we reparametrize γ by the arc length s . So we have the unit tangent vector t ( s ) = γ ˙ ( s ) with ‖ t ( s ) ‖ = 1 . In this case, we call γ a unit speed curve. Then, we have a unit normal vector field n along Σ = Π ( U ) defined by

n ( p ) = Π u 1 ( u ) ∧ Π u 2 ( u ) ∧ Π u 3 ( u ) ‖ Π u 1 ( u ) ∧ Π u 2 ( u ) ∧ Π u 3 ( u ) ‖ ,

for p = Π ( u ) , where Π u i = ∂ Π ∕ ∂ u i , i = 1 , 2 , 3 . A unit normal vector field n γ along γ is defined by n γ ( s ) = n ∘ γ ( s ) .

Under the assumption that ‖ t ′ ( s ) − ⟨ t ′ ( s ) , n γ ( s ) ⟩ n γ ( s ) ‖ ≠ 0 , we can construct

n 1 ( s ) = t ′ ( s ) − ⟨ t ′ ( s ) , n γ ( s ) ⟩ n γ ( s ) ‖ t ′ ( s ) − ⟨ t ′ ( s ) , n γ ( s ) ⟩ n γ ( s ) ‖ .

It follows that ⟨ t , n 1 ⟩ = 0 and ⟨ n γ , n 1 ⟩ = 0 . Moreover, we have a unit vector defined by n 2 ( s ) = t ( s ) ∧ n γ ( s ) ∧ n 1 ( s ) . Then, we have a orthonormal frame { t ( s ) , n γ ( s ) , n 1 ( s ) , n 2 ( s ) } . By standard arguments, we have the Frenet-Serret type formulae for the aforementioned frame as follows:

t ′ ( s ) = k n ( s ) n γ ( s ) + k g ( s ) n 1 ( s ) n γ ′ ( s ) = − k n ( s ) t ( s ) + τ 1 ( s ) n 1 ( s ) + τ 2 ( s ) n 2 ( s ) n 1 ′ ( s ) = − τ 1 ( s ) n γ ( s ) − k g ( s ) t ( s ) + τ g ( s ) n 2 ( s ) n 2 ′ ( s ) = − τ 2 ( s ) n γ ( s ) − τ g ( s ) n 1 ( s ) ,

where k n ( s ) = ⟨ n γ ( s ) , t ′ ( s ) ⟩ , τ 1 ( s ) = ⟨ n γ ′ ( s ) , n 1 ( s ) ⟩ , τ 2 ( s ) = ⟨ n γ ′ ( s ) , n 2 ( s ) ⟩ , k g ( s ) = ‖ t ′ ( s ) − ⟨ t ′ ( s ) , n γ ( s ) ⟩ n γ ( s ) ‖ = ‖ t ′ ( s ) − k n ( s ) n γ ( s ) ‖ , and τ g ( s ) = ⟨ n 1 ′ ( s ) , n 2 ( s ) ⟩ . The invariant k n is called a normal curvature, τ 1 is a first normal torsion, τ 2 is a second normal torsion, k g is a geodesic curvature, and τ g is a geodesic torsion. Under the assumption k g ( s ) = ‖ t ′ ( s ) − ⟨ t ′ ( s ) , n γ ( s ) ⟩ n γ ( s ) ‖ ≠ 0 , we have k g > 0 .

Definition 2.1

Let X : R 4 → R be a submersion and γ : I → Σ be a regular curve. We say that γ and X − 1 ( 0 ) have contact of order k at s 0 , if the function g ( s ) = X ∘ γ ( s ) satisfies g ( s 0 ) = g ′ ( s 0 ) = … = g ( k ) ( s 0 ) = 0 and g ( k + 1 ) ( s 0 ) ≠ 0 , i.e., g has an A k -singularity at s 0 .

Let G : R × R r , ( s 0 , x 0 ) → R be a family of germs of functions. We call G an r -parameter deformation of f if f ( s ) = G x 0 ( s ) . We assume that f has an A k -singularity ( k ≥ 1 ) at s 0 , we can write

j k − 1 ∂ G ∂ x i ( s , x 0 ) ( s 0 ) = ∑ j = 0 k − 1 a j i ( s − s 0 ) j ,

for i = 1 , … , r . Then G is a versal deformation if the k × r matrix of coefficients ( α j i ) has rank k ( k ≤ r ) .

The discriminant set of G is given by

D G = x ∈ ( R r , x 0 ) ∣ G = ∂ G ∂ s = 0 at ( s , x ) for some s ∈ ( R , s 0 ) ,

and the bifurcation set of G is given by

ℬ G = x ∈ ( R r , x 0 ) ∣ ∂ G ∂ s = ∂ 2 G ∂ s 2 = 0 at ( s , x ) for some s ∈ ( R , s 0 ) .

The next result is from [20].

Theorem 2.2

Let G : R × R r , ( s 0 , x 0 ) → R be an r -parameter deformation of f such that f has an A k -singularity at s 0 . We assume that G is a versal deformation, then D G is locally diffeomorphic to

  1. C × R r − 2 if k = 2 ,

  2. SW × R r − 3 if k = 3 ,

where C × R = { ( x 1 , x 2 , x 3 ) ∣ x 1 2 = x 2 3 } × R is the cuspidal edge and SW = { ( x 1 , x 2 , x 3 ) ∣ x 1 = 3 u 4 + u 2 v , x 2 = 4 u 3 + 2 u v , x 3 = v } is the swallowtail surface (Figure 1).

Figure 1 Cuspidal edge (left) and swallowtail (right).
Figure 1

Cuspidal edge (left) and swallowtail (right).

3 Tangential height functions

In this section, we define a family of functions on a curve in a hypersurface Σ .

Let γ : I → Σ ⊂ R 4 , we give the following definitions:

H : I × S 3 → R ; ( s , v ) ↦ ⟨ t ( s ) , v ⟩ .

The functions H are called a family of tangential height functions of γ . The meaning of H is that it measures the contact of the curve t with hyperplanes in R 4 . Generically, this contact can be of order k , k = 1 , 2 , 3 . For any fixed v ∈ S 3 , we denote h v ( s ) = H ( s , v ) .

In the following proposition, we find the conditions for characterizing the A k -singularity, k = 1 , 2 , 3 .

Proposition 3.1

Let γ : I → Σ be a unit speed curve, we assume that k n ≠ 0 , k g ≠ 0 , so k g > 0 , and ( k n τ 2 + k g τ g ) ( s ) ≠ 0 . Thus, we obtain the following:

  1. h v ( s ) = 0 if and only if there exists δ , ξ , φ ∈ R such that δ 2 + ξ 2 + φ 2 = 1 and

    v = δ n γ ( s ) + ξ n 1 ( s ) + φ n 2 ( s ) .

  2. h v ( s ) = h v ′ ( s ) = 0 if and only if there exists θ ∈ R such that

    v = cos θ k g 2 ( s ) + k n 2 ( s ) ( − k g ( s ) n γ ( s ) + k n ( s ) n 1 ( s ) ) + sin θ n 2 ( s ) .

  3. h v ( s ) = h v ′ ( s ) = h v ″ ( s ) = 0 if and only if

    v = cos θ k g 2 ( s ) + k n 2 ( s ) ( − k g ( s ) n γ ( s ) + k n ( s ) n 1 ( s ) ) + sin θ n 2 ( s )

    and tan θ = k n ′ k g − k g 2 τ 1 − k g ′ k n − k n 2 τ 1 k g 2 + k n 2 ( k n τ 2 + k g τ g ) ( s ) .

  4. h v ( s ) = h v ′ ( s ) = h v ″ ( s ) = h v ‴ ( s ) = 0 if and only if

    v = cos θ k g 2 ( s ) + k n 2 ( s ) ( − k g ( s ) n γ ( s ) + k n ( s ) n 1 ( s ) ) + sin θ n 2 ( s ) ,

    tan θ = k n ′ k g − k g 2 τ 1 − k g ′ k n − k n 2 τ 1 k g 2 + k n 2 ( k n τ 2 + k g τ g ) ( s ) and χ ( s ) = 0 , where

    χ ( s ) = ( ( − k n ″ k g + 2 k g k g ′ τ 1 + k g 2 τ 1 ′ + k g k n τ 2 2 + k g 2 τ g τ 2 + 2 k n k n ′ τ 1 + k n 2 τ 1 ′ + k n k g ″ − k n 2 τ 2 τ g − k n k g τ g 2 ) ( k n τ 2 + k g τ g ) + ( 2 k n ′ τ 2 − k g τ 2 τ 1 + k n τ 1 τ g + 2 k g ′ τ g + k n τ 2 ′ + k g τ g ′ ) ( k n ′ k g − k g 2 τ 1 − k n 2 τ 1 − k n k g ′ ) ) ( s ) .

  5. h v ( s ) = h v ′ ( s ) = h v ″ ( s ) = h v ‴ ( s ) = h v ( 4 ) ( s ) = 0 if and only if

    v = cos θ k g 2 ( s ) + k n 2 ( s ) ( − k g ( s ) n γ ( s ) + k n ( s ) n 1 ( s ) ) + sin θ n 2 ( s ) ,

    tan θ = k n ′ k g − k g 2 τ 1 − k g ′ k n − k n 2 τ 1 k g 2 + k n 2 ( k n τ 2 + k g τ g ) ( s ) and χ ( s ) = χ ′ ( s ) = 0 .

Proof

(1) We can know v ∈ S 3 , there are λ , δ , ξ , φ ∈ R with λ 2 + δ 2 + ξ 2 + φ 2 = 1 such that v = λ t ( s ) + δ n γ ( s ) + ξ n 1 ( s ) + φ n 2 ( s ) . According to h v ( s ) = ⟨ t ( s ) , v ⟩ = 0 , we have λ = 0 . So δ 2 + ξ 2 + φ 2 = 1 and v = δ n γ ( s ) + ξ n 1 ( s ) + φ n 2 ( s ) . Thus, (1) holds.

(2) Because h v ( s ) = h v ′ ( s ) = 0 , so we have

⟨ t ( s ) , v ⟩ = ⟨ t ′ ( s ) , v ⟩ = ⟨ k n ( s ) n γ ( s ) + k g ( s ) n 1 ( s ) , δ n γ ( s ) + ξ n 1 ( s ) + φ n 2 ( s ) ⟩ = δ k n ( s ) + ξ k g ( s ) = 0 .

we calculate that δ = − ξ k g ( s ) k n ( s ) , and k g 2 ( s ) + k n 2 ( s ) k n 2 ( s ) ξ 2 + φ 2 = 1 . It follows that δ = − cos θ k g ( s ) k g 2 ( s ) + k n 2 ( s ) , ξ = cos θ k n ( s ) k g 2 ( s ) + k n 2 ( s ) , φ = sin θ . Thus, (2) holds.

(3) When h v ( s ) = h v ′ ( s ) = h v ″ ( s ) = 0 , we have ⟨ t ( s ) , v ⟩ = ⟨ t ′ ( s ) , v ⟩ = ⟨ t ″ ( s ) , v ⟩ = 0 . Where v = cos θ k g 2 ( s ) + k n 2 ( s ) ( − k g ( s ) n γ ( s ) + k n ( s ) n 1 ( s ) ) + sin θ n 2 ( s ) , and we calculate that

t ″ ( s ) = ( − k n 2 ( s ) − k g 2 ( s ) ) t ( s ) + ( k n ′ ( s ) − k g ( s ) τ 1 ( s ) ) n γ ( s ) + ( k n ( s ) τ 1 ( s ) + k g ′ ( s ) ) n 1 ( s ) + ( k n ( s ) τ 2 ( s ) + k g ( s ) τ g ( s ) ) n 2 ( s ) .

Hence, there exists θ such that tan θ = k n ′ k g − k g 2 τ 1 − k g ′ k n − k n 2 τ 1 k g 2 + k n 2 ( k n τ 2 + k g τ g ) ( s ) . Thus, (3) holds.

(4) Based on (3) and h v ‴ ( s ) = 0 , we have ⟨ t ‴ ( s ) , v ⟩ = 0 . Then we calculate that

t ‴ ( s ) = ( − 3 k n ( s ) k n ′ ( s ) − 3 k g ( s ) k g ′ ( s ) ) t ( s ) + ( − k n 3 ( s ) − k g 2 ( s ) k n ( s ) + k n ″ ( s ) − 2 k g ′ ( s ) τ 1 ( s ) − k g ( s ) τ 1 ′ ( s ) − k n ( s ) τ 1 2 ( s ) − k n ( s ) τ 2 2 ( s ) − k g ( s ) τ g ( s ) τ 2 ( s ) ) n γ ( s ) + ( − k g 3 ( s ) − k n 2 ( s ) k g ( s ) + k g ″ ( s ) + 2 k n ′ ( s ) τ 1 ( s ) + k n ( s ) τ 1 ′ ( s ) − k g ( s ) τ 1 2 ( s ) − k g ( s ) τ g 2 ( s ) − k n ( s ) τ g ( s ) τ 2 ( s ) ) n 1 ( s ) + ( 2 k n ′ ( s ) τ 2 ( s ) − k g ( s ) τ 1 ( s ) τ 2 ( s ) + k n ( s ) τ g ( s ) τ 1 ( s ) + 2 k g ′ ( s ) τ g ( s ) + k n ( s ) τ 2 ′ ( s ) + k g ( s ) τ g ′ ( s ) ) n 2 ( s ) .

Thus, h v ( s ) = h v ′ ( s ) = h v ″ ( s ) = h v ‴ ( s ) = 0 if and only if

v = cos θ k g 2 ( s ) + k n 2 ( s ) ( − k g ( s ) n γ ( s ) + k n ( s ) n 1 ( s ) ) + sin θ n 2 ( s ) ,

tan θ = k n ′ k g − k g 2 τ 1 − k g ′ k n − k n 2 τ 1 k g 2 + k n 2 ( k n τ 2 + k g τ g ) ( s ) ,

and

χ ( s ) = ( ( − k n ″ k g + 2 k g k g ′ τ 1 + k g 2 τ 1 ′ + k g k n τ 2 2 + k g 2 τ g τ 2 + 2 k n k n ′ τ 1 + k n 2 τ 1 ′ + k n k g ″ − k n 2 τ 2 τ g − k n k g τ g 2 ) ( k n τ 2 + k g τ g ) + ( 2 k n ′ τ 2 − k g τ 2 τ 1 + k n τ 1 τ g + 2 k g ′ τ g + k n τ 2 ′ + k g τ g ′ ) ( k n ′ k g − k g 2 τ 1 − k n 2 τ 1 − k n k g ′ ) ) ( s ) = 0 .

(5) Based on (4), we know χ ( s ) = ⟨ t ‴ ( s ) , v ⟩ = 0 , so h v ( 4 ) ( s ) = ⟨ t ( 4 ) ( s ) , v ⟩ = χ ′ ( s ) = 0 . Thus, (5) holds. Proof completed.□

In the following proposition, we find that the family of tangential height functions on a curve in Σ is a versal deformation of an A k -singularity, k = 2 , 3 .

Proposition 3.2

Let γ : I → Σ be a unit speed curve with k g ≠ 0 , so k g > 0 , and ( k n τ 2 + k g τ g ) ( s ) ≠ 0 . Thus, we have

  1. If h v 0 has an A 2 -singularity at s 0 , then H is a versal deformation of h v 0 .

  2. If h v 0 has an A 3 -singularity at s 0 , then H is a versal deformation of h v 0 .

Proof

The family of tangential height functions is given by

H ( s , v ) = ⟨ t ( s ) , v ⟩ = v 0 x 0 ′ ( s ) + v 1 x 1 ′ ( s ) + v 2 x 2 ′ ( s ) + v 3 x 3 ′ ( s ) ,

where v = ( v 0 , v 1 , v 2 , v 3 ) , t ( s ) = ( x 0 ′ ( s ) , x 1 ′ ( s ) , x 2 ′ ( s ) , x 3 ′ ( s ) ) , and v 1 = 1 − v 0 2 − v 2 2 − v 3 2 . In order not to lose generality, we assume that v 1 ≠ 0 . So we have

∂ H ∂ v 0 ( s , v ) = x 0 ′ ( s ) − v 0 v 1 x 1 ′ ( s ) , ∂ 2 H ∂ s ∂ v 0 ( s , v ) = x 0 ″ ( s ) − v 0 v 1 x 1 ″ ( s ) , ∂ 3 H ∂ 2 s ∂ v 0 ( s , v ) = x 0 ‴ ( s ) − v 0 v 1 x 1 ‴ ( s ) , ∂ H ∂ v i ( s , v ) = x i ′ ( s ) − v i v 1 x 1 ′ ( s ) , ∂ 2 H ∂ s ∂ ν i ( s , v ) = x i ″ ( s ) − v i v 1 x 1 ″ ( s ) , ∂ 3 H ∂ 2 s ∂ v i ( s , v ) = x i ‴ ( s ) − v i v 1 x 1 ‴ ( s ) , ( i = 2 , 3 ) .

Therefore, the 1-jet of ∂ H ∂ v i ( s , v ) at s 0 is given by

x i ′ ( s 0 ) − v i v 1 x 1 ′ ( s 0 ) + ( x i ″ ( s 0 ) − v i v 1 x 1 ″ ( s 0 ) ) ( s − s 0 ) ,

and the 2-jet of ∂ H ∂ v i ( s , v ) at s 0 is given by

x i ′ ( s 0 ) − v i v 1 x 1 ′ ( s 0 ) + ( x i ″ ( s 0 ) − v i v 1 x 1 ″ ( s 0 ) ) ( s − s 0 ) + 1 2 ( x i ‴ ( s 0 ) − v i v 1 x 1 ‴ ( s 0 ) ) ( s − s 0 ) 2 ,

where i = 0 , 2 , 3 .

(1) If h v has an A 2 -singularity at s = s 0 . Let us consider the following matrix:

B = x 0 ′ ( s 0 ) − v 0 v 1 x 1 ′ ( s 0 ) x 2 ′ ( s 0 ) − v 2 v 1 x 1 ′ ( s 0 ) x 3 ′ ( s 0 ) − v 3 v 1 x 1 ′ ( s 0 ) x 0 ″ ( s 0 ) − v 0 v 1 x 1 ″ ( s 0 ) x 2 ″ ( s 0 ) − v 2 v 1 x 1 ″ ( s 0 ) x 3 ″ ( s 0 ) − v 3 v 1 x 1 ″ ( s 0 ) .

We calculate the Gram-Schmidt matrix of B ˜ = v 1 B . We denote the lines of B ˜ by

F = ( x 0 ′ ( s 0 ) v 1 − x 1 ′ ( s 0 ) v 0 , x 2 ′ ( s 0 ) v 1 − x 1 ′ ( s 0 ) v 2 , x 3 ′ ( s 0 ) v 1 − x 1 ′ ( s 0 ) v 3 ) ,

G = ( x 0 ″ ( s 0 ) v 1 − x 1 ″ ( s 0 ) v 0 , x 2 ″ ( s 0 ) v 1 − x 1 ″ ( s 0 ) v 2 , x 3 ″ ( s 0 ) v 1 − x 1 ″ ( s 0 ) v 3 ) .

Since ⟨ v , v ⟩ = 1 , ⟨ t ( s ) , t ( s ) ⟩ = 1 , ⟨ t ( s ) , v ⟩ = 0 , ⟨ t ′ ( s ) , v ⟩ = 0 , and ⟨ t ′ ( s ) , t ′ ( s ) ⟩ = ⟨ ( k n n γ + k g n 1 ) ( s ) , ( k n n γ + k g n 1 ) ( s ) ⟩ = k g 2 ( s ) + k n 2 ( s ) , we have the following Euclidean inner product

F ⋅ F = v 1 2 + ( x 1 ′ ) 2 , F ⋅ G = x 1 ′ x 1 ″ , G ⋅ G = ( k g 2 ( s ) + k n 2 ( s ) ) v 1 2 + ( x 1 ″ ) 2 .

Therefore, the Gram-Schmidt matrix of B ˜ is given by

G B ˜ = v 1 2 + ( x 1 ′ ) 2 x 1 ′ x 1 ″ x 1 ′ x 1 ″ v 1 2 ( k g 2 ( s ) + k n 2 ( s ) ) + ( x 1 ″ ) 2 .

We assume that n γ ( s 0 ) = ( 0 , 1 , 0 , 0 ) . In this case, we have x 1 ′ ( s 0 ) = 0 , x 1 ″ ( s 0 ) = k n ( s 0 ) and v 1 = − k g ( s 0 ) cos θ 0 k g 2 ( s ) + k n 2 ( s ) . Thus, the determinant of G B ˜ is

( ( x 1 ′ ) 2 + v 1 2 ) [ ( k g 2 ( s 0 ) + k n 2 ( s 0 ) ) v 1 2 + ( x 1 ″ ) 2 ] − ( x 1 ′ x 1 ″ ) 2 = k g 2 ( s 0 ) cos 2 θ 0 k g 2 ( s 0 ) + k n 2 ( s 0 ) ( k g 2 ( s 0 ) cos 2 θ 0 + k n 2 ( s 0 ) )

that is different from zero. Thus, the rank of the matrix B is equal to two and so the assertion (1) follows.

(2) We now assume that h v has an A 3 -singularity at s = s 0 . In this case, we show that the determinant of the matrix

A = x 0 ′ ( s 0 ) − v 0 v 1 x 1 ′ ( s 0 ) x 2 ′ ( s 0 ) − v 2 v 1 x 1 ′ ( s 0 ) x 3 ′ ( s 0 ) − v 3 v 1 x 1 ′ ( s 0 ) x 0 ″ ( s 0 ) − v 0 v 1 x 1 ″ ( s 0 ) x 2 ″ ( s 0 ) − v 2 v 1 x 1 ″ ( s 0 ) x 3 ″ ( s 0 ) − v 3 v 1 x 1 ″ ( s 0 ) x 0 ‴ ( s 0 ) − v 0 v 1 x 1 ‴ ( s 0 ) x 2 ‴ ( s 0 ) − v 2 v 1 x 1 ‴ ( s 0 ) x 3 ‴ ( s 0 ) − v 3 v 1 x 1 ‴ ( s 0 )

is nonzero. Denote

a i = x i ′ ( s 0 ) x i ″ ( s 0 ) x i ‴ ( s 0 ) ( i = 0 , 1 , 2 , 3 ) .

By a simple calculation, we have

det A = − v 0 v 1 det ( a 1 , a 2 , a 3 ) + v 1 v 1 det ( a 0 , a 2 , a 3 ) − v 2 v 1 det ( a 0 , a 1 , a 3 ) + v 3 v 1 det ( a 0 , a 1 , a 2 ) .

On the other hand,

γ ′ ( s 0 ) ∧ γ ″ ( s 0 ) ∧ γ ‴ ( s 0 ) = ( det ( a 1 , a 2 , a 3 ) , − det ( a 0 , a 2 , a 3 ) , det ( a 0 , a 1 , a 3 ) , − det ( a 0 , a 1 , a 2 ) ) .

Therefore, det A = − v 0 v 1 , v 1 v 1 , v 2 v 1 , v 3 v 1 , ( γ ′ ∧ γ ″ ∧ γ ‴ ) ( s 0 ) . We calculate that

γ ′ ( s 0 ) ∧ γ ″ ( s 0 ) ∧ γ ‴ ( s 0 ) = − k g ( k n τ 2 + k g τ g ) n γ + k n ( k n τ 2 + k g τ g ) n 1 + ( k g ( k n ′ − k g τ 1 ) + k n ( k n τ 1 + k g ′ ) ) n 2 ( s 0 ) .

If h v has an A 3 -singularity at s = s 0 , thus, using v = cos θ k g 2 ( s ) + k n 2 ( s ) ( − k g ( s ) n γ ( s ) + k n ( s ) n 1 ( s ) ) + sin θ n 2 ( s ) , and tan θ = k n ′ k g − k g 2 τ 1 − k g ′ k n − k n 2 τ 1 k g 2 + k n 2 ( k n τ 2 + k g τ g ) ( s ) , we have

det A = − v 0 v 1 , v 1 v 1 , v 2 v 1 , v 3 v 1 , ( γ ′ ∧ γ ″ ∧ γ ‴ ) ( s 0 ) = − − k g cos θ 0 k g 2 + k n 2 n γ + k n cos θ 0 k g 2 + k n 2 n 1 + sin θ 0 n 2 , − k g ( k n τ 2 + k g τ g ) n γ + k n ( k n τ 2 + k g τ g ) n 1 + ( k g ( k n ′ − k g τ 1 ) + k n ( k n τ 1 + k g ′ ) ) n 2 ⟩ ( s 0 ) = k g 2 + k n 2 k g cos θ 0 ( s 0 ) .

Therefore, if h v has an A 3 -singularity at s 0 , then det A ≠ 0 and H is a versal deformation of h v 0 . This completes the proof.□

We now define a deformation H ˜ : I × S 3 × R → R by H ˜ ( s , v , u ) = H ( s , v ) + u ( s − s 0 ) 2 = ⟨ t ( s ) , v ⟩ + u ( s − s 0 ) 2 . The germ at ( s 0 , v 0 , 0 ) represented by H ˜ is considered.

Proposition 3.3

If h v 0 has an A 3 -singularity at s 0 , then H ˜ is a versal deformation of h v 0 .

Proof

We have

H ˜ ( s , v , u ) = H ( s , v ) + u ( s − s 0 ) 2 = v 0 x 0 ′ + v 1 x 1 ′ + v 2 x 2 ′ + v 3 x 3 ′ + u ( s − s 0 ) 2 ,

where v = ( v 0 , v 1 , v 2 , v 3 ) , t ( s ) = ( x 0 ′ ( s ) , x 1 ′ ( s ) , x 2 ′ ( s ) , x 3 ′ ( s ) ) and v 1 = 1 − v 0 2 − v 2 2 − v 3 2 . Thus,

∂ H ˜ ∂ v i ( s , v , 0 ) = x i ′ ( s ) − v i v 1 x 1 ′ ( s ) ,

for i = 0 , 2 , 3 . The 2-jet of ∂ H ˜ ∂ v i ( s , v , 0 ) at s 0 is given by

x i ′ ( s 0 ) − v i v 1 x 1 ′ ( s 0 ) + ( x i ″ ( s 0 ) − v i v 1 x 1 ″ ( s 0 ) ) ( s − s 0 ) + 1 2 ( x i ‴ ( s 0 ) − v i v 1 x 1 ‴ ( s 0 ) ) ( s − s 0 ) 2 ,

and the 2-jet of ∂ H ˜ ∂ u ( s , v , 0 ) at s 0 is ( s − s 0 ) 2 . We assume that h v has an A 3 -singularity at s = s 0 . Then we can show that

rank x 0 ′ ( s 0 ) − v 0 v 1 x 1 ′ ( s 0 ) x 2 ′ ( s 0 ) − v 2 v 1 x 1 ′ ( s 0 ) x 3 ′ ( s 0 ) − v 3 v 1 x 1 ′ ( s 0 ) 0 x 0 ″ ( s 0 ) − v 0 v 1 x 1 ″ ( s 0 ) x 2 ″ ( s 0 ) − v 2 v 1 x 1 ″ ( s 0 ) x 3 ″ ( s 0 ) − v 3 v 1 x 1 ″ ( s 0 ) 0 x 0 ‴ ( s 0 ) − v 0 v 1 x 1 ‴ ( s 0 ) x 2 ‴ ( s 0 ) − v 2 v 1 x 1 ‴ ( s 0 ) x 3 ‴ ( s 0 ) − v 3 v 1 x 1 ‴ ( s 0 ) 1 = rank 0 0 1 x 0 ′ ( s 0 ) − v 0 v 1 x 1 ′ ( s 0 ) x 0 ″ ( s 0 ) − v 0 v 1 x 1 ″ ( s 0 ) 0 x 2 ′ ( s 0 ) − v 2 v 1 x 1 ′ ( s 0 ) x 2 ″ ( s 0 ) − v 2 v 1 x 1 ″ ( s 0 ) 0 x 3 ′ ( s 0 ) − v 3 v 1 x 1 ′ ( s 0 ) x 3 ″ ( s 0 ) − v 3 v 1 x 1 ″ ( s 0 ) 0 = 3 .

The rank of the last matrix has the same value as the rank of

1 0 1 x 0 ′ ( s 0 ) − v 0 v 1 x 1 ′ ( s 0 ) x 0 ″ ( s 0 ) − v 0 v 1 x 1 ″ ( s 0 ) 0 x 2 ′ ( s 0 ) − v 2 v 1 x 1 ′ ( s 0 ) x 2 ″ ( s 0 ) − v 2 v 1 x 1 ″ ( s 0 ) 0 x 3 ′ ( s 0 ) − v 3 v 1 x 1 ′ ( s 0 ) x 3 ″ ( s 0 ) − v 3 v 1 x 1 ″ ( s 0 ) 0 .

Consider

l 1 ( s 0 ) = 1 , x 0 ′ ( s 0 ) − v 0 v 1 x 1 ′ ( s 0 ) , x 2 ′ ( s 0 ) − v 2 v 1 x 1 ′ ( s 0 ) , x 3 ′ ( s 0 ) − v 3 v 1 x 1 ′ ( s 0 ) , l 2 ( s 0 ) = 0 , x 0 ″ ( s 0 ) − v 0 v 1 x 1 ″ ( s 0 ) , x 2 ″ ( s 0 ) − v 2 v 1 x 1 ″ ( s 0 ) , x 3 ″ ( s 0 ) − v 3 v 1 x 1 ″ ( s 0 ) ,

and l 3 ( s 0 ) = ( 1 , 0 , 0 , 0 ) . It is enough to show that l 1 ( s 0 ) , l 2 ( s 0 ) , and l 3 ( s 0 ) are linearly independent. Because, if l 1 ( s 0 ) , l 2 ( s 0 ) , l 3 ( s 0 ) are linearly dependent, then we have x 0 ′ ( s 0 ) = v 0 v 1 x 1 ′ ( s 0 ) , x 2 ′ ( s 0 ) = v 2 v 1 x 1 ′ ( s 0 ) , and x 3 ′ ( s 0 ) = v 3 v 1 x 1 ′ ( s 0 ) . That is, t ( s 0 ) and v are parallel, and so we have a contradiction because ⟨ t ( s 0 ) , v ⟩ = 0 .□

4 Spherical surface

In this section, we give the definition of a spherical surface. In addition, we study the classification of singular points on spherical surfaces.

Let γ : I → Σ be a unit speed curve with k g ( s ) ≠ 0 and ( k n τ 2 + k g τ g ) ≠ 0 , and a surface S γ : I × J → S 3 is given by

S γ ( s , θ ) = cos θ k g 2 ( s ) + k n 2 ( s ) ( − k g ( s ) n γ ( s ) + k n ( s ) n 1 ( s ) ) + sin θ n 2 ( s ) ,

where J = [ 0 , 2 π ] . We call S γ a spherical surface of γ .

Corollary 4.1

The spherical surface of γ is the discriminant set D H of the family of tangential height functions H.

The cuspidal beaks are defined to be a germ of surface diffeomorphic to CBK = { ( x 1 , x 2 , x 3 ) ∣ x 1 = v , x 2 = − 2 u 3 + v 2 u , x 3 = 3 u 4 − v 2 u 2 } . The cuspidal lips are defined to be a germ of surface diffeomorphic to C L K = { ( x 1 , x 2 , x 3 ) ∣ x 1 = v , x 2 = 2 u 3 + v 2 u , x 3 = 3 u 4 + v 2 u 2 } (Figure 2).

Figure 2 Cuspidal beaks (left) and cuspidal lips (right).
Figure 2

Cuspidal beaks (left) and cuspidal lips (right).

By using Theorem 2.2 and Propositions 3.2 and 3.3, we can obtain the diffeomorphism type of the spherical surface in the following theorem.

Theorem 4.2

Let γ : I → Σ be a unit speed curve with k g ≠ 0 and ( k n τ 2 + k g τ g ) ( s ) ≠ 0 , and S γ is the spherical surface of γ . We obtain the following:

  1. S γ is singular at ( s 0 , θ 0 ) if and only if

    tan θ 0 = k n ′ k g − k g 2 τ 1 − k g ′ k n − k n 2 τ 1 k g 2 + k n 2 ( k n τ 2 + k g τ g ) ( s 0 ) .

  2. The germ of S γ at ( s 0 , θ 0 ) is diffeomorphic to a cuspidal edge if

    tan θ 0 = k n ′ k g − k g 2 τ 1 − k g ′ k n − k n 2 τ 1 k g 2 + k n 2 ( k n τ 2 + k g τ g ) ( s 0 ) a n d χ ( s 0 ) ≠ 0 .

  3. The germ of S γ at ( s 0 , θ 0 ) is diffeomorphic to a swallowtail if

    tan θ 0 = k n ′ k g − k g 2 τ 1 − k g ′ k n − k n 2 τ 1 k g 2 + k n 2 ( k n τ 2 + k g τ g ) ( s 0 ) , χ ( s 0 ) = 0 a n d χ ′ ( s 0 ) ≠ 0 .

  4. The germ of S γ at ( s 0 , θ 0 ) is diffeomorphic to a cuspidal beaks if

    tan θ 0 = k n ′ k g − k g 2 τ 1 − k g ′ k n − k n 2 τ 1 k g 2 + k n 2 ( k n τ 2 + k g τ g ) ( s 0 ) , λ 1 ≠ 0 , χ ( s 0 ) = 0 a n d χ ′ ( s 0 ) ≠ 0 ,

    where λ 1 ( s 0 ) = ( k n ′ τ 2 + k n τ 2 ′ + k g ′ τ g + k g τ g ′ ) ( s 0 ) .

  5. A cuspidal lip does not appear.

Proof

(1) By using the definition of the spherical surface, we have

S γ ( s , θ ) = cos θ k g 2 ( s ) + k n 2 ( s ) ( − k g ( s ) n γ ( s ) + k n ( s ) n 1 ( s ) ) + sin θ n 2 ( s ) .

Taking the partial derivative of s , we obtain

∂ S γ ∂ s ( s , θ ) = cos θ ( − k n 2 k g ′ + k g k n k n ′ − k g 2 k n τ 1 − k n 3 τ 1 ) − sin θ τ 2 ( k g 2 + k n 2 ) k g 2 + k n 2 ( k g 2 + k n 2 ) k g 2 + k n 2 ( k g 2 + k n 2 ) n γ + cos θ ( k g 2 k n ′ − k g k n k g ′ − k n 2 k g τ 1 − k g 3 τ 1 ) − sin θ τ g ( k g 2 + k n 2 ) k g 2 + k n 2 ( k g 2 + k n 2 ) k g 2 + k n 2 ( k g 2 + k n 2 ) n 1 + cos θ ( k n τ g − k g τ 2 ) k g 2 + k n 2 n 2 ( s ) .

By taking the partial derivative of θ , we obtain

∂ S γ ∂ θ ( s , θ ) = sin θ k g ( s ) k g 2 ( s ) + k n 2 ( s ) n γ ( s ) − sin θ k n ( s ) k g 2 ( s ) + k n 2 ( s ) n 1 ( s ) + cos θ n 2 ( s ) .

Therefore, when ( s 0 , θ 0 ) is a singularity if and only if the vectors { ∂ S γ ∂ s ( s 0 , θ 0 ) , ∂ S γ ∂ θ ( s 0 , θ 0 ) } are linearly dependent. That is, the corresponding coefficients are proportional. Through calculation, we obtain if and only if

tan θ 0 = k g k n ′ − k g 2 τ 1 − k n 2 τ 1 − k n k g ′ k g 2 + k n 2 ( k g τ g + k n τ 2 ) ( s 0 ) .

Thus, (1) holds.

(2) It follows from assertions (3) and (4) of Proposition 3.1 that h v has an A 2 -singularity at s = s 0 if and only if

tan θ 0 = k g k n ′ − k g 2 τ 1 − k n 2 τ 1 − k n k g ′ k g 2 + k n 2 ( k g τ g + k n τ 2 ) ( s 0 ) and χ ( s 0 ) ≠ 0 .

Therefore, by (1) of Proposition 3.2 and Theorem 2.2, we have assertions (2).

(3) It also follows from assertions (4) and (5) of Proposition 3.1 that h v has an A 3 -singularity at s = s 0 if and only if

tan θ 0 = k g k n ′ − k g 2 τ 1 − k n 2 τ 1 − k n k g ′ k g 2 + k n 2 ( k g τ g + k n τ 2 ) ( s 0 ) , χ ( s 0 ) = 0 and χ ′ ( s 0 ) ≠ 0 .

Therefore, by Proposition 3.2 and Theorem 2.2, we have assertions (3).

(4) By using Proposition 7.5 in [21] and Proposition 3.3, we can obtain that H is a Morse family of hypersurfaces. We now calculate σ = ( ∂ 2 H ∕ ∂ s 2 ) ∣ D H . Then, we have

∂ 2 H ∂ s 2 ( s , θ ) = t ″ ( s ) , cos θ k g 2 ( s ) + k n 2 ( s ) ( − k g ( s ) n γ ( s ) + k n ( s ) n 1 ( s ) ) + sin θ n 2 ( s ) = − cos θ k g 2 ( s ) + k n 2 ( s ) ( k g k n ′ − k g 2 τ 1 − k n 2 τ 1 − k n k g ′ ) ( s ) + sin θ ( k n τ 2 + k g τ g ) ( s ) .

The Hessian matrix of

σ ( s , θ ) = − cos θ k g 2 ( s ) + k n 2 ( s ) ( k g k n ′ − k g 2 τ 1 − k n 2 τ 1 − k n k g ′ ) ( s ) + sin θ ( k n τ 2 + k g τ g ) ( s )

is

Hess ( σ ) ( s 0 , 0 ) = ∂ 2 σ ∂ s 2 ( s 0 , 0 ) λ 1 ( s 0 ) λ 1 ( s 0 ) 0 .

So when λ 1 ( s 0 ) ≠ 0 , we have det Hess ( σ ) ( s 0 , 0 ) ≠ 0 . By Lemma 7.7 in [21], H is P - K -equivalent to t 4 ± v 1 2 t 2 + v 2 t + v 3 . The singular set of D H is given by σ ( s , θ ) = 0 . Therefore, it consists of two curves that transversally intersect at ( s 0 , 0 ) . So the normal form is t 4 − v 1 2 t 2 + v 2 t + v 3 , and the surface is diffeomorphic to the cuspidal beaks. Thus, we obtain assertions (4) and (5).□

5 Examples

In this section, we give two examples of spherical surfaces.

Example 5.1

We suppose Σ = R 3 = { x = ( x 0 , x 1 , x 2 , x 3 ) ∈ R 4 ∣ x 0 = 0 } . For γ : I → R 3 , we have n γ = e 0 , t ( s ) = γ ′ ( s ) , n 1 ( s ) = n ( s ) and n 2 ( s ) = b ( s ) . Here, { t , n , b } is the ordinary Frenet frame. In this case, k n = τ 1 = τ 2 = 0 , k g = k and τ g = τ . The Frenet-Serret type formulae are the original Frenet-Serret formulae:

e 0 ′ ( s ) = 0 , t ′ ( s ) = k ( s ) n ( s ) , n ′ ( s ) = − k ( s ) t ( s ) + τ ( s ) b ( s ) , b ′ ( s ) = − τ ( s ) n ( s ) .

The spherical surface of γ is given by

S γ ( s , θ ) = − cos θ e 0 ( s ) + sin θ b ( s ) .

Let γ : I → R 3 be a curve defined by

γ ( s ) = 0 , cos 3 3 s , − sin 3 3 s , 6 3 s ,

we have

t ( s ) = γ ′ ( s ) = 0 , − 3 3 sin 3 3 s , − 3 3 cos 3 3 s , 6 3 , n ( s ) = 0 , − cos 3 3 s , sin 3 3 s , 0 , b ( s ) = 0 , − 6 3 sin 3 3 s , − 6 3 cos 3 3 s , − 3 3 .

Let sin θ = u , cos θ = 1 − u 2 . Thus, the spherical surface of γ is given by

S γ ( s , u ) = − 1 − u 2 , − 6 3 sin 3 3 s u , − 6 3 cos 3 3 s u , − 3 3 u .

We draw the projection of the image of the spherical surface to 3-space (Figure 3).

Figure 3 Projection of the image of the spherical surface on x 1 x 2 x 3 {x}_{1}{x}_{2}{x}_{3} -space.
Figure 3

Projection of the image of the spherical surface on x 1 x 2 x 3 -space.

Example 5.2

We suppose Σ = S 3 . For γ : I → S 3 , we have n γ = γ ( s ) , t ( s ) = γ ′ ( s ) , n 1 ( s ) and n 2 ( s ) . Here, { t , γ , n 1 , n 2 } is the orthonormal frame. In this case, k n ( s ) = − 1 , τ 1 ( s ) = τ 2 ( s ) = 0 , k g ( s ) = k b ( s ) and τ g ( s ) = τ b ( s ) .

γ ′ ( s ) = t ( s ) , t ′ ( s ) = − γ ( s ) + k b ( s ) n 1 ( s ) , n 1 ′ ( s ) = − k b ( s ) t ( s ) + τ b ( s ) n 2 ( s ) , n 2 ′ ( s ) = − τ b ( s ) n 1 ( s ) .

Therefore, the spherical surface of γ is given by

S γ ( s , θ ) = cos θ 1 + k b 2 ( s ) ( − k b ( s ) γ ( s ) − n 1 ( s ) ) + sin θ n 2 ( s ) .

Let γ : I → S 3 be a curve defined by

γ ( s ) = n γ ( s ) = 1 3 cos 2 2 3 s , 1 3 sin 2 2 3 s , 2 3 cos 1 6 s , 2 3 sin 1 6 s ,

we have

t ( s ) = γ ′ ( s ) = − 2 2 3 sin 2 2 3 s , 2 2 3 cos 2 2 3 s , − 1 3 sin 1 6 s , 1 3 cos 1 6 s , t ′ ( s ) = γ ″ ( s ) = − 8 3 3 cos 2 2 3 s , − 8 3 3 sin 2 2 3 s , − 1 3 6 cos 1 6 s , − 1 3 6 sin 1 6 s .

By the direct computation, we obtain ‖ γ ( s ) ‖ = 1 , ∥ t ( s ) ∥ = 1 , and k n ( s ) = ⟨ t ′ ( s ) , n γ ( s ) ⟩ = − 1 . Thus, we obtain

t ′ ( s ) + n γ ( s ) = − 5 3 3 cos 2 2 3 s , − 5 3 3 sin 2 2 3 s , 5 3 6 cos 1 6 s , 5 3 6 sin 1 6 s ,

and k g ( s ) = ‖ t ′ ( s ) + n γ ( s ) ‖ = 5 3 2 . We obtain the normal vector n 1 ( s ) , which is given by

n 1 ( s ) = − 2 3 cos 2 2 3 s , − 2 3 sin 2 2 3 s , 1 3 cos 1 6 s , 1 3 sin 1 6 s .

The other normal vector n 2 ( s ) is given by

n 2 ( s ) = t ( s ) ∧ n γ ( s ) ∧ n 1 ( s ) = 1 3 sin 2 2 3 s , − 1 3 cos 2 2 3 s , − 2 2 3 sin 1 6 s , 2 2 3 cos 1 6 s .

Let sin θ = u , cos θ = 1 − u 2 . Thus, the spherical surface of γ is given by

S γ ( u , s ) = ( x 1 ( u , s ) , x 2 ( u , s ) , x 3 ( u , s ) , x 4 ( u , s ) ) ,

where

x 1 ( u , s ) = 1 129 1 − u 2 cos 2 2 3 s + 1 3 u sin 2 2 3 s , x 2 ( u , s ) = 1 129 1 − u 2 sin 2 2 3 s − 1 3 u cos 2 2 3 s , x 3 ( u , s ) = − 8 2 129 1 − u 2 cos 1 6 s − 2 2 3 u sin 1 6 s , x 4 ( u , s ) = − 8 2 129 1 − u 2 sin 1 6 s + 2 2 3 u cos 1 6 s .

The points ( s , θ ( s ) ) = ( s , 0 ) are the cuspidal edge-type of singularities of S γ , where s ∈ I . We draw the projection of the image of the spherical surface S γ (in green) and its critical value set S γ ( s , 0 ) (in red) to 3-space (Figures 4, 5, 6, 7).

Figure 4 Projection of the image of the spherical surface on x 1 x 2 x 3 {x}_{1}{x}_{2}{x}_{3} -space (in green) and its critical value set (in red).
Figure 4

Projection of the image of the spherical surface on x 1 x 2 x 3 -space (in green) and its critical value set (in red).

Figure 5 Projection of the image of the spherical surface on x 2 x 3 x 4 {x}_{2}{x}_{3}{x}_{4} -space (in green) and its critical value set (in red).
Figure 5

Projection of the image of the spherical surface on x 2 x 3 x 4 -space (in green) and its critical value set (in red).

Figure 6 Projection of the image of the spherical surface on x 1 x 3 x 4 {x}_{1}{x}_{3}{x}_{4} -space (in green) and its critical value set (in red).
Figure 6

Projection of the image of the spherical surface on x 1 x 3 x 4 -space (in green) and its critical value set (in red).

Figure 7 Projection of the image of the spherical surface on x 1 x 2 x 4 {x}_{1}{x}_{2}{x}_{4} -space (in green) and its critical value set (in red).
Figure 7

Projection of the image of the spherical surface on x 1 x 2 x 4 -space (in green) and its critical value set (in red).

  1. Funding information: This research was funded by the Project of Science and Technology of Heilongjiang Provincial Education Department (Grant No. 1453ZD019), the Special Fund for Scientific and Technological Innovation of Graduate Students in Mudanjiang Normal University (Grant No. kjcx2022-021mdjnu), and the Reform and Development Foundation for Local Colleges and Universities of the Central Government (Grant No. ZYQN2019071).

  2. Author contributions: All the authors contributed to this work equally and should be regarded as co-first authors.

  3. Conflict of interest: The authors state no conflicts of interest.

References

[1] Z. Yang, Y. Li, M. Erdoğdu, and Y. Zhu, Evolving evolutoids and pedaloids from viewpoints of envelope and singularity theory in Minkowski plane, J. Geom. Phys. 176 (2022), 104513, DOI: https://doi.org/10.1016/j.geomphys.2022.104513. 10.1016/j.geomphys.2022.104513Search in Google Scholar

[2] Y. Li and Z. Wang, Lightlike tangent developables in de Sitter 3-space, J. Geom. Phys. 164 (2021), 104188, DOI: https://doi.org/10.1016/j.geomphys.2021.104188. 10.1016/j.geomphys.2021.104188Search in Google Scholar

[3] Y. Li, Z. Wang, and T. Zhao, Geometric algebra of singular ruled surfaces, Adv. Appl. Clifford Algebr. 31 (2021), no. 2, 19, DOI: https://doi.org/10.1007/s00006-020-01097-1. 10.1007/s00006-020-01097-1Search in Google Scholar

[4] Y. Li, Y. Zhu, and Q. Sun, Singularities and dualities of pedal curves in pseudo-hyperbolic and de Sitter space, Int. J. Geom. Methods Mod. Phys. 18 (2021), 2150008, DOI: https://doi.org/10.1142/S0219887821500080. 10.1142/S0219887821500080Search in Google Scholar

[5] L. Chen, On spacelike surfaces in Anti de Sitter 3-space from the contact viewpoint, Hokkaido Math. J. 38 (2009), 701–720, DOI: https://doi.org/10.14492/hokmj/1258554241. 10.14492/hokmj/1258554241Search in Google Scholar

[6] X. Lian, Z. Wang, and H. Li, Singularities of worldsheets in spherical space-times, Int. J. Modern Phys. A 33 (2018), 1850114, DOI: https://doi.org/10.1142/S0217751X18501142. 10.1142/S0217751X18501142Search in Google Scholar

[7] Z. Wang and M. He, Singularities of dual hypersurfaces and hyperbolic focal surfaces along spacelike curves in light cone in Minkowski 5-space, Mediterr. J. Math. 16 (2019), no. 4, 96, DOI: https://doi.org/10.1007/s00009-019-1355-5. 10.1007/s00009-019-1355-5Search in Google Scholar

[8] L. Chen and M. Takahashi, Dualities and evolutes of fronts in hyperbolic and de Sitter space, J. Math. Anal. Appl. 437 (2016), 133–159, DOI: https://doi.org/10.1016/j.jmaa.2015.12.029. 10.1016/j.jmaa.2015.12.029Search in Google Scholar

[9] Y. Wang, D. Pei, and X. Cui, Pseudo-spherical normal Darboux images of curves on a lightlike surface, Math. Method Appl. Sci. 40 (2017), 7151–7161, DOI: https://doi.org/10.1002/mma.4519. 10.1002/mma.4519Search in Google Scholar

[10] X. Song and D. Pei, Nullcone dual worldsheets and caustics of spacelike fronts in null de Sitter sphere, Int. J. Modern Phys. A 36 (2021), 2150240, DOI: https://doi.org/10.1142/S0217751X21502407. 10.1142/S0217751X21502407Search in Google Scholar

[11] X. Zhao and D. Pei, Pedal curves of the mixed-type curves in the Lorentz-Minkowski plane, Mathematics 9 (2021), 2852, DOI: https://doi.org/10.3390/math9222852. 10.3390/math9222852Search in Google Scholar

[12] X. Zhao and D. Pei, Evolutoids of the mixed-type curves, Adv. Math. Phys. 2021 (2021), no. 1, 9330963, DOI: https://doi.org/10.1155/2021/9330963. 10.1155/2021/9330963Search in Google Scholar

[13] J. Sun and D. Pei, Singularity analysis of Lorentzian Hypersurfaces on pseudo n-spheres, Math. Methods Appl. Sci. 38 (2015), 2561–2573, DOI: https://doi.org/10.1002/mma.3242. 10.1002/mma.3242Search in Google Scholar

[14] S. Izumiya, M. Kossowski, D. Pei, and M. C. R. Fuster, Singularities of lightlike hypersurfaces in Minkowski four-space, Tohoku Math. J. Sec. Ser. 58 (2006), 71–88, DOI: https://doi.org/10.2748/tmj/1145390206. 10.2748/tmj/1145390206Search in Google Scholar

[15] D. Pei, L. Kong, J. Sun, and Q. Wang, Singularities of lightlike hypersurface in semi-Euclidean 4-space with index 2, Sci. China Math. 53 (2010), 3243–3254, DOI: https://doi.org/10.1007/s11425-010-4121-5. 10.1007/s11425-010-4121-5Search in Google Scholar

[16] S. Izumiya, A. C. Nabarro, A. J. Sacramento, Curves in a spacelike hypersurface in Minkowski space-time, Osaka J. Math. 58 (2021), 947–966, DOI: https://doi.org/10.18910/84959. Search in Google Scholar

[17] M. Rexer and C. Hirt, Ultra-high-degree surface spherical harmonic analysis using the Gauss-Legendre and the Driscoll/Healy quadrature theorem and application to planetary topography models of Earth, Mars and Moon, Surv. Geophys. 36 (2015), 803–830, DOI: https://doi.org/10.1007/s10712-015-9345-z. 10.1007/s10712-015-9345-zSearch in Google Scholar

[18] B. W. Li, H. G. Chen, and J. H. Zhou, The spherical surface symmetrical equal dividing angular quadrature scheme for discrete ordinates method, J. Heat Mass Tran. 124 (2002), 482–490, DOI: https://doi.org/10.1115/1.1459731. 10.1115/1.1459731Search in Google Scholar

[19] M. A. J. Chaplain, M. Ganesh, and I. G. Graham, Spatio-temporal pattern formation on spherical surfaces: numerical simulation and application to solid tumour growth, J. Math. Biol. 42 (2001), 387–423, DOI: https://doi.org/10.1007/s002850000067. 10.1007/s002850000067Search in Google Scholar PubMed

[20] J. W. Bruce and P. J. Giblin, Curves and Singularities: A Geometrical Introduction to Singularity Theory, Cambridge University Press, Cambridge, 1992. 10.1017/CBO9781139172615Search in Google Scholar

[21] S. Izumiya, K. Saji, and M. Takahashi, Horospherical flat surfaces in hyperbolic 3-space, J. Math. Soc. Japan 62 (2010), 789–849, DOI: https://doi.org/10.2969/jmsj/06230789. 10.2969/jmsj/06230789Search in Google Scholar
Received: 2023-10-10
Accepted: 2024-06-26
Published Online: 2024-08-10

© 2024 the author(s), published by De Gruyter

This work is licensed under the Creative Commons Attribution 4.0 International License.

Articles in the same Issue

  1. Special Issue on Contemporary Developments in Graph Topological Indices
  2. On the maximum atom-bond sum-connectivity index of graphs
  3. Upper bounds for the global cyclicity index
  4. Zagreb connection indices on polyomino chains and random polyomino chains
  5. On the multiplicative sum Zagreb index of molecular graphs
  6. The minimum matching energy of unicyclic graphs with fixed number of vertices of degree two
  7. Special Issue on Convex Analysis and Applications - Part I
  8. Weighted Hermite-Hadamard-type inequalities without any symmetry condition on the weight function
  9. Scattering threshold for the focusing energy-critical generalized Hartree equation
  10. (p, q)-Compactness in spaces of holomorphic mappings
  11. Characterizations of minimal elements of upper support with applications in minimizing DC functions
  12. Some new Hermite-Hadamard-type inequalities for strongly h-convex functions on co-ordinates
  13. Global existence and extinction for a fast diffusion p-Laplace equation with logarithmic nonlinearity and special medium void
  14. Extension of Fejér's inequality to the class of sub-biharmonic functions
  15. On sup- and inf-attaining functionals
  16. Regularization method and a posteriori error estimates for the two membranes problem
  17. Rapid Communication
  18. Note on quasivarieties generated by finite pointed abelian groups
  19. Review Articles
  20. Amitsur's theorem, semicentral idempotents, and additively idempotent semirings
  21. A comprehensive review of the recent numerical methods for solving FPDEs
  22. On an Oberbeck-Boussinesq model relating to the motion of a viscous fluid subject to heating
  23. Pullback and uniform exponential attractors for non-autonomous Oregonator systems
  24. Regular Articles
  25. On certain functional equation related to derivations
  26. The product of a quartic and a sextic number cannot be octic
  27. Combined system of additive functional equations in Banach algebras
  28. Enhanced Young-type inequalities utilizing Kantorovich approach for semidefinite matrices
  29. Local and global solvability for the Boussinesq system in Besov spaces
  30. Construction of 4 x 4 symmetric stochastic matrices with given spectra
  31. A conjecture of Mallows and Sloane with the universal denominator of Hilbert series
  32. The uniqueness of expression for generalized quadratic matrices
  33. On the generalized exponential sums and their fourth power mean
  34. Infinitely many solutions for Schrödinger equations with Hardy potential and Berestycki-Lions conditions
  35. Computing the determinant of a signed graph
  36. Two results on the value distribution of meromorphic functions
  37. Zariski topology on the secondary-like spectrum of a module
  38. On deferred f-statistical convergence for double sequences
  39. About j-Noetherian rings
  40. Strong convergence for weighted sums of (α, β)-mixing random variables and application to simple linear EV regression model
  41. On the distribution of powered numbers
  42. Almost periodic dynamics for a delayed differential neoclassical growth model with discontinuous control strategy
  43. A new distributionally robust reward-risk model for portfolio optimization
  44. Asymptotic behavior of solutions of a viscoelastic Shear beam model with no rotary inertia: General and optimal decay results
  45. Silting modules over a class of Morita rings
  46. Non-oscillation of linear differential equations with coefficients containing powers of natural logarithm
  47. Mutually unbiased bases via complex projective trigonometry
  48. Hyers-Ulam stability of a nonlinear partial integro-differential equation of order three
  49. On second-order linear Stieltjes differential equations with non-constant coefficients
  50. Complex dynamics of a nonlinear discrete predator-prey system with Allee effect
  51. The fibering method approach for a Schrödinger-Poisson system with p-Laplacian in bounded domains
  52. On discrete inequalities for some classes of sequences
  53. Boundary value problems for integro-differential and singular higher-order differential equations
  54. Existence and properties of soliton solution for the quasilinear Schrödinger system
  55. Hermite-Hadamard-type inequalities for generalized trigonometrically and hyperbolic ρ-convex functions in two dimension
  56. Endpoint boundedness of toroidal pseudo-differential operators
  57. Matrix stretching
  58. A singular perturbation result for a class of periodic-parabolic BVPs
  59. On Laguerre-Sobolev matrix orthogonal polynomials
  60. Pullback attractors for fractional lattice systems with delays in weighted space
  61. Singularities of spherical surface in R4
  62. Variational approach to Kirchhoff-type second-order impulsive differential systems
  63. Convergence rate of the truncated Euler-Maruyama method for highly nonlinear neutral stochastic differential equations with time-dependent delay
  64. On the energy decay of a coupled nonlinear suspension bridge problem with nonlinear feedback
  65. The limit theorems on extreme order statistics and partial sums of i.i.d. random variables
  66. Hardy-type inequalities for a class of iterated operators and their application to Morrey-type spaces
  67. Solving multi-point problem for Volterra-Fredholm integro-differential equations using Dzhumabaev parameterization method
  68. Finite groups with gcd(χ(1), χc (1)) a prime
  69. Small values and functional laws of the iterated logarithm for operator fractional Brownian motion
  70. The hull-kernel topology on prime ideals in ordered semigroups
  71. ℐ-sn-metrizable spaces and the images of semi-metric spaces
  72. Strong laws for weighted sums of widely orthant dependent random variables and applications
  73. An extension of Schweitzer's inequality to Riemann-Liouville fractional integral
  74. Construction of a class of half-discrete Hilbert-type inequalities in the whole plane
  75. Analysis of two-grid method for second-order hyperbolic equation by expanded mixed finite element methods
  76. Note on stability estimation of stochastic difference equations
  77. Trigonometric integrals evaluated in terms of Riemann zeta and Dirichlet beta functions
  78. Purity and hybridness of two tensors on a real hypersurface in complex projective space
  79. Classification of positive solutions for a weighted integral system on the half-space
  80. A quasi-reversibility method for solving nonhomogeneous sideways heat equation
  81. Higher-order nonlocal multipoint q-integral boundary value problems for fractional q-difference equations with dual hybrid terms
  82. Noetherian rings of composite generalized power series
  83. On generalized shifts of the Mellin transform of the Riemann zeta-function
  84. Further results on enumeration of perfect matchings of Cartesian product graphs
  85. A new extended Mulholland's inequality involving one partial sum
  86. Power vector inequalities for operator pairs in Hilbert spaces and their applications
  87. On the common zeros of quasi-modular forms for Γ+0(N) of level N = 1, 2, 3
  88. One special kind of Kloosterman sum and its fourth-power mean
  89. The stability of high ring homomorphisms and derivations on fuzzy Banach algebras
  90. Integral mean estimates of Turán-type inequalities for the polar derivative of a polynomial with restricted zeros
  91. Commutators of multilinear fractional maximal operators with Lipschitz functions on Morrey spaces
  92. Vector optimization problems with weakened convex and weakened affine constraints in linear topological spaces
  93. The curvature entropy inequalities of convex bodies
  94. Brouwer's conjecture for the sum of the k largest Laplacian eigenvalues of some graphs
  95. High-order finite-difference ghost-point methods for elliptic problems in domains with curved boundaries
  96. Riemannian invariants for warped product submanifolds in Q ε m × R and their applications
  97. Generalized quadratic Gauss sums and their 2mth power mean
  98. Euler-α equations in a three-dimensional bounded domain with Dirichlet boundary conditions
  99. Enochs conjecture for cotorsion pairs over recollements of exact categories
  100. Zeros distribution and interlacing property for certain polynomial sequences
  101. Random attractors of Kirchhoff-type reaction–diffusion equations without uniqueness driven by nonlinear colored noise
  102. Study on solutions of the systems of complex product-type PDEs with more general forms in ℂ2
  103. Dynamics in a predator-prey model with predation-driven Allee effect and memory effect
  104. A note on orthogonal decomposition of 𝔰𝔩n over commutative rings
  105. On the δ-chromatic numbers of the Cartesian products of graphs
  106. Binomial convolution sum of divisor functions associated with Dirichlet character modulo 8
  107. Commutator of fractional integral with Lipschitz functions related to Schrödinger operator on local generalized mixed Morrey spaces
  108. System of degenerate parabolic p-Laplacian
  109. Stochastic stability and instability of rumor model
  110. Certain properties and characterizations of a novel family of bivariate 2D-q Hermite polynomials
  111. Stability of an additive-quadratic functional equation in modular spaces
  112. Monotonicity, convexity, and Maclaurin series expansion of Qi's normalized remainder of Maclaurin series expansion with relation to cosine
  113. On k-prime graphs
  114. On the existence of tripartite graphs and n-partite graphs
  115. Classifying pentavalent symmetric graphs of order 12pq
  116. Almost periodic functions on time scales and their properties
  117. Some results on uniqueness and higher order difference equations
  118. Coding of hypersurfaces in Euclidean spaces by a constant vector
  119. Cycle integrals and rational period functions for Γ0+(2) and Γ0+(3)
  120. Degrees of (L, M)-fuzzy bornologies
  121. A matrix approach to determine optimal predictors in a constrained linear mixed model
  122. On ideals of affine semigroups and affine semigroups with maximal embedding dimension
  123. Solutions of linear control systems on Lie groups
  124. A uniqueness result for the fractional Schrödinger-Poisson system with strong singularity
  125. On prime spaces of neutrosophic extended triplet groups
  126. On a generalized Krasnoselskii fixed point theorem
  127. On the relation between one-sided duoness and commutators
  128. Non-homogeneous BVPs for second-order symmetric Hamiltonian systems
  129. Erratum
  130. Erratum to “Infinitesimals via Cauchy sequences: Refining the classical equivalence”
  131. Corrigendum
  132. Corrigendum to “Matrix stretching”
  133. Corrigendum to “A comprehensive review of the recent numerical methods for solving FPDEs”
https://doi.org/10.1515/math-2024-0033
Keywords for this article
spherical surface; hypersurface; cuspidal beaks
Creative Commons
BY 4.0

Articles in the same Issue

  1. Special Issue on Contemporary Developments in Graph Topological Indices
  2. On the maximum atom-bond sum-connectivity index of graphs
  3. Upper bounds for the global cyclicity index
  4. Zagreb connection indices on polyomino chains and random polyomino chains
  5. On the multiplicative sum Zagreb index of molecular graphs
  6. The minimum matching energy of unicyclic graphs with fixed number of vertices of degree two
  7. Special Issue on Convex Analysis and Applications - Part I
  8. Weighted Hermite-Hadamard-type inequalities without any symmetry condition on the weight function
  9. Scattering threshold for the focusing energy-critical generalized Hartree equation
  10. (p, q)-Compactness in spaces of holomorphic mappings
  11. Characterizations of minimal elements of upper support with applications in minimizing DC functions
  12. Some new Hermite-Hadamard-type inequalities for strongly h-convex functions on co-ordinates
  13. Global existence and extinction for a fast diffusion p-Laplace equation with logarithmic nonlinearity and special medium void
  14. Extension of Fejér's inequality to the class of sub-biharmonic functions
  15. On sup- and inf-attaining functionals
  16. Regularization method and a posteriori error estimates for the two membranes problem
  17. Rapid Communication
  18. Note on quasivarieties generated by finite pointed abelian groups
  19. Review Articles
  20. Amitsur's theorem, semicentral idempotents, and additively idempotent semirings
  21. A comprehensive review of the recent numerical methods for solving FPDEs
  22. On an Oberbeck-Boussinesq model relating to the motion of a viscous fluid subject to heating
  23. Pullback and uniform exponential attractors for non-autonomous Oregonator systems
  24. Regular Articles
  25. On certain functional equation related to derivations
  26. The product of a quartic and a sextic number cannot be octic
  27. Combined system of additive functional equations in Banach algebras
  28. Enhanced Young-type inequalities utilizing Kantorovich approach for semidefinite matrices
  29. Local and global solvability for the Boussinesq system in Besov spaces
  30. Construction of 4 x 4 symmetric stochastic matrices with given spectra
  31. A conjecture of Mallows and Sloane with the universal denominator of Hilbert series
  32. The uniqueness of expression for generalized quadratic matrices
  33. On the generalized exponential sums and their fourth power mean
  34. Infinitely many solutions for Schrödinger equations with Hardy potential and Berestycki-Lions conditions
  35. Computing the determinant of a signed graph
  36. Two results on the value distribution of meromorphic functions
  37. Zariski topology on the secondary-like spectrum of a module
  38. On deferred f-statistical convergence for double sequences
  39. About j-Noetherian rings
  40. Strong convergence for weighted sums of (α, β)-mixing random variables and application to simple linear EV regression model
  41. On the distribution of powered numbers
  42. Almost periodic dynamics for a delayed differential neoclassical growth model with discontinuous control strategy
  43. A new distributionally robust reward-risk model for portfolio optimization
  44. Asymptotic behavior of solutions of a viscoelastic Shear beam model with no rotary inertia: General and optimal decay results
  45. Silting modules over a class of Morita rings
  46. Non-oscillation of linear differential equations with coefficients containing powers of natural logarithm
  47. Mutually unbiased bases via complex projective trigonometry
  48. Hyers-Ulam stability of a nonlinear partial integro-differential equation of order three
  49. On second-order linear Stieltjes differential equations with non-constant coefficients
  50. Complex dynamics of a nonlinear discrete predator-prey system with Allee effect
  51. The fibering method approach for a Schrödinger-Poisson system with p-Laplacian in bounded domains
  52. On discrete inequalities for some classes of sequences
  53. Boundary value problems for integro-differential and singular higher-order differential equations
  54. Existence and properties of soliton solution for the quasilinear Schrödinger system
  55. Hermite-Hadamard-type inequalities for generalized trigonometrically and hyperbolic ρ-convex functions in two dimension
  56. Endpoint boundedness of toroidal pseudo-differential operators
  57. Matrix stretching
  58. A singular perturbation result for a class of periodic-parabolic BVPs
  59. On Laguerre-Sobolev matrix orthogonal polynomials
  60. Pullback attractors for fractional lattice systems with delays in weighted space
  61. Singularities of spherical surface in R4
  62. Variational approach to Kirchhoff-type second-order impulsive differential systems
  63. Convergence rate of the truncated Euler-Maruyama method for highly nonlinear neutral stochastic differential equations with time-dependent delay
  64. On the energy decay of a coupled nonlinear suspension bridge problem with nonlinear feedback
  65. The limit theorems on extreme order statistics and partial sums of i.i.d. random variables
  66. Hardy-type inequalities for a class of iterated operators and their application to Morrey-type spaces
  67. Solving multi-point problem for Volterra-Fredholm integro-differential equations using Dzhumabaev parameterization method
  68. Finite groups with gcd(χ(1), χc (1)) a prime
  69. Small values and functional laws of the iterated logarithm for operator fractional Brownian motion
  70. The hull-kernel topology on prime ideals in ordered semigroups
  71. ℐ-sn-metrizable spaces and the images of semi-metric spaces
  72. Strong laws for weighted sums of widely orthant dependent random variables and applications
  73. An extension of Schweitzer's inequality to Riemann-Liouville fractional integral
  74. Construction of a class of half-discrete Hilbert-type inequalities in the whole plane
  75. Analysis of two-grid method for second-order hyperbolic equation by expanded mixed finite element methods
  76. Note on stability estimation of stochastic difference equations
  77. Trigonometric integrals evaluated in terms of Riemann zeta and Dirichlet beta functions
  78. Purity and hybridness of two tensors on a real hypersurface in complex projective space
  79. Classification of positive solutions for a weighted integral system on the half-space
  80. A quasi-reversibility method for solving nonhomogeneous sideways heat equation
  81. Higher-order nonlocal multipoint q-integral boundary value problems for fractional q-difference equations with dual hybrid terms
  82. Noetherian rings of composite generalized power series
  83. On generalized shifts of the Mellin transform of the Riemann zeta-function
  84. Further results on enumeration of perfect matchings of Cartesian product graphs
  85. A new extended Mulholland's inequality involving one partial sum
  86. Power vector inequalities for operator pairs in Hilbert spaces and their applications
  87. On the common zeros of quasi-modular forms for Γ+0(N) of level N = 1, 2, 3
  88. One special kind of Kloosterman sum and its fourth-power mean
  89. The stability of high ring homomorphisms and derivations on fuzzy Banach algebras
  90. Integral mean estimates of Turán-type inequalities for the polar derivative of a polynomial with restricted zeros
  91. Commutators of multilinear fractional maximal operators with Lipschitz functions on Morrey spaces
  92. Vector optimization problems with weakened convex and weakened affine constraints in linear topological spaces
  93. The curvature entropy inequalities of convex bodies
  94. Brouwer's conjecture for the sum of the k largest Laplacian eigenvalues of some graphs
  95. High-order finite-difference ghost-point methods for elliptic problems in domains with curved boundaries
  96. Riemannian invariants for warped product submanifolds in Q ε m × R and their applications
  97. Generalized quadratic Gauss sums and their 2mth power mean
  98. Euler-α equations in a three-dimensional bounded domain with Dirichlet boundary conditions
  99. Enochs conjecture for cotorsion pairs over recollements of exact categories
  100. Zeros distribution and interlacing property for certain polynomial sequences
  101. Random attractors of Kirchhoff-type reaction–diffusion equations without uniqueness driven by nonlinear colored noise
  102. Study on solutions of the systems of complex product-type PDEs with more general forms in ℂ2
  103. Dynamics in a predator-prey model with predation-driven Allee effect and memory effect
  104. A note on orthogonal decomposition of 𝔰𝔩n over commutative rings
  105. On the δ-chromatic numbers of the Cartesian products of graphs
  106. Binomial convolution sum of divisor functions associated with Dirichlet character modulo 8
  107. Commutator of fractional integral with Lipschitz functions related to Schrödinger operator on local generalized mixed Morrey spaces
  108. System of degenerate parabolic p-Laplacian
  109. Stochastic stability and instability of rumor model
  110. Certain properties and characterizations of a novel family of bivariate 2D-q Hermite polynomials
  111. Stability of an additive-quadratic functional equation in modular spaces
  112. Monotonicity, convexity, and Maclaurin series expansion of Qi's normalized remainder of Maclaurin series expansion with relation to cosine
  113. On k-prime graphs
  114. On the existence of tripartite graphs and n-partite graphs
  115. Classifying pentavalent symmetric graphs of order 12pq
  116. Almost periodic functions on time scales and their properties
  117. Some results on uniqueness and higher order difference equations
  118. Coding of hypersurfaces in Euclidean spaces by a constant vector
  119. Cycle integrals and rational period functions for Γ0+(2) and Γ0+(3)
  120. Degrees of (L, M)-fuzzy bornologies
  121. A matrix approach to determine optimal predictors in a constrained linear mixed model
  122. On ideals of affine semigroups and affine semigroups with maximal embedding dimension
  123. Solutions of linear control systems on Lie groups
  124. A uniqueness result for the fractional Schrödinger-Poisson system with strong singularity
  125. On prime spaces of neutrosophic extended triplet groups
  126. On a generalized Krasnoselskii fixed point theorem
  127. On the relation between one-sided duoness and commutators
  128. Non-homogeneous BVPs for second-order symmetric Hamiltonian systems
  129. Erratum
  130. Erratum to “Infinitesimals via Cauchy sequences: Refining the classical equivalence”
  131. Corrigendum
  132. Corrigendum to “Matrix stretching”
  133. Corrigendum to “A comprehensive review of the recent numerical methods for solving FPDEs”
Sign up now to receive a 20% welcome discount
Institutional Access
How does access work?
Have an idea on how to improve our website?
Please write us.
© 2025 De Gruyter Brill
Downloaded on 7.9.2025 from https://www.degruyterbrill.com/document/doi/10.1515/math-2024-0033/html
Scroll to top button