Surfaces of revolution of frontals in the Euclidean space
Masatomo Takahashi Keisuke Teramoto








Surfaces of revolution of frontals in the Euclidean space
    Masatomo Takahashi and Keisuke Teramoto
    December 30, 2023
=========================================================

 
For Legendre curves, we consider surfaces of revolution of frontals. 
The surface of revolution of a frontal can be considered as a framed base surface. 
We give the curvatures and basic invariants for surfaces of revolution by using the curvatures of Legendre curves.
Moreover, we give properties of surfaces of revolution with singularities and cones.



[2010 Mathematics Subject classification: 57R45, 53A05, 58K05]
[Key Words and Phrases. surface of revolution, frontal, Legendre curve, framed surface]




§ INTRODUCTION


The surface of revolution is one of classical object in differential geometry (cf. <cit.>). 
It has been known that if the profile curve (the plane curve) across the axis of revolution, then the surface of revolution has cone-type singularity.
On the other hand, the profile curve is not regular, then the surface of revolution must have singularities. 

In <cit.>, the surfaces of revolution of regular curves are investigated. 
Especially, Kenmotsu gave concrete construction of the surface of revolution with prescribed mean curvature. 
In <cit.>, the surfaces of revolution of singular curves are investigated. 
They construct a given the unbounded mean curvature of the surface of revolution. 

In this paper, we consider more general situation. 
We consider frontals (Legendre curves) as singular plane curves and framed base surfaces (framed surfaces) as singular surfaces. 
In <cit.>, we give the curvature of Legendre curves in order to analyze Legendre curves. 
In <cit.>, we give the basic invariants of framed surfaces so as to analyze framed surfaces. 
In 2, we review the theories of Legendre curves in the unit tangent bundle over the Euclidean plane ^2 
and framed surfaces in the Euclidean space ^3.
The surface of revolution of a frontal is a framed base surface.
We can deal with surfaces of revolution with singular points more directly.
In fact, we give the curvatures and basic invariants for surfaces of revolution by using the curvatures of Legendre curves in 3. 
Moreover, we give profile curves for given information of the curvatures, for instance, the Gauss curvature or the mean curvature. 
For the cases of constant Gauss and mean curvature surfaces of revolution, see <cit.>.
We also consider relations between right-left equivalence relations among profile curves and right-left equivalence relations among the surface of revolution.
In particular, we give characterizations of j/i-cusps ((i,j)=(2,3),(2,5),(3,4),(3,5)) appearing on profile curves of surfaces of revolution in terms of the  curvatures of Legendre curves. 
Further, applying such characterizations to the case of surfaces of revolution with constant Gauss or constant mean curvature. 
We give conditions that constant Gauss curvature surfaces of revolution 
have j/i-cusps as ((i,j)=(2,3),(3,4),(3,5)) singularities by the construction data, 
and show that there are no constant Gauss curvature surfaces of revolution with 5/2-cusp.
In contrast, we show that profile curves of constant mean curvature surfaces of revolution do not have j/i-cusps as singularities. 
Further, we classify cone-type singularities in some cases. 
In 4, we give concrete examples.

All maps and manifolds considered here are differentiable of class C^∞ unless stated otherwise. 


Acknowledgement. 
The first author was partially supported by JSPS KAKENHI Grant Number JP 17K05238 and the second author was partially supported by  JSPS KAKENHI Grant Number JP 17J02151. 




§ PRELIMINARY

We quickly review the theories of Legendre curves in the unit tangent bundle over the Euclidean plane (cf. <cit.>) and framed surface in the Euclidean space (cf. <cit.>).




 §.§ Legendre curves


Let γ:I→^2 and ν:I→ S^1 be smooth mappings, where I is an interval of  and S^1 is the unit circle.
We say that (γ,ν):I →^2 × S^1 is a Legendre curve if (γ,ν)^*θ=0 for all t ∈ I, where θ is a canonical contact form on the unit tangent bundle T_1 ^2=^2 × S^1 over ^2 (cf. <cit.>). 
This condition is equivalent to γ̇(t) ·ν(t)=0 for all t ∈ I, 
where γ̇(t)=(dγ/dt)(t) and ·=a_1b_1+a_2b_2 for any =(a_1,a_2), =(b_1,b_2)∈^2. 
A point t_0∈ I is called a singular point of γ if γ̇(t_0)=0. 
When a Legendre curve (γ,ν):I→^2× S^1 gives an immersion, it is called a Legendre immersion.
We say that γ:I →^2 is a frontal (respectively, a front) if there exists ν:I → S^1 such that (γ,ν) is a Legendre curve (respectively, a Legendre immersion). 
Examples of Legendre curves see <cit.>.
We have the Frenet type formula of a frontal γ as follows.
We put on (t)=J(ν (t)), where J is the anticlockwise rotation by angle π/2 in ^2.
Then {ν(t), (t) } is a moving frame of the frontal γ(t) in ^2 and we have the Frenet type formula,

    (
    [ ν̇(t);   (t) ])
    =
    (
    [     0  ℓ(t); -ℓ(t)     0 ])
    (
    [ ν(t);  (t) ]),  γ̇(t) = β(t) (t),

where ℓ(t)=ν̇(t) ·(t) and β(t)=γ̇(t) ·(t).
We call the pair (ℓ,β) the curvature of the Legendre curve. 
By (<ref>), we see that (γ,ν) is a Legendre immersion if and only if (ℓ,β)≠(0,0). 


Let (γ,ν) and (γ,ν):I →^2 × S^1 be Legendre curves.
We say that (γ,ν) and (γ,ν) are congruent as Legendre curves if there exist a constant rotation A ∈ SO(2) and a translation  on ^2 such that γ(t)=A(γ(t))+ and ν(t)=A (ν(t)) for all t ∈ I. 




 
Let (ℓ,β):I →^2 be a smooth mapping. 
There exists a Legendre curve (γ,ν):I →^2 × S^1 whose curvature of the Legendre curve is (ℓ, β).


Actually, we have the following.


    γ(t)    =(-∫β(t) sin( ∫ℓ(t) dt) dt, ∫β(t) cos(∫ℓ(t) dt) dt),
    ν(t)    = (cos(∫ℓ(t)  dt),  sin(∫ℓ(t) dt) ).




 
Let (γ,ν) and (γ,ν):I →^2 × S^1 be Legendre curves with the curvatures of Legendre curves (ℓ,β) and (ℓ,β), respectively.
Then (γ,ν) and (γ,ν) are congruent as Legendre curves if and only if (ℓ,β) and (ℓ,β) coincide.



Let (γ,ν):I →^2 × S^1 be a Legendre curve with the curvature (ℓ,β). 
We define the parallel curves of (γ,ν) by γ^λ:I →^2, γ^λ(t)=γ(t)+λν(t) for λ∈. 
Then (γ^λ,ν):I →^2 × S^1 is also a Legendre curve with the curvature (ℓ,β+λℓ).

Moreover, we define the evolute of (γ,ν) by ℰv(γ):I →^2, 
ℰv(γ)(t)=γ(t)+(β(t)/ℓ(t))ν(t), where ℓ(t) ≠0 for all t ∈ I (cf. <cit.>). 





 §.§ Framed surfaces


Let ^3 be the 3-dimensional Euclidean space equipped with the inner product · = a_1 b_1 + a_2 b_2 + a_3 b_3, 
where = (a_1, a_2, a_3) and = (b_1, b_2, b_3) ∈^3. 
The norm of  is given by || = √(·). 
We also define the vector product

    ×=(
    [  _1  _2  _3; a_1 a_2 a_3; b_1 b_2 b_3 ]),

where {_1, _2, _3 } is the canonical basis of ^3. 
Let U be a simply connected domain of ^2 and S^2 be the unit sphere in ^3, that is, S^2={∈^3| ||=1}.
We denote a 3-dimensional smooth manifold {(,) ∈ S^2 × S^2| ·=0} by Δ.
We say that (,,):U →^3 ×Δ is a framed surface if _u (u,v) · (u,v)=0 and _v(u,v) ·(u,v)=0 for all (u,v) ∈ U, where _u(u,v)=(∂/∂ u)(u,v) and _v(u,v)=(∂/∂ v)(u,v). 
We say that :U →^3 is a framed base surface if there exists (,):U →Δ such that (,,) is a framed surface.

Similarly to the case of Legendre curves, 
the pair (,):U→^3× S^2 is said to be a Legendre surface 
if _u (u,v) · (u,v)=0 and _v(u,v) ·(u,v)=0 for all (u,v) ∈ U. 
Moreover, when a Legendre surface (,):U→^3× S^2 gives an immersion, this is called a Legendre immersion. 
We say that :U→^3 be a frontal (respectively, a front) if there exists a map :U→ S^2 such that 
the pair (,):U→^3× S^2 is a Legendre surface (respectively, a Legendre immersion).
By definition, the framed base surface is a frontal. 
At least locally, the frontal is a framed base surface. 
For a framed surface (,,), we say that a point p∈ U is a singular point of  
if  is not an immersion at p.

We denote (u,v)=(u,v) ×(u,v). 
Then {(u,v),(u,v),(u,v)} is a moving frame along (u,v).
Thus, we have the following systems of differential equations:

    ([ _u; _v ])
    =
    ([ a_1 b_1; a_2 b_2 ])
    ([ ;  ]),


    ([ _u; _u; _u ])
    =
    ([    0  e_1  f_1; -e_1    0  g_1; -f_1 -g_1    0 ])
    ([ ; ;  ])
    ,  ([ _v; _v; _v ])
    =
    ([    0  e_2  f_2; -e_2    0  g_2; -f_2 -g_2    0 ])
    ([ ; ;  ]),

where a_i,b_i,e_i,f_i,g_i:U →, i=1,2 are smooth functions and we call the functions basic invariants of the framed surface. 
We denote the matrices (<ref>) and (<ref>) by 𝒢, ℱ_1, ℱ_2, respectively. 
We also call the matrices (𝒢, ℱ_1, ℱ_2)  basic invariants of the framed surface (,,).
Since the integrability condition _uv=_vu and ℱ_2,u-ℱ_1,v
=ℱ_1ℱ_2-ℱ_2ℱ_1, 
the basic invariants should satisfy the following conditions:

    a_1,v-b_1g_2 = a_2,u-b_2g_1, 
    
    b_1,v-a_2g_1 = b_2,u-a_1g_2, 
    
    a_1 e_2 + b_1 f_2 = a_2 e_1 + b_2 f_1,
    
    e_1,v-f_1g_2 = e_2,u-f_2g_1, 
    
    f_1,v-e_2g_1 = f_2,u-e_1g_2, 
    
    g_1,v-e_1f_2 = g_2,u-e_2f_1.


We give fundamental theorems for framed surfaces, that is, 
the existence and uniqueness theorem of framed surfaces for basic invariants.

Let (,,), (,,):U →^3 ×Δ be framed surfaces.
We say that (,,) and (,,) are congruent as framed surfaces if there exist a constant rotation A ∈ SO(3) and a translation ∈^3 such that 

    (u,v)=A((u,v))+,  (u,v)=A((u,v)),  (u,v)=A((u,v))
 
for all (u,v) ∈ U.




Let U be a simply connected domain in ^2 and let a_i,b_i,e_i,f_i,g_i:U →, i=1,2 be smooth functions with the integrability conditions (<ref>).
Then there exists a framed surface (,,):U →^3 ×Δ whose associated basic invariants is a_i,b_i,e_i,f_i,g_i, i=1,2.




Let (,,), (,,):U →^3 ×Δ be framed surfaces 
with basic invariants (𝒢,ℱ_1,ℱ_2) and (𝒢,ℱ_1,ℱ_2), respectively.
Then (,,) and (,,) are congruent as framed surfaces if and only if (𝒢,ℱ_1,ℱ_2) and (𝒢,ℱ_1,ℱ_2)  coincide.



Let (,,):U →^3 ×Δ be a framed surface with basic invariants (𝒢,ℱ_1,ℱ_2).


We define a smooth mapping C_F=(J_F,K_F,H_F):U →^3 by 

    J_F =
    ([ a_1 b_1; a_2 b_2 ]), 
    K_F =
    ([ e_1 f_1; e_2 f_2 ]), 
    H_F = -1/2{([ a_1 f_1; a_2 f_2 ])
    -
    ([ b_1 e_1; b_2 e_2 ])
    }.

We call C_F = (J_F,K_F,H_F) a curvature of the framed surface.
We also define a smooth mapping I_F:U →^8 by 

    I_F=(C_F,
    ([ a_1 g_1; a_2 g_2 ]),
    ([ b_1 g_1; b_2 g_2 ]),.
    .
    ([ e_1 g_1; e_2 g_2 ]), 
    ([ f_1 g_1; f_2 g_2 ]),
    ([ a_1 e_1; a_2 e_2 ])
    ).

We call the mapping I_F:U →^8 a concomitant mapping of the framed surface (,,). 



If the surface  is regular, then we have K=K_F/J_F and H=H_F/J_F, where 
K is the Gauss curvature and H is the mean curvature of the regular surface (cf. <cit.>). 
For relations between behaviour of the Gauss curvature, the mean curvature of fronts at non-degenerate singular points and geometric invariants of fronts see <cit.>. 


We say that (,,):U →^3 ×Δ is a framed immersion if (,,) is an immersion.



Let (,,):U →^3 ×Δ be a framed surface and p ∈ U.
(1)  is an immersion (a regular surface) around p if and only if J_F (p) ≠0.
(2) (,) is a Legendre immersion around p if and only if C_F (p) ≠0.
(3) (,,) is a framed immersion around p if and only if I_F (p) ≠0.



Let (,,):U →^3 ×Δ be a framed surface with basic invariants (𝒢,ℱ_1,ℱ_2). 
We define the parallel surface ^λ : U →ℝ^3 of the framed surface (,,) by  ^λ(u,v) = (u,v) + λ(u,v), where λ∈ℝ. 
Then (^λ,,):U →^3 ×Δ is also a framed surface and  basic invariants (𝒢^λ,  ℱ_1^λ, ℱ_2^λ) are given by  

    𝒢^λ = 𝒢 + λ([ e_1 f_1; e_2 f_2 ]),  ℱ_1^λ =
    ℱ_1,  ℱ_2^λ =ℱ_2.

By a direct calculation, we have the curvature (J^λ_F,K^λ_F,H^λ_F) of the framed surface (^λ,,) is given by 

    J^λ_F=J_F-2H_Fλ+K_Fλ^2,  K^λ_F=K_F,   H^λ_F=H_F-K_Fλ.


We also define the evolute (or, the focal surface) of the framed surface (,,) by ℰv():U →^3, ℰv()(u,v)=(u,v)+λ(u,v), where 
λ is a solution of the equation

    K_F(u,v) λ^2-2H_F(u,v) λ+J_F(u,v)=0

for all (u,v) ∈ U. 
The study of focal surfaces of fronts from a different viewpoint is known in <cit.>.

Even if K_F(u,v)=0 for all (u,v) ∈ U, we can define an evolute when H_F(u,v) ≠0. 
In this case, we have only one evolute. 



Moreover, if we consider a similar surface (r,,):U →^3 ×Δ for non-zero constant r ∈, 
Then the curvature (J^r_F,K^r_F,H^r_F) of the framed surface (r,,) is given by 

    J^r_F=r^2 J_F,  K^r_F=K_F,   H^r_F=r H_F.





§ SURFACES OF REVOLUTION OF FRONTALS


Let (γ,ν):I →^2 × S^1 be  a Legendre curve with the curvature (ℓ,β). 
We denote γ(t)=(x(t),z(t)) and ν(t)=(a(t),b(t)). 
By definition and the Frenet type formula (<ref>), we have ẋ(t)a(t)+ż(t)b(t)=0, a^2(t)+b^2(t)=1, 

    ([ ẋ(t); ż(t) ])=
    β(t)
    ([ -b(t);  a(t) ]), 
    ([ ȧ(t); ḃ(t) ])=
    ℓ(t)
    ([ -b(t);  a(t) ])

for all t ∈ I. 
Set a smooth function φ:I → such that a(t)=cosφ(t) and b(t)=sinφ(t). 

We consider ^2 ⊂^3 as (x,z)-plane into (x,y,z)-space. 
We give the two surfaces of revolution, that is, around x-axis and z-axis of the frontal γ, respectively. 
We call γ a profile curve (cf. <cit.>).

First we consider the surface of revolution around x-axis.
We denote the surface of revolution of γ around x-axis by :I × [0,2π) →^3,

    (t,θ)=(x(t),z(t) cosθ, z(t) sinθ).

Then the surface of revolution around x-axis of the frontal is a framed base surface. 


Under the above notation, 
(,^x,^x):I × [0,2π) →^3 ×Δ is a framed surface with basic invariants,

    𝒢^x=
    ([     0 -β(t); -z(t)     0 ]),
    ℱ^x_1=
    ([     0     0  ℓ(t);     0     0     0; -ℓ(t)     0     0 ]),
    ℱ^x_2=
    ([     0  b(t)     0; -b(t)     0 -a(t);     0  a(t)     0 ]),

where 
^x(t,θ)=(-a(t),-b(t) cosθ,-b(t) sinθ), 
^x(t,θ)=(0,sinθ, -cosθ).


By a direct calculation, 
^x(t,θ)=^x(t,θ) ×^x(t,θ)=(b(t),-a(t)cosθ, -a(t)sinθ).
Since 

    _t(t,θ)   = (-β(t)b(t),β(t)a(t)cosθ,β(t)a(t)sinθ)=-β(t)^x(t,θ),
    _θ(t,θ)   =(0,-z(t)sinθ,z(t)cosθ)=-z(t)^x(t,θ),
    ^x_t(t,θ)   =(ℓ(t)b(t),-ℓ(t)a(t)cosθ,-ℓ(t)a(t)sinθ)=ℓ(t)^x(t,θ),
    ^x_θ(t,θ)   =(0,b(t)sinθ,-b(t)cosθ)=b(t)^x(t,θ),
    ^x_t(t,θ)   =0,
    ^x_θ(t,θ)   =(0,cosθ,sinθ)=-b(t)^x(t,θ)-a(t)^x(t,θ),
    ^x_t(t,θ)   =(ℓ(t)a(t), ℓ(t)b(t)cosθ, ℓ(t)b(t)sinθ)=-ℓ(t)^x(t,θ),
    ^x_θ(t,θ)   =(0,b(t)sinθ,b(t)cosθ)=a(t)^x(t,θ),

we have basic invariants (𝒢^x,ℱ^x_1,ℱ^x_2).

By a direct calculation, we have 

    J^x_F(t,θ)=-β(t)z(t), K^x_F(t,θ)=-b(t)ℓ(t), H^x_F(t,θ)=-1/2(z(t)ℓ(t)+β(t)b(t)),
       ([ a_1 g_1; a_2 g_2 ])(t,θ)=0,
    ([ b_1 g_1; b_2 g_2 ])(t,θ)=β(t)a(t),
    ([ e_1 g_1; e_2 g_2 ])(t,θ)=0,
       ([ f_1 g_1; f_2 g_2 ])(t,θ)=-ℓ(t)a(t),
    ([ a_1 e_1; a_2 e_2 ])(t,θ)=0.


Next, we consider the surface of revolution around  z-axis.
We denote the surface of revolution of γ around z-axis by :I × [0,2π) →^3,

    (t,θ)=(x(t) cosθ,x(t) sinθ, z(t)).

Then the surface of revolution around  z-axis of the frontal is also a framed base surface. 
By the similar calculation in Proposition <ref>, we have the following result.


Under the above notation, 
(,^z,^z):I × [0,2π) →^3 ×Δ is a framed surface with  basic invariants,

    𝒢^z=
    ([     0 -β(t); -x(t)     0 ]),
    ℱ^z_1=
    ([     0     0 -ℓ(t);     0     0     0;  ℓ(t)     0     0 ]),
    ℱ^z_2=
    ([     0 -a(t)     0;  a(t)     0  b(t);     0 -b(t)     0 ]),

where 
^z(t,θ)=(a(t) cosθ,a(t)sinθ,b(t)), 
^z(t,θ)=(sinθ,-cosθ,0).





By a direct calculation, we have 

    J^z_F(t,θ)=-β(t)x(t), K^z_F(t,θ)=-a(t)ℓ(t), H^z_F(t,θ)=1/2(x(t)ℓ(t)+β(t)a(t)),
       ([ a_1 g_1; a_2 g_2 ])(t,θ)=0,
    ([ b_1 g_1; b_2 g_2 ])(t,θ)=-β(t)b(t),
    ([ e_1 g_1; e_2 g_2 ])(t,θ)=0,
       ([ f_1 g_1; f_2 g_2 ])(t,θ)=-ℓ(t)b(t),
    ([ a_1 e_1; a_2 e_2 ])(t,θ)=0.



 
Under the above notation, we have the following.
(1) The surfaces of revolution  and  are frontals if and only if γ is a frontal. 
(2) The surface of revolution  or  is a front if and only if γ is a front. 


(1) If  and  are frontals, then there exists a point (t,θ) such that C^x_F(t,θ)=0 and C^z_F(t,θ)=0 by Proposition <ref>. 
Since K^x_F(t,θ)=K^z_F(t,θ)=0, we have ℓ(t)=0.
Moreover, since H^x_F(t,θ)=H^z_F(t,θ)=0, we have β(t)=0. 
It follows that γ is a frontal. 
Conversely, let γ be a frontal.
If t is a singular point of (γ,ν), then ℓ(t)=β(t)=0. 
By a direct calculation, we have C^x_F(t,θ)=0 and C^z_F(t,θ)=0. 
Therefore,  and  are frontals.
(2) By (1), we have the result.



 
(,^x,^x) and (,^z,^z) are congruent as framed surfaces if and only if the Legendre curve (γ,ν):I →^2 × S^1 is given by 

    γ(t)=( ±∫1/√(2)β(t) dt, ±∫1/√(2)β(t) dt), 
    ν(t)=(±1/√(2), ∓1/√(2)).



By Theorem <ref>, 
(,^x,^x) and (,^z,^z) are congruent as framed surface if and only if (𝒢^x,ℱ^x_1,ℱ^x_2)(t,θ)=(𝒢^z,ℱ^z_1,ℱ_2)(t,θ) for all (t,θ) ∈ I ×[0,2π).
It follows that z(t)=x(t), ℓ(t)=0, a(t)=-b(t) for all t ∈ I.
Since ℓ(t)=0, a(t) and b(t) are constants.
By a^2(t)+b^2(t)=1, we have ν(t)=(±1/√(2), ∓ 1/√(2)).
Moreover, ẋ(t)=-β(t) b(t), ż(t)=β(t) a(t), we have γ(t)=(x(t),z(t)) by integration.






Suppose that (𝒢^x,ℱ^x_1,ℱ^x_2) and (𝒢^z,ℱ^z_1,ℱ^z_2) are given by the forms of (<ref>) and of (<ref>) with 
ẋ(t)=-β(t)b(t), ẏ(t)=β(t)a(t), ȧ(t)=-ℓ(t)b(t), ḃ(t)=ℓ(t)a(t), a^2(t)+b^2(t)=1. 
Then there exists a unique Legendre curve (γ,ν):I →^2 × S^1 with the curvature (ℓ,β) such that basic invariants of surfaces of revolution around x-axis and z-axis are (𝒢^x,ℱ^x_1,ℱ^x_2) and (𝒢^z,ℱ^z_1,ℱ^z_2), respectively.


By the forms of (<ref>) and of (<ref>), we define (γ,ν):I →^2 × S^1  by γ(t)=(x(t),z(t)) and ν(t)=(a(t),b(t)).
Then (γ,ν) is a Legendre curve with the curvature (ℓ,β). 
Moreover, basic invariants of the surfaces of x-axis revolution (,^x,^x) and z-axis revolution (,^z,^z) of the Legendre curve (γ,ν) are (𝒢^x,ℱ^x_1,ℱ^x_2) and (𝒢^z,ℱ^z_1,ℱ^z_2), respectively. 


We now concentrate on the case of surfaces of revolution around z-axis. 
Let (γ,ν):I →^2 × S^1 be a Legendre curve with the curvature (ℓ,β). We denote γ(t)=(x(t),z(t)) and ν(t)=(a(t),b(t)). 

The mapping f:U →^3 is a cone if f is given by f(u,v)=(u cos v,u sin v,u).

 
If x(t_0)=0, β(t_0) ≠ 0, a(t_0) ≠0 and b(t_0) ≠0, then the surface of revolution :I × [0,2π) →^3 at (t_0,θ_0) ∈ I × [0,2π) is diffeomorphic to the cone at (0,θ_0).


By the assumption, we have ẋ(t_0) ≠0 and ż(t_0) ≠0. 
The surface of revolution :I × [0,2π) →^3 is given by 
(t,θ)=(x(t) cosθ,x(t) sinθ, z(t)).
By using the diffeomorphisms of the source (t,θ) ↦ (x(t),θ) and the target (X,Y,Z) ↦ (X,Y,z(Z)), then 
(t,θ) at (t_0,θ_0) is diffeomorphic to the cone (t,θ) ↦ (tcosθ,tsinθ, t) at (0,θ_0).

For more general cases see Theorem <ref>.


 
If K^z_F(t,θ)=0 for all (t,θ) ∈ I × [0,2π), then γ is a part of a line. 


If K^z_F(t,θ)=0 for all (t,θ) ∈ I × [0,2π), then -a(t)ℓ(t)=0 for all t ∈ I.
By differentiating, we have b(t)ℓ^2(t)+a(t)ℓ̇(t)=0.
If ℓ(t_0) ≠0 at a point t_0 ∈ I, then a(t)=0 and b(t)=0 around t_0.
This is a contradict the fact that (a(t),b(t)) ∈ S^1. 
It follows that ℓ(t)=0 for all t ∈ I. 
This is equivalent to a(t) and b(t) are constants. 
Hence, γ is a part of a line.

By Proposition <ref>, the surfaces of revolution of a frontal satisfying 
K^z_F(t,θ)=0 for all (t,θ) ∈ I × [0,2π) are given by a part of a cone, a cylinder, a plane, a line, a circle or a point. 

We consider general cases. 
We denote J(t)=J^z_F(t,θ), K(t)=K^z_F(t,θ) and H(t)=H^z_F(t,θ).


Let (γ,ν):(I,t_0) →^2 × S^1 be a Legendre curve of the form γ(t)=(x(t),z(t)) and ν(t)=(cosφ(t), sinφ(t)) 
with the curvature (φ̇,β). 
(1) Suppose that β is a real analytic around t_0 and there exists a  function α:(I,t_0) → such that K(t)=α(t)J(t) and α(t)β^2(t)(t-t_0)^2 is a real analytic around t_0. 
Then x(t) is a solution of 

    β(t)ẍ(t)-β̇(t)ẋ(t)+α(t)β^3(t)x(t)=0

around t_0, 


    z(t) = ±∫β(t) (1-(∫α(t) β(t) x(t) dt)^2 )^1/2 dt

and 

    cosφ(t)= ±(1-(∫α(t)β(t)x(t) dt)^2)^1/2,  sinφ(t)=∫α(t)β(t)x(t) dt.

(2) Suppose that x(t_0)>0 and given smooth functions J and K: (I,t_0) →. 
Then  (γ,ν) is given by 

    x(t) = (-2 ∫ J(t) (∫ K(t) dt) dt )^1/2,  
    z(t) = ∓∫J(t)/x(t)(1-(∫ K(t) dt)^2 )^1/2 dt

and 

    cosφ(t)=±(1-(∫ K(t) dt)^2)^1/2,  sinφ(t)=-∫ K(t) dt.



(1) Since K(t)=α(t)J(t), we have φ̇(t)cosφ(t)=α(t)β(t)x(t).
By 

(<ref>), ẋ(t)=-β(t)sinφ(t). 
It follows that ẍ(t)=-β̇(t)sinφ(t)-α(t)β^2(t)x(t) and β(t) ẍ(t)=β̇(t)ẋ(t)-α(t)β^3(t)x(t). 
Then x(t) is satisfied a second order ordinary linear differential equation of (<ref>). 
By the assumption, t_0 is a regular singularity of (<ref>). 
Then there exists a solution of x(t) around t_0 by using the method of Frobenius (cf. <cit.>). 
Moreover, (d/dt)(sinφ(t))=φ̇(t)cosφ(t)=α(t)β(t)x(t), we have 

    cosφ(t)= ±(1-(∫α(t)β(t)x(t) dt)^2)^1/2, sinφ(t)=∫α(t)β(t)x(t) dt.

Since ż(t)=β(t) cosφ(t), we have 

    z(t)=∫β(t)cosφ(t) dt =±∫β(t) (1-(∫α(t) β(t) x(t) dt)^2 )^1/2 dt.

(2) We have J(t)=-β(t)x(t) and K(t)=-φ̇(t)cosφ(t). 
By (d/dt)(sinφ(t))=φ̇(t)cosφ(t)=-K(t), we have 

    cosφ(t)=±(1-(∫ K(t) dt)^2)^1/2,  sinφ(t)=-∫ K(t) dt.

By the Frenet type formula (<ref>), ẋ(t)=-β(t)sinφ(t). 
It follows that 

    d/dt x^2(t)=2 x(t) ẋ(t)=-2β(t)x(t)sinφ(t)
    =-2 J(t) ∫ K(t) dt.

Then 

    x^2(t)=-2 ∫ J(t) (∫ K(t) dt) dt.

Since x(t_0)>0, we have x(t) around t_0. 
Further, by ż(t)=β(t)cosφ(t)=-(J(t)/x(t)) cosφ(t), we have

    z(t)=∓∫J(t)/x(t)(1-(∫ K(t) dt)^2)^1/2 dt.



By the similar calculation in <cit.>, we have the following result for H. 


Let (γ,ν):(I,t_0) →^2 × S^1 be a Legendre curve of the form γ(t)=(x(t),z(t)) and ν(t)=(cosφ(t), sinφ(t)) 
with the curvature (φ̇,β). 
Suppose that x(t_0)>0, we give β and there exists a smooth function α:(I,t_0) → such that H(t)=α(t)J(t).
Then (γ,ν) is given by


    x(t) = (F(t)^2+G(t)^2)^1/2,  
    z(t) = ∫β(t)/x(t)( F(t)sinη(t)-G(t)cosη(t)) dt

and 

    cosφ(t)=F(t)sinη(t)-G(t)cosη(t)/x(t), sinφ(t)=F(t)cosη(t)+G(t)sinη(t)/x(t),

where 

    F(t)=-∫β(t)cosη(t) dt,  G(t)=-∫β(t)sinη(t) dt,  η(t)=2 ∫α(t)β(t) dt.



Since H(t)=α(t)J(t), we have φ̇(t)x(t)+β(t) cosφ(t)=-2α(t)β(t)x(t).
It follows that 

    φ̇(t) cosφ(t) x(t)+β(t)cosφ^2(t)   =-2α(t)β(t) x(t) cosφ(t),
    φ̇(t) sinφ(t) x(t)+β(t)cosφ(t) sinφ(t)   =-2α(t)β(t)x(t) sinφ(t).

We define X(t)=x(t)sinφ(t)+i x(t)cosφ(t), where i is the imaginary unit. 
Then 

    Ẋ(t)=φ̇(t)x(t) cosφ(t)-β(t)(1-cos^2φ(t))-i (φ̇(t)x(t)+β(t)sinφ(t)).

By a direct calculation, we have Ẋ(t)-2iα(t)β(t)X(t)=-β(t).
A solution of the first order ordinary linear differential equation is given by X(t)=(F(t)-i G(t))(cosη(t)+isinη(t)), where 

    F(t)=-∫β(t)cosη(t) dt,  G(t)=-∫β(t)sinη(t) dt,  η(t)=2 ∫α(t)β(t) dt.

It follows that 

    x(t) cosφ(t)   =F(t)sinη(t)-G(t)cosη(t), 
    
    x(t) sinφ(t)   =F(t)cosη(t) +G(t) sinη(t).

Then we have x^2(t)=F(t)^2+G(t)^2.
Since x(t_0)>0, we have x(t)=(F(t)^2+G(t)^2)^1/2, sinφ(t) and cosφ(t) around t_0.
Moreover, by ż(t)=β(t) cosφ(t), we have 

    z(t) = ∫β(t)/x(t)( F(t)sinη(t)-G(t)cosη(t)) dt.


Let (γ,ν) be a Legendre curve constructed by Theorem <ref>. 
Then the curvature (φ̇,β) of (γ,ν) is calculated as 

    (φ̇(t),β(t))=(-β(t)X(t),β(t))  (X(t)=F(t)sinη(t)-G(t)cosη(t)F(t)^2+G(t)^2+2α(t)).
 
Thus the curvature (φ̇(t),β(t)) as in (<ref>) vanishes at a singular point t_0 of γ. 
This implies that the curve γ constructed by (<ref>) is a frontal, but not a front at t_0 because (φ̇(t_0),β(t_0))≠(0,0) 
when (γ,ν) is a Legendre immersion (see <cit.>). 
Therefore the surface of revolution  of γ is a frontal but not a front by Proposition <ref>.
For the case of γ being a front, see <cit.>.





Let (γ,ν):(I,t_0) →^2 × S^1 be a Legendre curve of the form γ(t)=(x(t),z(t)) and ν(t)=(cosφ(t), sinφ(t)) 
with the curvature (φ̇,β). 
(1) Suppose that x(t_0)>0 and given a smooth function J:(I,t_0) →. 
Then  γ is given by 

    x(t) = (2 ∫ J(t) sinφ(t) dt )^1/2,  
    z(t) = - ∫J(t)/x(t)cosφ(t) dt.

(2) Suppose that cosφ(t_0) ≠0 and given a smooth function  H: (I,t_0) →. 
Then  γ is given by 

    x(t) = 1/cosφ(t)(-∫ 2H(t) sinφ(t) dt ),  
    z(t) = ∫(2H(t)-φ̇(t) x(t) ) dt.



(1) By the assumption, we have J(t)=-β(t)x(t). 
Since the differential ẋ(t) of x(t) by t is ẋ(t)=-β(t)sinφ(t), we have

    d/dt x^2(t)=2x(t)ẋ(t)=-2x(t)β(t)sinφ(t)=2J(t)sinφ(t).

Hence, 

    x^2(t)=2 ∫ J(t) sinφ(t) dt.
 
By x(t_0)>0, we have 

    x(t)=(2∫^t_t_0 J(t) sinφ(t) dt)^1/2.

Since ż(t)=β(t)cosφ(t) and β(t)=-J(t)/x(t), we have 

    z(t) = -∫J(t)/x(t)cosφ(t) dt.

(2) By the assumption, we have H(t)=(1/2)(φ̇(t)x(t)+β(t)cosφ(t)). 
Since ẋ(t)=-β(t)sinφ(t), we have

    (cosφ(t))ẋ(t)=-β(t)cosφ(t)sinφ(t)
    =(φ̇(t)sinφ(t)) x(t)-2H(t)sinφ(t).

By cosφ(t_0) ≠0, a solution of the first order ordinary linear differential equation 
(cosφ(t))ẋ(t)-(φ̇(t)sinφ(t)) x(t)=-2H(t)sinφ(t) is given by 

    x(t) = 1/cosφ(t)(-∫ 2H(t) sinφ(t) dt ).

Since ż(t)=β(t)cosφ(t) and β(t)=(2H(t)-φ̇(t)x(t))/cosφ(t), we have 

    z(t) = ∫(2H(t)-φ̇(t) x(t) ) dt.



We have already known that the surface of revolution of a parallel curve of a Legendre curve is a parallel surface of the surface of revolution of the frontal (cf. <cit.>).



Let (γ,ν):I →^2 × S^1 be a Legendre curve.
The parallel surface ^λ:I × [0,2π) →^3 of the surface of revolution of the frontal γ:I →^2 is the surface of revolution of a parallel curve γ^λ:I →^2 of the Legendre curve (γ,ν).


By definition, the parallel surface of (,^z,^z):I × [0,2π) →^3 ×Δ is given by ^λ:I × [0,2π) →^3,

    ^λ(t,θ)=(t,θ)+λ^z(t,θ)=
    ((x(t)+λ a(t)) cosθ,(x(t)+λ a(t))sinθ, z(t)+λ a(t)).
 
On the other hand, the parallel curve of the Legendre curve (γ,ν):I →^2 × S^1 is given by γ^λ:I →^2, γ^λ(t)=γ(t)+λν(t)=(x(t)+λ a(t),z(t)+λ b(t)). 
It follows that the surface of revolution of the parallel curve γ^λ is given by ^λ:I × [0,2π) →^3.


Let (γ,ν):I →^2 × S^1 be a Legendre curve and (,^z,^z):I × [0,2π) →^3 ×Δ be the surface of revolution. 
We consider the evolute ℰv() of the surface of revolution  of the frontal γ.
In the following, we denote by ℰv^z the evolute of  instead of ℰv().



Let (γ,ν):I →^2 × S^1 be a Legendre curve with the curvature (ℓ,β).
(1) Suppose that K^z_F(t,θ) ≠0 for all (t,θ) ∈ I × [0,2π).
One of the evolute of the surface of revolution of the front is the surface of revolution of the evolute of the front.
The other evolute of the surface of revolution of the front is given by 
ℰv^z(t,θ)=(0,0,z(t)-x(t)b(t)/a(t)).
(2) Suppose that ℓ(t) ≠0 for all t ∈ I.
At least an evolute of the surface of revolution of the front is the surface of revolution of the evolute of the front.
(3) Suppose that a(t) ≠0 for all t ∈ I.
At least an evolute of the surface of revolution of the front is given by ℰv^z(t,θ)=(0,0,z(t)-x(t)b(t)/a(t)).


(1) Since K^z_F(t,θ)=-a(t)ℓ(t) ≠0, we have a(t) ≠0 and ℓ(t) ≠0 for all t ∈ I. 
Since 


    K^z_F(t,θ)λ^2-2H^z_F(t,θ)λ+J^z_F(t,θ)
       =-a(t)ℓ(t)λ^2-(x(t)ℓ(t)+β(t)a(t))λ-x(t)β(t)
       =-(a(t)λ+x(t))(ℓ(t)λ+β(t)),


the solutions of the equation K^z_F(t,θ)λ^2-2H^z_F(t,θ)λ+J^z_F(t,θ)=0 are given by λ=-β(t)/ℓ(t) and λ=-x(t)/a(t).
Therefore, one of the evolute is given by 

    ℰv^z(t,θ)   =(t,θ)-β(t)/ℓ(t)^z(t,θ)
       =((x(t) cosθ,x(t)sinθ,z(t))-β(t)/ℓ(t)(a(t) cosθ,a(t)sinθ,b(t))
       =( (x(t)-β(t)/ℓ(t)a(t))cosθ, (x(t)-β(t)/ℓ(t)a(t))sinθ, z(t)-β(t)/ℓ(t)b(t)).

Hence it is the surface of revolution of the evolute ℰv(γ)(t)=γ(t)-(β(t)/ℓ(t))ν(t).
The other evolute is given by 

    ℰv^z(t,θ)   =(t,θ)-x(t)/a(t)^z(t,θ)
       =((x(t) cosθ,x(t)sinθ,z(t))-x(t)/a(t)(a(t) cosθ,a(t)sinθ,b(t))
       =(0,0,z(t)-x(t)/a(t)b(t)).

(2) Suppose that ℓ(t) ≠0 for all t ∈ I. 
Then we have at least a solution λ=-β(t)/ℓ(t) of the equation K^z_F(t,θ)λ^2-2H^z_F(t,θ)λ+J^z_F(t,θ)=0. 
Therefore, by the same calculation of (1), 
at least an evolute of the surface of revolution of the front is the surface of revolution of evolute of the front. 
(3) By the same argument as in (2), we have the result.


If ℓ(t)=0, a(t) ≠0 and β(t) ≠0 for all t ∈ I, then  we have the evolute, see Remark <ref> and Proposition <ref> (3). 
Moreover, even if β(t)=0 for isolated points, we also define an evolute of the surface of revolution of the frontal as the same by the continuous property. 




We consider classification problems. 
We use notations x_0=x(t_0), z_0=z(t_0) and _0=(t_0,θ_0)=(x_0cosθ_0,x_0sinθ_0,z_0) for 
smooth curve γ=(x,z):(I,t_0)→(^2,(x_0,z_0)) and its surface of revolution  around z-axis, respectively.
We also use a notation γ_0=γ(t_0)=(x_0,z_0).

First we consider the case of x_0>0.



Let γ:(I,t_0) → (^2,(x_0,z_0)) and γ:(I,t_0) → (^2,(x_0,z_0)) be smooth curves 
with x_0, x_0>0, 
let :(I × [0,2π),(t_0,θ_0)) → (^2 ∖{0}×,_0) 
and :(I× [0,2π),(t_0,θ_0)) → (^2 ∖{0}×,_0) 
be surfaces of revolution of the frontals around z-axis, respectively. 
(1) If there exist diffeomorphism germs ϕ:(I,t_0) → (I,t_0) 
and ψ: (^2,γ_0)→ (^2,γ_0), ψ(X,Z)=(ψ_1(X,Z),ψ_2(X,Z)) such that ψ∘γ=γ∘ϕ, 
then there exist diffeomorphism germs Φ:(I × [0,2π),(t_0,θ_0)) → (I× [0,2π),(t_0,θ_0)) of the form Φ(t,θ)=(ϕ(t),θ) 
and Ψ:(^2 ∖{0}×,_0) →  (^2 ∖{0}×,_0) of the form 

    Ψ(X,Y,Z)=(Xψ_1(√(X^2+Y^2),Z)/√(X^2+Y^2), Yψ_1(√(X^2+Y^2),Z)/√(X^2+Y^2), ψ_2(√(X^2+Y^2),Z))
 
such that Ψ∘=∘Φ.
(2) If there exist diffeomorphism germs Φ:(I × [0,2π),(t_0,θ_0)) → (I× [0,2π),(t_0,θ_0)) of the form Φ(t,θ)=(ϕ(t),θ) 
and Ψ:(^2 ∖{0}×,_0) →  (^2 ∖{0}×,_0), Ψ(X,Y,Z)=(Ψ_1(X,Y,Z),Ψ_2(X,Y,Z),Ψ_3(X,Y,Z)) 
such that Ψ∘=∘Φ, 
then there exists a diffeomorphism germ ψ: (^2,γ_0)→ (^2,γ_0), ψ(X,Z)=(ψ_1(X,Z),ψ_2(X,Z)) of the form 

    ψ_1(X,Z)   = Ψ_1(Xcosθ_0,Xsinθ_0,Z)cosθ_0+Ψ_2(Xcosθ_0,Xsinθ_0,Z)sinθ_0,
    ψ_2(X,Z)   =Ψ_3(Xcosθ_0,Xsinθ_0,Z)

such that ψ∘γ=γ∘ϕ


(1) Since ϕ is a diffeomorphism germ, Φ is a diffeomorphism germ. 
We show that Ψ(X,Y,Z)=(Ψ_1(X,Y,Z),Ψ_2(X,Y,Z),Ψ_3(X,Y,Z)) is a diffeomorphism germ. 
By a direct calculation, we have 


    Ψ_1X(_0)   =ψ_1(x_0,z_0)sin^2 θ_0+x_0 ψ_1X(x_0,z_0) cos^2 θ_0/x_0,
    Ψ_1Y(_0)   =(x_0ψ_1X(x_0,z_0)- ψ_1(x_0,z_0))cosθ_0 sinθ_0/x_0,
    Ψ_1Z(_0)   =ψ_1Z(x_0,z_0)cosθ_0,
    Ψ_2X(_0)   =(x_0ψ_1X(x_0,z_0)- ψ_1(x_0,z_0))cosθ_0 sinθ_0/x_0,
    Ψ_2Y(_0)   =ψ_1(x_0,z_0)cos^2 θ_0+x_0 ψ_1X(x_0,z_0) sin^2 θ_0/x_0,
    Ψ_2Z(_0)   =ψ_1Z(x_0,z_0)sinθ_0, 
    Ψ_3X(_0)   =ψ_2X(x_0,z_0)cosθ_0, 
    Ψ_3Y(_0)   =ψ_2X(x_0,z_0)sinθ_0, 
    Ψ_3Z(_0)   =ψ_2Z(x_0,z_0).


Then we can show that the determinant of the Jacobi matrix of Ψ at _0 is non-zero. 
Hence Ψ is a diffeomorphism germ.
By the assumption, we have 

    (x(ϕ(t)),z(ϕ(t)))=(ψ_1(x(t),z(t)),ψ_2(x(t),z(t))).

It follows that 

    Ψ∘(t,θ)   =Ψ (x(t)cosθ,x(t)sinθ,z(t))
       =(ψ_1(x(t),z(t)) cosθ ,ψ_1(x(t),z(t)) sinθ,ψ_2(x(t),z(t)))
       =(x(ϕ(t))cosθ, x(ϕ(t))sinθ,z(ϕ(t)))
       =∘Φ(t,θ).

(2) 
Since Φ is a diffeomorphism germ, ϕ is a diffeomorphism germ. 
We show that ψ(X,Z)=(ψ_1(X,Z),ψ_2(X,Z)) is a diffeomorphism germ. 
By a direct calculation, we have 


    ψ_1X(x_0,z_0)   =Ψ_1X(_0)cos^2 θ_0+Ψ_1Y(_0)cosθ_0 sinθ_0
         +Ψ_2X(_0)cosθ_0 sinθ_0+Ψ_2Y(_0) sin^2 θ_0,
    ψ_1Z(x_0,z_0)   =Ψ_1Z(_0)cosθ_0+Ψ_2Z(_0)sinθ_0,
    ψ_2X(x_0,z_0)   =Ψ_3X(_0)cosθ_0+Ψ_3Y(_0)sinθ_0,
    ψ_2Z(x_0,z_0)   =Ψ_3Z(_0).


Since Ψ∘=∘Φ, we have 

    x(ϕ(t)) cosθ   = Ψ_1(x(t)cosθ,x(t) sinθ,z(t)),
    x(ϕ(t)) sinθ   = Ψ_2(x(t)cosθ,x(t) sinθ,z(t)),
    z(ϕ(t))    = Ψ_3(x(t)cosθ,x(t) sinθ,z(t))

for all (t, θ). 
By differentiating with respect to θ, we have 

    -x(ϕ(t)) sinθ   = Ψ_1X(x(t)cosθ,x(t) sinθ,z(t))(-x(t)sinθ)
         +Ψ_1Y(x(t)cosθ,x(t) sinθ,z(t))x(t)cosθ
       =-Ψ_2(x(t)cosθ,x(t) sinθ,z(t)),
    x(ϕ(t)) cosθ   = Ψ_2X(x(t)cosθ,x(t) sinθ,z(t))(-x(t)sinθ)
         +Ψ_2Y(x(t)cosθ,x(t) sinθ,z(t))x(t)cosθ
       =Ψ_1(x(t)cosθ,x(t) sinθ,z(t)),
    
    0    = Ψ_3X(x(t)cosθ,x(t) sinθ,z(t))(-x(t) sinθ)
         +Ψ_3Y(x(t)cosθ,x(t) sinθ,z(t))x(t) cosθ.

By using the above, we can show that the determinant of the Jacobi matrix of ψ at γ_0=(x_0,z_0) is non-zero.
Hence ψ is a diffeomorphism germ.
Moreover, 


    ψ∘γ(t)   =(ψ_1(x(t),z(t)),ψ_2(x(t),z(t)))
       =(Ψ_1(x(t)cosθ_0,x(t)sinθ_0,z(t))cosθ_0
       +Ψ_2(x(t)cosθ_0,x(t)sinθ_0,z(t))sinθ_0, 
    Ψ_3(x(t)cosθ_0,x(t)sinθ_0,z(t)))
       =(x(ϕ(t))cos^2 θ_0+x(ϕ(t))sin^2 θ_0, z(ϕ(t)))
       =γ∘ϕ(t).


This completes the proof of Theorem <ref>.


To characterize singularities of surfaces of revolution applying Theorem <ref>, 
we give some definitions (cf. <cit.>).

(1) Let f and g:(^m,0)→(^n,0) be smooth map-germs. 
Then f is 𝒜-equivalent to g if there exist diffeomorphism germs ϕ:(^m,0)→(^m,0) and Φ:(^n,0)→(^n,0) 
such that g=Φ∘ f∘ϕ^-1 holds. 
If the diffeomorphism germ Φ (respectively, ϕ) appeared in above is the identity map, 
we say that f is ℛ-equivalent (respectively, ℒ-equivalent) to g.
(2) Let γ:(I,t_0)→(^2,0) be a smooth curve. 
We say that γ at t_0 is a j/i-cusp, where (i,j)=(2,3),(2,5),(3,4),(3,5) if γ is 𝒜-equivalent to the germ t↦(t^i,t^j) at the origin.
(3) Let f:(^2,0)→(^3,0) be a smooth map. 
We say that f at 0 is a j/i-cuspidal edge, where (i,j)=(2,3),(2,5),(3,4),(3,5) if f is 𝒜-equivalent to the germ (u,v)↦(u,v^i,v^j) at the origin.



We note that curves with j/i-cusps are frontal (curves). 
Moreover, surfaces with j/i-cuspidal edges are not only frontal (surfaces), 
but also framed base surfaces (see <cit.>).
In particular, 3/2-cusps and 4/3-cusps (respectively, 3/2-cuspidal edges and 4/3-cuspidal edges) are front singularities. 
As a corollary of Theorem <ref>, we have the following.



Let γ=(x,z):(I,t_0)→(^2,(x_0,z_0)) be a smooth curve with x_0>0. 
Then γ at t_0 is a j/i-cusp 
if and only if the surface of revolution (t,θ) of γ 
around the z-axis at (t_0,θ_0) is a j/i-cuspidal edge for any θ_0.



For curves with j/i-cusps, the following criteria are known (cf. <cit.>). 

Let γ:I→^2 be a smooth curve and t_0∈ I a singular point of γ, 
namely, γ̇(t_0)=0. 
Then the following assertions hold.
(1) γ has a 3/2-cusp at t_0 if and only if (γ̈,⃛γ)(t_0)≠0.
(2) γ has a 5/2-cusp at t_0 if and only if 
γ̈(t_0)≠0, ⃛γ(t_0)=Cγ̈(t_0) for some constant C∈ 
and (γ̈,3γ^(5)-10Cγ^(4))(t_0)≠0.
(3) γ has a 4/3-cusp at t_0 if and only if γ̈(t_0)=0 and 
(⃛γ,γ^(4))(t_0)≠0.
(4) γ has a 5/3-cusp at t_0 if and only if γ̈(t_0)=0, 
(⃛γ,γ^(4))(t_0)=0 and (⃛γ,γ^(5))(t_0)≠0.

Using Proposition <ref>, we show the following.


Let (γ,ν):(I,t_0)→^2× S^1 
be a Legendre curve of the form γ(t)=(x(t),z(t)), ν(t)=(cosφ(t),sinφ(t)) with the curvature (φ̇,β). 
Assume that β(t_0)=0. 
Then we have the following.
(1) γ is a 3/2-cusp at t_0 if and only if β̇(t_0)φ̇(t_0)≠0.
(2) γ is a 5/2-cusp at t_0 if and only if β̇(t_0)≠0, φ̇(t_0)=0 and 
β̈(t_0)φ̈(t_0)-β̇(t_0)⃛φ(t_0)≠0.
(3) γ is a 4/3-cusp at t_0 if and only if β̇(t_0)=0 and β̈(t_0)φ̇(t_0)≠0.
(4) γ is a 5/3-cusp at t_0 if and only if β̇(t_0)=φ̇(t_0)=0 and β̈(t_0)φ̈(t_0)≠0.

By the Frenet type formula (<ref>), we have γ̇(t)=β(t)(t), ν̇(t)=φ̇(t)(t) and (t)=-φ̇(t)ν(t), where (t)=(-sinφ(t),cosφ(t)). 
We consider differentials of γ(t). 
By direct calculations, we see that 

    γ̈(t)   =β̇(t)(t)-β(t)φ̇(t)ν(t),
    ⃛γ(t)   =(β̈(t)-β(t)(φ̇)^2(t)(t)-(2β̇(t)φ̇(t)+β(t)φ̈(t))ν(t),
    γ^(4)(t)   =(⃛β(t)-3β̇(t)(φ̇)^2(t)-3β(t)φ̇(t)φ̈(t))(t)
          -(3β̈(t)φ̇(t)+3β̇(t)φ̈(t)+β(t)⃛φ(t)-β(t)(φ̇)^3(t)ν(t).


Since β(t_0)=0, we have (γ̈,⃛γ)(t_0)=2(β̇)^2(t_0)φ̇(t_0). 
Thus the assertion (1) holds by Proposition <ref> (1).
 
We show (3) by using Proposition <ref> (3). 
The condition for γ̈(t_0)=0 is β̇(t_0)=0. 
In this case, ⃛γ(t_0)=β̈(t_0)(t_0), 
and hence ⃛γ(t_0)≠0 if and only if β̈(t_0)≠0. 
Moreover, (⃛γ,γ^(4))(t_0)=3(β̈)^2(t_0)φ̇(t_0) holds. 
Thus we have (3).  

We next consider (2). 
Since γ̈(t_0)≠0, we note that β̇(t_0)≠0 holds. 
Two vectors γ̈(t_0) and ⃛γ(t_0) are parallel 
if and only if (γ̈,⃛γ)(t_0)=0. 
This is equivalent to φ̇(t_0)=0, 
and hence we see that ⃛γ(t_0)=Cγ̈(t_0), where C=β̈(t_0)/β̇(t_0). 
Further, γ^(4)(t_0) and γ^(5)(t_0) are calculated as

    γ^(4)(t_0)   =⃛β(t_0)(t_0)-3β̇(t_0)φ̈(t_0)ν(t_0),
    γ^(5)(t_0)   =β^(4)(t_0)(t_0)-(6β̈(t_0)φ̈(t_0)+4β̇(t_0)⃛φ(t_0))ν(t_0).

Thus 

    (γ̈,3γ^(5)-10Cγ^(4))(t_0)
    =-12β̇(t_0)(β̈(t_0)φ̈(t_0)-β̇(t_0)⃛φ(t_0))

holds. 
By Proposition <ref> (2), we see that the assertion (2) holds.

Finally we show (4). 
Since γ̈(t_0)=(⃛γ,γ^(4))(t_0)=0 and ⃛γ(t_0)≠0, 
we see that β̇(t_0)=φ̇(t_0)=0 and β̈(t_0)≠0. 
Under these conditions, γ^(5)(t_0) is given by 


    γ^(5)(t_0)=β^(4)(t_0)(t_0)-6β̈(t_0)φ̈(t_0)ν(t_0).


Hence we have 


    (⃛γ,γ^(5))(t_0)=6(β̈)^2(t_0)φ̈(t_0).


Thus we have the assertion by Proposition <ref> (4).
If we use the notation for a Legendre curve (γ,ν) with the curvature (ℓ,β), 
assertions in Theorem <ref> also hold by replacing φ̇ with ℓ. 

Let f:(I,t_0)→ be a function germ and k be a non-negative integer. 
Then f has a zero of order (k+1) at t_0 if 

    f(t_0)=ḟ(t_0)=f̈(t_0)=⋯=f^(k)(t_0)=0,   f^(k+1)(t_0)≠0,

where ḟ=df/dt and f^(n)=d^nf/dt^n for a positive integer n. 
In this case, we write (f)(t_0)=k+1. 
For a Legendre curve (γ,ν) with the curvature (φ̇,β), 
we remark that β satisfies (β)(t_0)≥1 at a singular point t_0 of γ since β(t_0)=0.

We now use notations ν(t)=(a(t),b(t)) for the unit normal and (ℓ(t),β(t)) for the curvature 
instead of ν(t)=(cosφ(t),sinφ(t)) and (φ̇(t),β(t)), respectively. 



Let (γ,ν):I→^2× S^1, γ(t)=(x(t),z(t)), ν(t)=(a(t),b(t)) be a Legendre curve which is given by 
(1) in Theorem <ref>. 
Suppose that x(t_0)≠0 and α(t_0)≠0 at a singular point t_0 of γ. 
Then we have the following.
(1) If a(t_0)≠0, then γ is a frontal but not a front at t_0.
(2) Suppose that (a)(t_0)=m+1 and (β)(t_0)=n+1. 
Then γ is a front at t_0 if and only if (a)(t_0)=(β)(t_0).


Without loss of generality, we may assume that t_0=0. 
By the proof of (1) in Theorem <ref>, the function ℓ(t) satisfies

    ℓ(t)a(t)=α(t)β(t)x(t).

If a(0)≠0, the function ℓ can be given as 

    ℓ(t)=α(t)β(t)x(t)a(t)

around 0. 
Since β(0)=0, we have (ℓ(0),β(0))=(0,0). 
Hence, γ is a frontal but not a front at t=0. 
Thus the assertion (1) holds.

We show the assertion (2). 
We now assume that (a)(0)=m+1 and (β)(0)=n+1.  
Then by the division lemma, there exist smooth functions â(t) and β̂(t) around t=0 such that 
a(t)=t^m+1â(t) and β(t)=t^n+1β̂(t).
We note that â(0)≠0 and β̂(0)≠0. 
If m>n, then we have t^m-nâ(t) ℓ(t)=α(t) β̂(t) x(t). 
It follows that α(0) β̂(0) x(0)=0. 
This contradicts the fact that α(0) β̂(0) x(0) ≠0.
Therefore, m≤ n. 
Then the function ℓ(t) can be expressed as 

    ℓ(t)=t^n-mα(t)β̂(t)x(t)â(t).

Thus ℓ(0)≠0 if and only if n=m. 
This implies that (a)(0)=(β)(0). 
Therefore we have the assertion (2).


We have the following characterizations of singularities 
for the case of constant Gauss curvature surfaces of revolution (see (1) in Theorem <ref>). 


Under the same assumptions as in Theorem <ref> (1) with conditions that the Legendre curve (γ,ν):(I,t_0) →^2 × S^1,  γ(t)=(x(t),z(t)), ν(t)=(a(t),b(t)) satisfy x(t_0) ≠ 0 and (a)(t_0)=m+1 ≤(β)(t_0)=n+1. 
If the smooth function α:(I,t_0)→ satisfying K(t)=α(t)J(t) is a non-zero constant c∈, namely, α(t)=c, 
then the curve γ given by (1) in Theorem <ref> has
(1) a 3/2-cusp at t_0 if and only if (a)(t_0)=(β)(t_0)=1,
(2) a 4/3-cusp at t_0 if and only if (a)(t_0)=(β)(t_0)=2,
(3) a 5/3-cusp at t_0 if and only if (a)(t_0)=1 and (β)(t_0)=2.

Moreover, γ cannot have a 5/2-cusp at t_0.

By a parallel translation on the source, we may assume that t_0=0. 
By (<ref>), the function ℓ(t) is given by 

    ℓ(t)=ct^n-mβ̂(t)x(t)â(t),

where a(t)=t^m+1â(t) with â(0)≠0 and β(t)=t^n+1β̂(t) with β̂(0)≠0. 
We first consider the case of γ to be a front at 0, that is, m+1=(a)(0)=(β)(0)=n+1 by Proposition <ref>.
Then β̇(0)≠0 if and only if n=0. 
By (1) in Theorem <ref>, we have the assertion (1). 
Moreover, β̇(0)=0 and β̈(0)≠0 if and only if n=1, and hence we have the assertion (2) 
by (3) in Theorem <ref>.

We next consider the case of 0≤ m<n. 
In this case, γ at t=0 is a frontal but not a front. 
Moreover, the functions ℓ and β are expressed as 

    ℓ(t)=t^n-mA(t),  β(t)=t^n+1β̂(t)  (A(t)=cβ̂(t)x(t)â(t)).
 
Thus β̇(0)≠0 if and only if n=0. 
Since m<n, it implies that m<0. 
This contradicts the fact that m≥0. 
Therefore, γ cannot have a 5/2-cusp at t=0 by (2) in Theorem <ref>. 
We assume that β̇(0)=0, that is, n≥1. 
Under this situation, β̈(0)≠0 if and only if n=1. 
Since 0≤ m<1=n, we have m=0. 
Then it holds that (ℓ(t),β(t))=(tA(t),t^2β̂(t)). 
Noting A(0)≠0 and β̂(0)≠0, this pair (ℓ(t),β(t)) satisfy the conditions of (4) in Theorem <ref>. 
Therefore we have the assertion. 


We next consider singularities of curves obtained by Theorem <ref>. 
Although for the similar case of Theorem <ref>, criteria for j/i-cusps 
are obtained by using the data η and β (<cit.>), 

in our setting, the following assertions hold. 

Under the same assumptions as in Theorem <ref>, we have the following. 
(1) Suppose that β(t_0)=0 and the function α:(I,t_0)→ satisfying H(t)=α(t)J(t) is not a constant function. 
Then γ given by (<ref>) does not have neither a 3/2-cusp nor a 4/3-cusp at t_0. 
Moreover, γ has a 5/2-cusp (respectively, 5/3-cusp) at t_0
 if and only if β̇(t_0)α̇(t_0)≠0 (respectively, β̇(t_0)=0 and β̈(t_0)α̇(t_0)≠0).
(2) Suppose that the function α:(I,t_0)→ satisfying H(t)=α(t)J(t) is a constant c∈, i.e., α(t)=c. 
Then γ given by (<ref>) does not have j/i-cusps ((i,j)=(2,3),(2,5),(3,4),(3,5)).

(1) By Remark <ref>, a curve γ obtained by Theorem <ref> is a frontal but not a front. 
Thus we see that 3/2-cusps and 4/3-cusps do not appear on the curve γ. 
By (<ref>), the curvature (φ̇(t),β(t)) of a Legendre curve given by Theorem <ref> is 

    (φ̇(t),β(t))
    =(-β(t)X(t),β(t))  (X(t)=F(t)sinη(t)-G(t)cosη(t)F(t)^2+G(t)^2+2α(t)).

By the definitions of F(t), G(t) and η(t), we see that Ḟ(t_0)=Ġ(t_0)=η̇(t_0)=0 since β(t_0)=0. 
Thus Ẋ(t_0)=2α̇(t_0) holds. 
On the other hand, it follows that φ̈(t)=-β̇(t)X(t)-β(t)Ẋ(t) 
and ⃛φ(t)=-β̈(t)X(t)-2β̇(t)Ẋ(t)-β(t)Ẍ(t). 
Therefore φ̈(t_0)=-β̇(t_0)X(t_0) and ⃛φ(t_0)=-β̈(t_0)X(t_0)-2β̇(t_0)Ẋ(t_0) hold. 
By (2) and (4) of Theorem <ref>, we have the assertion. 

(2) By the proof of the assertion (1) and the Theorem <ref>, 
we see that the curvature (φ̇,β) of a Legendre curve (γ,ν) given by Theorem <ref> 
cannot satisfy the conditions for j/i-cusps ((i,j)=(2,3),(2,5),(3,4),(3,5)) when α(t)=c. 
Thus we have the conclusion.



The next, we consider the case of x_0=0.


Let γ:(I,t_0) → (^2,0) and γ:(I,t_0) → (^2,0) be smooth curves, 
let :(I × [0,2π),(t_0,θ_0)) → (^3,0) and  :(I× [0,2π),(t_0,θ_0)) → (^3,0) 
be surfaces of revolution of the frontals  around z-axis, respectively. 
(1) If there exist diffeomorphism germs ϕ:(I,t_0) → (I,t_0) and ψ: (^2,0) → (^2,0) of the form 
ψ(X,Z)=(X,φ(X^2,Z)) such that ψ∘γ=γ∘ϕ, then 
there exist diffeomorphism germs Φ:(I × [0,2π),(t_0,θ_0)) → (I× [0,2π),(t_0,θ_0)) of the form Φ(t,θ)=(ϕ(t),θ) and Ψ:(^3,0) →  (^3,0) of the form 
Ψ(X,Y,Z)=(X,Y,φ (X^2+Y^2,Z)) 
such that Ψ∘=∘Φ.
(2) If there exist diffeomorphism germs Φ:(I × [0,2π),(t_0,θ_0)) → (I× [0,2π),(t_0,θ_0)) of the form Φ(t,θ)=(ϕ(t),θ) and Ψ:(^3,0) →  (^3,0) of the form Ψ(X,Y,Z)=(X,Y,φ(X^2+Y^2,Z)) such that Ψ∘=∘Φ, then there exists a diffeomorphism germ ψ: (^2,0) → (^2,0) of the form ψ(X,Z)=(X,φ(X^2,Z))
such that ψ∘γ=γ∘ϕ


(1) It is easy to see that Φ(t,θ)=(ϕ(t),θ) and Ψ(X,Y,Z)=(X,Y,φ(X^2+Y^2,Z)) are diffeomorphism germs. 
By the assumption of equivalence relation between curves, 
we have (x(t),φ(x^2(t),z(t)))=(x(ϕ(t)),z(ϕ(t))). 
Therefore, 

    Ψ∘(t,θ)   =Ψ(x(t)cosθ,x(t) sinθ,z(t))
       =(x(t) cosθ,x(t)sinθ, φ(x^2(t),z(t)))
       =(x(ϕ(t))cosθ, x(ϕ(t))sinθ, z(ϕ(t)))
       =∘Φ(t,θ).

(2) By the assumption, ϕ and ψ are diffeomorphism germs.
Since Ψ∘(t,θ)=∘Φ(t,θ), we have 

    (x(t)cosθ, x(t)sinθ,φ(x^2(t),z(t)))=(x(ϕ(t))cosθ,x(ϕ(t))sinθ,z(ϕ(t))).

It follows that x(t)=x(ϕ(t)) and φ(x^2(t),z(t))=z(ϕ(t)) and hence ψ∘γ(t)=γ∘ϕ(t).


By the definition, we have the following result. 



Let γ=(x,z), γ=(x,z):(I,t_0) → (^2,0) be smooth curves.
If x is ℛ-equivalent to x and z is ℒ-equivalent to z at t_0, then γ is equivalent to γ 
in the sense of Theorem <ref> (1).



In detail for ℒ-equivalence see <cit.>.

If (f)(t_0)=k+1 (see, (<ref>)) then f is ℛ-equivalent to ± t^k+1 (cf. <cit.>).



Let γ=(x,z):(I,t_0) → (^2,0) be a smooth curve.
If  x satisfies (x)(t_0) =k+1 and z is regular at t_0, then γ is equivalent to t ↦ (± t^k+1,t) in the sense of Theorem <ref> (1).




Since x is ℛ-equivalent to ± t^k+1 and z is ℒ-equivalent to t, we have the result by Proposition <ref>.





§ EXAMPLES


We give concrete examples of surfaces of revolution with singular points.

Let x: I→ be a function given by 

    x(t)=exp(∫t dt)=exp(∫cos t/sin t  dt)=exp (logsin t)=sint,
 
where I=(0,π). 
This is a solution of (<ref>) when α(t)=-1 and β(t)=t. 
Thus we obtain the constant Gauss curvature -1 surface of revolution with singular point at t=π/2. 
By Theorem <ref>, we have z(t) as 
z(t)=cost+log(tan (t/2)), 
and hence the surface of revolution (t,θ)=(x(t)cosθ,x(t)sinθ,z(t)) 
of γ(t)=(x(t),z(t)) is a pseudo-sphere, see Figure <ref> (cf. <cit.>). 
In this case, we can take ν(t)=(cost,-sint) and (t)=(sint,cost). 
Thus the curvature of (γ,ν) is (ℓ(t),β(t))=(-1,t). 
Since (cost)(π/2)=(t)(π/2)=1, 
γ actually has a 3/2-cusp at t=π/2 (cf. Proposition <ref>). 
Moreover, since ℓ(t) ≠ 0 for all t∈(0,π), 
one can consider at least an evolute ℰv^z(t,θ) of (t,θ) (cf. Proposition <ref> (2)). 
On the other hand, 
x(t)=sint≠0 on I and a(t)=cost vanishes at t=π/2. 
Thus the function x(t)/a(t)=tant is defined on Ĩ=I∖{π/2}. 


We now take ^z(t,θ) as (costcosθ,costsinθ,-sint). 
Then an evolute ℰv^z(t,θ) is given by

    ℰv^z(t,θ)   =(t,θ)-β(t)ℓ(t)^z(t,θ)
    =(tcosθ,tsinθ,log(tant2)).

Setting w(t)=log(tan(t/2)), then dw/dt=t≠0 for any t∈(0,π). 
Thus this gives a parameter change and we have t=2arctan(exp(w)). 
Hence it follows that 

    ℰv^z(w,θ)=(coshwcosθ,coshwsinθ,w).

This is a catenoid. 
On the other hand, we can define another evolute ℰv^z on Ĩ×[0,2π) by 

    ℰv^z(t,θ)=(t,θ)-x(t)a(t)^z(t,θ)=(0,0,log(tant2)+t).

Since Ĩ×[0,2π) is dense in I× [0,2π), the evolute ℰv^z can be extended on I×[0,2π) continuously. 
This evolute ℰv^z degenerates to the line (see Figure <ref>).






Let α(t) be a function satisfying H_F^z=α J_F^z for a surface of revolution (t,θ) 
of the curve γ(t)=(x(t),z(t)). 
Let c_1 and c_2 be initial values of F(t) and G(t), respectively. 
Then γ is given by (<ref>). 
We put α(t)=0, β(t)=t (0≤ t<2π), then γ(t) and ν(t) are given by  

    γ(t)   =(√(c_2^2 + (c_1 - t^2/2)^2),-c_2 log(-2 c_1 + t^2 + √(4 c_2^2 + (-2 c_1 + t^2)^2))),
    ν(t)   =(-c_1 -t^2/2√(c_2^2 + (c_1 - t^2/2)^2),-c_2√(c_2^2 + (c_1 - t^2/2)^2))

If we take c_1=1/5 and c_2=3/10, then we have a catenoid with singularity, see Figure <ref>. 


We put α(t)=-1/2 and β(t)=t (0≤ t <2π). 
Then the surface of revolution is a constant mean curvature with singular point at t=0. 
In this case, γ(t) is given by 


    γ(t)=(√((c_1-sin (t^2/2))^2+(c_2-cos (t^2/2))^2).,
    .∫tx(t)(1-2c_2cos (t^2/2)-2c_1sin (t^2/2))dt),


where x(t) is the first component of γ(t).
We note that the integral of the z component can be expressed concretely by using the Appell hypergeometric function F_1 (cf. <cit.>). 
However it is too long, so we omit to write down here. 
Moreover, we see that 


    ν(t)=(-1+c_1sin (t^2/2)+c_2cos (t^2/2)√((c_1-sin (t^2/2))^2+(c_2-cos (t^2/2))^2),.
    .-c_1cos (t^2/2)+c_2sin (t^2/2)√((c_1-sin (t^2/2))^2+(c_2-cos (t^2/2))^2)).


If we take c_1=1/5 and c_2=3/10, then we have an unduloidal surface (left-hand side of Figure <ref>). 
On the other hand, if we take c_1=1/5 and c_2=3/4, we obtain a nodoidal surface (center of Figure <ref>).
Moreover, we put α(t)=t, β(t)=sint (0<t<2π) and c_1=c_2=1/10. 
Then functions F(t) and G(t) are written as 


    F(t)=c_1 - ∫cos(2 (-t cos t + sin t)) sin t dt,   
    G(t)=c_2 - ∫sin(2 (-t cos t + sin t)) sin t dt.


In this case, by F(t), G(t) above and (1) of Proposition <ref>, we have 
the surface of  revolution with a 5/2-cusp (at t=π) whose mean curvature is bounded, see right-hand side of Figure <ref>. 







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Masatomo Takahashi, 


Muroran Institute of Technology, Muroran 050-8585, Japan,


E-mail address: mailto:masatomo@mmm.muroran-it.ac.jpmasatomo@mmm.muroran-it.ac.jp




Keisuke Teramoto,


Faculity of Science, Kobe University, Rokko 1-1, Kobe 657-8501, Japan


E-mail address: mailto:teramoto@math.kobe-u.ac.jpteramoto@math.kobe-u.ac.jp

