Concircular Hypersurfaces and Concircular Helices in Space Forms

Abstract

In this paper, we find a full description of concircular hypersurfaces in space forms as a special family of ruled hypersurfaces. We also characterize concircular helices in 3-dimensional space forms by means of a differential equation involving the concircular factor and their curvature and torsion, and we show that the concircular helices are precisely the geodesics of the concircular surfaces.

1 Introduction

Generalized helices, slant helices, and rectifying curves are well-known examples of curves satisfying a certain condition with respect to a special vector field. Generalized helices are defined by the property that their tangents make a constant angle with a fixed direction. Slant helices are defined by the property that their principal normals make a constant angle with a constant vector field, [5], and rectifying curves are defined as the curves whose position vector is orthogonal to its principal normal vector field (i.e., the position vector lies in the rectifying plane), [1]. Moreover, these curves are characterized as the geodesics of some special ruled surfaces: generalized helices in cylinders, slant helices in helix surfaces, [8], and rectifying curves in conical surfaces, [2]. Motivated by these examples of curves and surfaces, the authors in [9] have extended the above conditions, and have introduced the notion of concircular submanifold in the Euclidean space $\mathbb {R}^{n}$. In particular, they characterize concircular helices in $\mathbb {R}^{3}$ by means of a differential equation involving their curvature and torsion. Moreover, they also find a full description of concircular surfaces in $\mathbb {R}^{3}$ as a special family of ruled surfaces and characterize the concirculares helices in $\mathbb {R}^{3}$ as the geodesics of these surfaces.

In this paper, we generalize the results obtained in [9] to space forms of nonzero constant curvature. Recall that a vector field $V\in \mathfrak {X}(M)$ on a Riemannian manifold M, with Levi–Civita connection $\nabla$, is said to be concircular if $\nabla V=\mu I$, where $\mu \in \mathcal {C}^\infty (M)$ is a differentiable function called the concircular factor, [3, 4, 10]. We denote by ${{\,\textrm{Con}\,}}(M)$ the set of concircular vector fields of M. The following definition extends the one given in [9]:

Definition 1

Let $\mathbb {M}^n(C)$ be an n-dimensional space form of constant curvature C. A submanifold $M^m\subset \mathbb {M}^n(C)$ is said to be a concircular submanifold if there exists a concircular vector field $V\in {{\,\textrm{Con}\,}}(\mathbb {M}^n(C))$ (called the axis of $M^m$) such that $\left\langle \textbf{n},V \right\rangle$ is a constant function along $M^m$, $\textbf{n}$ being any unit vector field in the first normal space of $M^m$.

In the particular case of a hypersurface, $M^{n-1}$ is said to be a concircular hypersurface (with axis V) if $\left\langle N,V \right\rangle$ is a constant function along $M^{n-1}$, N being a unit normal vector field. Another very interesting case appears when $m=1$: a (non-geodesic) unit speed curve $\gamma$ in $\mathbb {M}^n(C)$ is said to be a concircular helix (with axis V) if $\left\langle N_\gamma ,V \right\rangle$ is a constant function along $\gamma$, $N_\gamma$ being the principal normal vector field of $\gamma$.

This paper is organized as follows: In Sect. 2, we characterize concircular vector fields in $\mathbb {M}^n(C)$, see Theorem 1. In Sect. 3, we present several properties of concircular hypersurfaces in $\mathbb {M}^n(C)$, see Propositions 4 and 5, and we finish this section with the characterization of all concircular hypersurfaces in $\mathbb {M}^n(C)$, see Theorem 6. Section 4 contains a characterization of all concircular helices in $\mathbb {M}^3(C)$, see Proposition 9 and Theorem 10. Finally, Sect. 5 contains the characterization of geodesics curves of concircular surfaces, see Proposition 11, and this characterization is used to show that concircular helices in $\mathbb {M}^3(C)$ can be described as the geodesics of the concircular surfaces, see Theorem 12.

2 Concircular Vector Fields in Space Forms

Let $\mathbb {M}^n(C)$ denote the n-dimensional space form of nonzero constant curvature C. Then $\mathbb {M}^n(C)$ stands for a sphere $\mathbb {S}^{n}\subset \mathbb {R}^{n+1}$ or a hyperbolic space $\mathbb {H}^{n}\subset \mathbb {R}^{n+1}_{1}$ according to $C>0$ or $C<0$, respectively. Put $C=\varepsilon /R^2$, with $\varepsilon =(-1)^\nu$, where $\nu \in \{0,1\}$ is the index of the ambient space $\mathbb {R}^{n+1}_{\nu }$ that contains $\mathbb {M}^n(C)$. $\mathbb {M}^n(C)$ can be described as follows:

$$\begin{aligned} \mathbb {M}^n(C)=\{p=(x_1,x_2,\ldots ,x_{n+1})\in \mathbb {R}^{n+1}_\nu \;|\;\left\langle p,p \right\rangle =1/C\}, \end{aligned}$$

where as usual $\mathbb {R}^{n+1}_\nu$ is the space $\mathbb {R}^{n+1}$ endowed with the flat metric

$$\begin{aligned} \left\langle , \right\rangle =\varepsilon \,\text {d}x_1^2+\text {d}x_2^2+\cdots +\text {d}x_{n+1}^2, \end{aligned}$$

$(x_1,x_2,\ldots ,x_{n+1})$ being the usual rectangular coordinates of $\mathbb {R}^{n+1}$.

Let us write $\nabla ^0$ and $\overline{\nabla }$ to denote the Levi–Civita connections of $\mathbb {R}^{n+1}_\nu$ and $\mathbb {M}^n(C)$, respectively. If $\phi :\mathbb {M}^n(C)\rightarrow \mathbb {R}^{n+1}_{\nu }$ denotes the usual isometric immersion (the position vector), then the Gauss formula is

$$\begin{aligned} \nabla ^0_XY=\overline{\nabla }_XY-C\left\langle X,Y \right\rangle \phi , \end{aligned}$$

(1)

for any vector fields X and Y tangent to $\mathbb {M}^n(C)$.

Given a point $p\in \mathbb {M}^n(C)$ and a unit vector $w\in T_p\mathbb {M}^n(C)$, the exponential map $\exp _p$ is given by

$$\begin{aligned} \exp _{p}(tw)=f\Big (\frac{t}{R}\Big )p+Rg\Big (\frac{t}{R}\Big )w, \end{aligned}$$

(2)

where functions f and g are given by $f(t)=\cos t$ and $g(t)=\sin t$ when $C>0$, or $f(t)=\cosh t$ and $g(t)=\sinh t$ when $C<0$. Note that $f^2+\varepsilon g^2=1$, $f'=-\varepsilon g$ and $g'=f$.

The following result characterizes the concircular vector fields.

Theorem 1

A vector field $V\in \mathfrak {X}(\mathbb {M}^n(C))$ is concircular if and only if V is the tangential part of a constant vector field $p_0$ in $\mathbb {R}^{n+1}_\nu$. Moreover, if $\mu$ is the concircular factor of V, then $V= p_0+\mu \phi$, where $\mu =-C\left\langle p_0,\phi \right\rangle$.

Proof

The curvature tensor of $\mathbb {M}^n(C)$ is given by

$$\begin{aligned} R_{XY}Z=\overline{\nabla }_{[X,Y]}Z-\overline{\nabla }_X(\overline{\nabla }_YZ)+\overline{\nabla }_Y(\overline{\nabla }_XZ). \end{aligned}$$

Then, if V is a concircular vector field with concircular factor $\mu$, we have

$$\begin{aligned} R_{XV}V=V(\mu )X-X(\mu )V. \end{aligned}$$

(3)

On the other hand, since $\mathbb {M}^n(C)$ is a space of constant curvature C, its curvature tensor is given by

$$\begin{aligned} R_{XV}V=C\{\left\langle V,X \right\rangle V-\left\langle V,V \right\rangle X\}. \end{aligned}$$

(4)

By assuming that X and V are two linearly independent vector fields, from (3) and (4), we get $-C\left\langle V,X \right\rangle =X(\mu )$ and $-C\left\langle V,V \right\rangle =V(\mu )$, and therefore

$$\begin{aligned} -CV=\nabla \mu . \end{aligned}$$

(5)

Take the vector field $\psi =V-\mu \phi$, then

$$\begin{aligned} \nabla ^0_X\psi =\overline{\nabla }_XV-C\left\langle X,V \right\rangle \phi -X(\mu )\phi -\mu X. \end{aligned}$$

From here and again (5) we get $\psi$ is constant, and so there exists a constant vector field $p_0\in \mathbb {R}^{n+1}_\nu$ such that

$$\begin{aligned} p_0= V-\mu \phi ,\quad \text {with }\mu =-C\left\langle p_0,\phi \right\rangle . \end{aligned}$$

(6)

Conversely, let $V=\{p_0\}^\top$ be the tangential part of a constant vector in $\mathbb {R}^{n+1}_\nu$. Then we have (6), and by derivating there, we get $0=\overline{\nabla }_XV-C\left\langle X,V \right\rangle \phi -X(\mu )\phi -\mu X$, where X is any tangent vector field in $\mathbb {M}^n(C)$. Hence $\overline{\nabla }_XV=\mu X$ for any X, so that V is a concircular vector field with concircular factor $\mu$. $\square$

As a consequence of (5), we have the following result:

Corollary 2

In a space form $\mathbb {M}^n(C)$ of nonzero curvature C, the concircular factor is a nonconstant function.

Proposition 3

The set ${{\,\textrm{Con}\,}}(\mathbb {M}^n(C))$ of all concircular vector fields of $\mathbb {M}^n(C)$ is a real vector space of dimension $n+1$.

Proof

It is sufficient to show that each concircular vector field V is determined by one, and only one, constant vector $p_0$. Let us suppose that V is determined by two constant vectors $p_0$ and $q_0$, that is, $V=p_0+\mu _1\phi$ and $V=q_0+\mu _2\phi$, for certain differentiable functions $\mu _1$ and $\mu _2$. Then $p_0-q_0= (\mu _2-\mu _1)\phi$, and so $\mu _1=\mu _2$ and $p_0=q_0$. $\square$

3 Concircular Hypersurfaces

To begin with, let us show some examples of concircular hypersurfaces in $\mathbb {M}^n(C)$.

Example 1

A totally umbilical hypersurface $Q^{n-1}(c)$ in $\mathbb {M}^n(C)$ can be obtained as the intersection $\mathbb {M}^n(C)\cap H(p_0)$, where $H(p_0)$ is a hyperplane in $\mathbb {R}^{n+1}_{\nu }$ orthogonal to a constant vector $p_0\in \mathbb {R}^{n+1}_{\nu }$. If N denotes the unit vector field normal to $Q^{n-1}(c)$ in $\mathbb {M}^n(C)$, then $\left\langle N,V \right\rangle =\left\langle N,p_0\right\rangle$, where $V=p_0-C\left\langle p_0,\phi \right\rangle \phi$. By derivating here, we get $X\left\langle N,p_0\right\rangle =\left\langle -AX,p_0\right\rangle =0$, for any vector field X tangent to $Q^{n-1}(c)$, where A denotes the shape operator associated to N. Hence $\left\langle N,V \right\rangle$ is constant and so $Q^{n-1}(c)$ is a concircular hypersurface. We will say a totally umbilical hypersurface is a trivial concircular hypersurface.

Example 2

A conical hypersurface M in $\mathbb {M}^n(C)$ (with vertex at $p_0\in \mathbb {M}^n(C)$) can be described as follows: Let $P^{n-2}$ be an $(n-2)$-dimensional hypersurface in the unit hypersphere $S^{n-1}(1)$ of the tangent space $T_{p_0}\mathbb {M}^n(C)$. For $\epsilon >0$ sufficiently small, the map $\Psi :P^{n-2}\times (-\epsilon ,\epsilon )\rightarrow \mathbb {M}^n(C)$ given by

$$\begin{aligned} \Psi (v,t)=\exp _{p_0}(tv)=f\Big (\frac{t}{R}\Big )p_0+Rg\Big (\frac{t}{R}\Big )v, \end{aligned}$$

defines an immersion. The image $M=\Psi (P^{n-2}\times (-\epsilon ,\epsilon ))$ is said to be a conical hypersurface in $\mathbb {M}^n(C)$ (see [6, 7] in the case $n=3$). We can identify in a natural way $P^{n-2}$ with $P^{n-2}\times \{0\}$ and $P^{n-2}\times (-\epsilon ,\epsilon )$ with M, and then the unit vector field normal to M in $\mathbb {M}^n(C)$ is given without loss of generality by $N(v,t)=\eta (v)$, $\eta$ being the unit vector field normal to $P^{n-2}$ in $S^{n-1}(1)$. Hence, $\left\langle N,V \right\rangle =0$, for $V=p_0-C\left\langle p_0,\phi \right\rangle \phi$, showing that M is a concircular hypersurface.

Before addressing the characterization of the concircular hypersurfaces, we will present a couple of results.

Proposition 4

Given a hypersurface $M\subset \mathbb {M}^n(C)$, then there exists a concircular vector field parallel to its normal vector field along M if and only if M is a totally umbilical hypersurface in $\mathbb {M}^n(C)$.

Proof

Suppose there exists a concircular vector field V such that $V|_M=\lambda \,N$, for a nonzero differentiable function $\lambda$. Then we get $\mu \,X=X(\lambda )\,N-\lambda \,AX$, for any vector field X tangent to M, A being the shape operator associated to N. From that equation, we have that $\lambda$ is a nonzero constant and $AX=-(\mu /\lambda )\,X$. Now, from (5), we get $X(\mu )=0$ for any tangent vector field X, and so M is a totally umbilical hypersurface.

Conversely, if M is a totally umbilical hypersurface, then there exists a constant m such that $\overline{\nabla }_XN=-AX=-m\,X$, and then the vector field $N+m\phi$ is constant along M. This shows that N is collinear with a concircular vector field along M. $\square$

Let $M\subset \mathbb {M}^n(C)$ be a nontrivial concircular hypersurface with axis V, and write $\left\langle V,N \right\rangle =\lambda$, $\lambda$ being a constant. By decomposing V in its tangential and normal components, we have

$$\begin{aligned} V|_M=\alpha \,T+\lambda \,N, \end{aligned}$$

(7)

where T is a unit vector field tangent to M and $\alpha \ne 0$ (otherwise, M would be a trivial concircular hypersurface).

Proposition 5

The integral curves of T are geodesics in $\mathbb {M}^n(C)$.

Proof

By derivating (7) and using the Gauss and Weingarten equations, we obtain

$$\begin{aligned} \mu X=X(\alpha )T+\alpha \nabla _XT+\alpha \sigma (X,T)-\lambda AX, \end{aligned}$$

(8)

where $\nabla$ and $\sigma$ denote the Levi–Civita connection and the second fundamental form of M. From here, we get $\sigma (X,T)=0$, for any tangent vector field X, or equivalently

$$\begin{aligned} AT=0. \end{aligned}$$

(9)

By putting $X=T$ in (8) and using (9), we obtain

$$\begin{aligned} \nabla _TT=0. \end{aligned}$$

(10)

From here and (9), we deduce the result. $\square$

In what follows, we will characterize nontrivial concircular hypersurfaces in $\mathbb {M}^n(C)$. Let $Q^{n-1}(c)=H(p_0)\cap \mathbb {M}^n(C)$ be a totally umbilical hypersurface of constant curvature c, where $H(p_0)$ is a hyperplane in $\mathbb {R}^{n+1}_\nu$ orthogonal to a unit vector $p_0\in \mathbb {R}^{n+1}_\nu$. Let $P^{n-2}$ be a hypersurface of $Q^{n-1}(c)$, and denote by $\eta _1$ and $\eta _2$ the unit vector fields normal to $P^{n-2}$ in $Q^{n-1}(c)$ and normal to $Q^{n-1}(c)$ in $\mathbb {M}^n(C)$, respectively. For a real number a, define the unit vector field $W_a(p)=\cos (a)\;\eta _1(p)+\sin (a)\;\eta _2(p)$, where $p\in P^{n-2}$. The map $\Psi _a:P^{n-2}\times I_0\rightarrow \mathbb {M}^n(C)$ given by

$$\begin{aligned} \Psi _a(p,z)=\exp _p(zW_a(p))=f\Big (\frac{z}{R}\Big )p+Rg\Big (\frac{z}{R}\Big )W_a(p), \end{aligned}$$

defines an immersion, for an enough small interval $I_0$ around the origin. Let M denote the ruled hypersurface in $\mathbb {M}^n(C)$ given by $\Psi _a(P^{n-2}\times I_0)$. We can identify, in a natural way, $P^{n-2}$ with $\Psi _a(P^{n-2}\times \{0\})$ and $P^{n-2}\times I_0$ with M. Without loss of generality, we can assume that the unit vector field normal to M in $\mathbb {M}^n(C)$ is given by

$$\begin{aligned} N(p,z)=-\sin (a)\;\eta _1(p)+\cos (a)\;\eta _2(p). \end{aligned}$$

Since $P^{n-2}\subset H(p_0)\cap \mathbb {M}^n(C)$, it is not difficult to see that $p_0\in \text {span}\{\eta _2(p),\phi (p)\}$, for any $p\in P^{n-2}$, so we can write $p_0=A\eta _2+B\phi$, for two differentiable functions A and B. By taking derivative here, and using that $p_0$ is constant, we get that A and B are also constants. Let us consider the concircular vector field in $\mathbb {M}^n(C)$ given by $V=p_0-C\left\langle p_0,\phi \right\rangle \phi$. It is not difficult to see that $\left\langle V,N \right\rangle =\left\langle p_0,N \right\rangle =A\cos (a)$ is constant, and so M is a concircular hypersurface.

Note that a hypersurface M in $\mathbb {M}^n(C)$ is concircular if and only if there exists a point $p_0\in \mathbb {R}^{n+1}_\nu$ such that $\left\langle N,p_0 \right\rangle$ is constant, N being the unit normal vector field of M in $\mathbb {M}^n(C)$.

The main result of this section is the following. We will show that every nontrivial concircular hypersurface in $\mathbb {M}^n(C)$ can be obtained by the construction described above.

Theorem 6

Let $M\subset \mathbb {M}^n(C)$ be a nontrivial concircular hypersurface with axis V. Then there exists a hypersurface $P^{n-2}$ in a totally umbilical hypersurface $Q^{n-1}(c)\subset \mathbb {M}^n(C)$, such that M can be locally described by

$$\begin{aligned} \Psi _a(p,z)=\exp _p(zW_a(p))=f\Big (\frac{z}{R}\Big )p+Rg\Big (\frac{z}{R}\Big )W_a(p), \end{aligned}$$

(11)

where $a\in \mathbb {R}^{}$ and $(p,z)\in P^{n-2}\times I_0$, $I_0$ being an interval around the origin.

Proof

Let us suppose that the axis V of M is given by $V=p_0-C\left\langle p_0,\phi \right\rangle \phi$, for a constant vector $p_0\in \mathbb {R}^{n+1}_\nu$, and assume $\left\langle V,N \right\rangle =\lambda$. Pick a point q in M and let $Q^{n-1}(c)=H(p_0)\cap \mathbb {M}^n(C)$ be a totally umbilical hypersurface containing q, $H(p_0)$ being a hyperplane in $\mathbb {R}^{n+1}_\nu$ orthogonal to $p_0$. Since M is a nontrivial concircular hypersurface, then there is an $(n-2)$-dimensional submanifold $P^{n-2}\subset Q^{n-1}(c)\cap M$ with $q\in P^{n-2}$.

Let T be the unit vector field tangent to M which is collinear with the tangential component of V. From Proposition 5 we deduce that there exists a neighborhood U(q) of q in M given by

$$\begin{aligned} \textstyle U(q)=\{f\left( \frac{z}{R}\right) p+R\, g\left( \frac{z}{R}\right) T(p)\;|\;p\in U_1(q),\ z\in (-\varepsilon _2,\varepsilon _2)\}, \end{aligned}$$

where $U_1(q)$ is an neighborhood of q in $P^{n-2}$. Since T is orthogonal to $P^{n-2}$, but tangent to M, then there exist a differentiable function $a\in {\mathcal C}^\infty (P^{n-2})$ such that

$$\begin{aligned} T(p)=\cos (a(p))\;\eta _1(p) +\sin (a(p))\;\eta _2(p), \end{aligned}$$

and, up to the sign, the unit normal vector field N along $P^{n-2}$ is given by

$$\begin{aligned} N(p)=-\sin (a(p))\;\eta _1(p) +\cos (a(p))\;\eta _2(p). \end{aligned}$$

Then we have $\lambda =\left\langle V,N \right\rangle (p)=\left\langle p_0,N \right\rangle (p)=A\cos (a(p))$ is constant and so a is a constant function. Hence the open set U(q) can be rewritten as in (11).

$\square$

In Example 2, we have seen that the conical hypersurfaces are concircular hypersurfaces associated to the constant value $\lambda =0$. Now, we will prove that these hypersurfaces are the only ones that satisfy this property.

Proposition 7

Let $M\subset \mathbb {M}^n(C)$ be a concircular hypersurface, with axis V, such that $\left\langle V,N \right\rangle =0$, N being the unit normal vector field. Then M is a conical hypersurface.

Proof

From Theorem 6, we know that M can be locally described by

$$\begin{aligned} \Psi _a(p,z)=f\Big (\frac{z}{R}\Big )p+Rg\Big (\frac{z}{R}\Big )W_a(p),\quad W_a(p)=\cos (a)\;\eta _1(p) +\sin (a)\;\eta _2(p), \end{aligned}$$

where $(p,z)\in P^{n-2}\times I_0$, for certain $P^{n-2}\subset Q^{n-1}(c)\subset \mathbb {M}^n(C)$, $Q^{n-1}(c)=H(p_0)\cap \mathbb {M}^n(C)$ and $I_0\subset \mathbb {R}^{}$. Suppose that the interval $I_0$ is the largest possible interval.

The point $p_0$ can be written as $p_0=A\eta _2+B\phi$, for certain constants A and B related by $Ak-B=0$, where $k=\sqrt{|c-C|}$. From the equality $\left\langle V,N \right\rangle =A\cos (a)$, we get $\cos (a)=0$ (since A cannot vanish) and then $W_a(p)=\eta _2(p)$. Take $z_0\in \mathbb {R}^{}$ such that $f(z_0/R)=kRg(z_0/R)$, and define a differentiable function $\varphi :P^{n-2}\rightarrow \mathbb {R}^{n+1}_\nu$ by $\varphi (p)=\Psi _a(p,z_0)$. A straightforward computation yields

$$\begin{aligned} d\varphi _p(v)=\left[ f\Big (\frac{z_0}{R}\Big )-kRg\Big (\frac{z_0}{R}\Big )\right] v=0, \end{aligned}$$

for any $p\in P^{n-2}$ and $v\in T_pP^{n-2}$. Hence, $\varphi$ is a constant $q_0\in \mathbb {R}^{n+1}_\nu$ and this shows that M is a conical hypersurface with vertex at $q_0$. $\square$

4 Concircular Helices in $\mathbb {M}^3(C)$

We begin this section by showing a couple of examples of concircular helices in $\mathbb {M}^3(C)$. Let $\gamma :I\rightarrow \mathbb {M}^3(C)$ be an arclength parametrized curve satisfying the following Frenet–Serret equations:

$$\begin{aligned} T_\gamma '(s)= & {} \nabla ^0_{T_\gamma }T_\gamma (s)= -C\gamma (s)+\kappa _\gamma (s)\,N_\gamma (s),\nonumber \\ N_\gamma '(s)= & {} \nabla ^0_{T_\gamma }N_\gamma (s) = -\kappa _\gamma (s)\,T_\gamma (s)+\tau _\gamma (s)\,B_\gamma (s),\nonumber \\ B_\gamma '(s)= & {} \nabla ^0_{T_\gamma }B_\gamma (s) = -\tau _\gamma (s)\,N_\gamma (s), \end{aligned}$$

(12)

where $\nabla ^0$ denotes the Levi–Civita connection on $\mathbb {R}^{4}_\nu$. As usual, $\kappa _\gamma$ and $\tau _\gamma$ are called the curvature and torsion of $\gamma$.

Example 3

Planar curves, i.e., curves $\gamma$ with zero torsion. These curves live in a surface $M^2(C)$ totally geodesic in $\mathbb {M}^3(C)$. That means there is a constant vector $p_0$ in $\mathbb {R}^{4}_\nu$ such that $M^2(C)=\mathbb {M}^3(C)\cap H_0(p_0)$, where $H_0(p_0)$ is the hyperplane through the origin orthogonal to $p_0$. Since $\left\langle N_\gamma , V \right\rangle =0$, for $V=p_0-C\left\langle p_0,\phi \right\rangle \phi$, $\gamma$ is a concircular helix.

Example 4

Rectifying curves, i.e., curves for which there exists a point $p_0$ in $\mathbb {M}^3(C)$ such that the geodesics connecting $p_0$ with $\gamma (s)$ are orthogonal to the principal normal geodesic starting from $\gamma (s)$, see [6, 7]. The nonplanar rectifying curves are characterized by the condition $\left\langle N_\gamma ,p_0\right\rangle =0$. Then $\gamma$ is a concircular helix since $\left\langle N_\gamma ,V \right\rangle =0$ for the concircular vector field $V=p_0-C\left\langle p_0,\phi \right\rangle \phi$.

In the following result, we show that the restriction to a curve $\gamma$ of a concircular vector field V is a vector field along $\gamma$ satisfying a property similar to that of concircular vector fields. For this reason, such a vector field will be called a concircular vector field along a curve.

Proposition 8

Let $\gamma :I\rightarrow \mathbb {M}^n(C)$ be a differentiable curve and consider a vector field $\textbf{v}$ along $\gamma$. Then $\textbf{v}$ is the restriction to $\gamma$ of a concircular vector field V on $\mathbb {M}^n(C)$ if and only if $\frac{\overline{D}\textbf{v}}{dt}=\omega \,T_\gamma$, where $\omega :I\rightarrow \mathbb {R}^{}$ is a differentiable function with $\omega '=-C\left\langle \textbf{v},T_\gamma \right\rangle$, $\overline{D}$ being the covariant derivative along $\gamma$.

Proof

Let us assume that V is an extension of $\textbf{v}$ to $\mathbb {M}^n(C)$ which is a concircular vector field. Then there exists a constant vector $p_0\in \mathbb {R}^{n+1}_\nu$ such that $p_0=V+C\left\langle p_0,\phi \right\rangle \phi$. If X is a local extension of the tangent vector $T_\gamma$ then

$$\begin{aligned} \frac{\overline{D}\textbf{v}}{dt}=\overline{\nabla }_XV|_\gamma =(\mu X)|_\gamma =\omega T_\gamma , \end{aligned}$$

where the differentiable function $\omega :I\rightarrow \mathbb {R}^{}$ is the restriction of $\mu$ along $\gamma$. On the other hand, from Theorem 1 we have $\mu =-C\left\langle p_0,\phi \right\rangle$, and then $\omega =-C\left\langle p_0,\gamma \right\rangle$ and $\omega '=-C\left\langle \textbf{v},T_\gamma \right\rangle$.

To prove the converse, let us consider the vector field Y along $\gamma$ given by $Y=\textbf{v}-\omega \gamma$. By derivating here with the Euclidean derivative, we obtain that Y is a constant vector $p_0\in \mathbb {R}^{n+1}_{\nu }$ along $\gamma$. By defining $V=p_0-C\left\langle p_0,\phi \right\rangle \phi$ and bearing Theorem 1 in mind, we deduce $\textbf{v}$ is the restriction to $\gamma$ of a concircular vector field. $\square$

For simplicity, and since $\omega (s)=\mu (\gamma (s))$, in what follows we will use $\mu (s)$ instead of $\omega (s)$. In the following, we will characterize the concircular helices in $\mathbb {M}^3(C)$. Note that $\gamma$ is a concircular helix if and only if there exists a point $p_0\in \mathbb {R}^{4}_\nu$ such that $\left\langle N_\gamma ,p_0 \right\rangle =\lambda$ constant.

Since the case $\lambda =0$ reduces to planar or rectifying curves, we will exclude this case from our study. A concircular helix $\gamma$ (with axis V) in $\mathbb {M}^3(C)$ is said to be proper if $\gamma$ is a nonplanar curve with $\lambda \ne 0$, $\lambda$ being the constant function $\left\langle N_\gamma ,V \right\rangle$.

Let $\gamma (s)\subset \mathbb {M}^3(C)\subset \mathbb {R}^{4}_\nu$ be an arclength parametrized concircular helix, and suppose it is a proper one. From Proposition 8, we can write

$$\begin{aligned} \textbf{v}(s)=V(\gamma (s))=t(s)\,T_\gamma (s)+\lambda \,N_\gamma (s)+z(s)\,B_\gamma (s), \end{aligned}$$

(13)

for certain differentiable functions $t$ and $z$. To simplify the writing, we will eliminate the s parameter. By derivating in (13), we get

$$\begin{aligned} \mu \,T_\gamma =(t'-\lambda \kappa _\gamma )\,T_\gamma + (t\kappa _\gamma -z\tau _\gamma )\,N_\gamma + (z'+\lambda \tau _\gamma )\,B_\gamma , \end{aligned}$$

and then

$$\begin{aligned} t'-\lambda \kappa _\gamma =\mu ,\quad t\kappa _\gamma -z\tau _\gamma =0,\quad z'+\lambda \tau _\gamma =0. \end{aligned}$$

(14)

Note that $z\ne 0$ since $\gamma$ is a proper concircular helix. Now we distinguish two cases, according to $t/z$ (called the rectifying slope of $\gamma$) is a constant or not.

Case 1: the rectifying slope $t/z$ of $\gamma$ is a constant function. Then the Lancret curvature $\rho =\tau _\gamma /\kappa _\gamma$ is constant, and using the first and third equations of (14), we get

$$\begin{aligned} \kappa _\gamma =\frac{-1}{\lambda (1+\rho ^2)}\mu \qquad \text {and}\qquad \tau _\gamma =\rho \kappa _\gamma , \end{aligned}$$

(15)

where the function $\mu$ satisfies

$$\begin{aligned} \mu ''+C\frac{\rho ^2}{1+\rho ^2}\mu =0. \end{aligned}$$

(16)

Now we will show that Eqs. (15) and (16) characterize proper concircular helices with constant rectifying slope. Let $\gamma$ be a curve satisfying these two equations. Define a function $z$ by

$$\begin{aligned} z=-\frac{\mu '}{C\rho }, \end{aligned}$$

and consider the vector field $\textbf{v}=z\,D_\gamma + \lambda N_\gamma$, where $D_\gamma =\rho T_\gamma +B_\gamma$ is the modified Darboux vector and $\lambda =-1/(m(1+\rho ^2))$. Then

$$\begin{aligned} \overline{\nabla }_{T_\gamma }\textbf{v}=\frac{\rho }{1+\rho ^2}\mu \,(\rho T_\gamma +B_\gamma )-\lambda m \mu T_\gamma +\lambda \rho m \mu B_\gamma =\mu T_\gamma . \end{aligned}$$

Since $\mu '=-C\left\langle \textbf{v},T_\gamma \right\rangle$, from Proposition 8 we get $\gamma$ is a concircular helix.

Hence, we have shown the following result:

Proposition 9

Let $\gamma$ be an arclength parametrized nonplanar curve in $\mathbb {M}^3(C)$. Then $\gamma$ is a proper concircular helix with constant rectifying slope if and only if its curvature and torsion are given by $\kappa _\gamma =m\mu$ and $\tau _\gamma =\rho \kappa _\gamma$, where m and $\rho$ are nonzero constants and the function $\mu$ satisfies (16).

Case 2: the rectifying slope $t/z$ is a nonconstant function. From the first equation of (14), bearing in mind that $\mu '=-C\left\langle \textbf{v},T_\gamma \right\rangle$, we have

$$\begin{aligned} \mu ''+C\mu =-C\lambda \kappa _\gamma , \end{aligned}$$

(17)

and then the second and third equations of (14) lead to

$$\begin{aligned} \Big (\frac{\mu '}{\rho }\Big )'=C\lambda \tau _\gamma . \end{aligned}$$

(18)

As before, we will show that equations (17) and (18) characterize concircular helices (when $\rho$ is a nonconstant function). Let $\gamma$ be an arclength parametrized curve satisfying (17) and (18), for a constant $\lambda$ and a differentiable function $\mu$. Define the functions

$$\begin{aligned} t=-\frac{\mu '}{C}\qquad \text {and}\qquad z=\frac{t}{\rho }, \end{aligned}$$

(19)

and consider the vector field $\textbf{v}$ along $\gamma$ given by

$$\begin{aligned} \textbf{v}=zD_\gamma +\lambda N_\gamma =t\,T_\gamma +\lambda \,N_\gamma +z\,B_\gamma . \end{aligned}$$

(20)

From (17), we get $t'-\lambda \kappa _\gamma =\mu$, and from (18), we obtain $z'=-\lambda \tau _\gamma$. Then by derivating in (20), we have

$$\begin{aligned} \overline{\nabla }_{T_\gamma }\textbf{v}= (t'-\lambda \kappa _\gamma )T_\gamma +(t\kappa _\gamma -z\tau _\gamma )N_\gamma +(\lambda \tau _\gamma +z')B_\gamma =\mu \,T_\gamma . \end{aligned}$$

Therefore, since $\mu '=-C\left\langle \textbf{v},T_\gamma \right\rangle$, from Proposition 8, we deduce $\gamma$ is a concircular helix.

In conclusion, putting cases 1 and 2 together, we have proved the following result. Note that equations (17) and (18) are equivalent to (15) and (16) in the case $\rho$ constant.

Theorem 10

Let $\gamma$ be an arclength parametrized nonplanar curve in $\mathbb {M}^3(C)$. Then $\gamma$ is a proper concircular helix if and only if Eqs. (17) and (18) are satisfied, for a constant $\lambda \in \mathbb {R}^{}$ and a differentiable function $\mu$. Moreover, the axis V of $\gamma$ is the extension of the vector field $\textbf{v}$ given in (20), t and z being the differentiable functions given in (19).

5 Geodesics of Concircular Surfaces

Let M be a nontrivial concircular surface in $\mathbb {M}^3(C)$ with axis V, and let us consider $\gamma (s)$ an arclength parametrized geodesic of M. From Theorem 6, we can assume that $\gamma$ is locally written as X(t(s), z(s)), where X is the parametrization (11). Here, the submanifold $P^{n-2}$ of Theorem 6 is a curve $\delta (t)$. Since the principal normal vector field $N_\gamma$ of $\gamma$ is collinear with the unit normal vector field N of M in $\mathbb {M}^3(C)$, then $\left\langle N_\gamma ,V|_\gamma \right\rangle$ is constant, and then $\gamma$ is a concircular helix in $\mathbb {M}^3(C)$. The goal of this section is to prove the converse.

First, we are going to obtain the equations of geodesics in a concircular surface M. Let $\gamma (s)=X\big (t(s),z(s)\big )$ be an arclength parametrized geodesic of M, with $\kappa _\gamma >0$. Then $T_\gamma (s)=t'(s)X_t(t(s),z(s))+z'(s)X_z(t(s),z(s))$ and so there exists a differentiable function $\theta$ such that

$$\begin{aligned} t'(s)\sqrt{E(t(s),z(s))}= & {} \sin \theta (s), \end{aligned}$$

(21)

$$\begin{aligned} z'(s)= & {} \cos \theta (s). \end{aligned}$$

(22)

Hence $T_\gamma (s)=\sin \theta (s)\,T_\delta (t(s))+\cos \theta (s)\,X_z(t(s),z(s))$, and by taking derivative here, we get

$$\begin{aligned} -C\gamma (s)+\kappa _\gamma (s)\,N_\gamma (s)= & {} \;\theta '(s)\big (\cos \theta (s) T_\delta (t(s))-\sin \theta (s) X_z(t(s),z(s))\big )\nonumber \\{} & {} +\;\sin \theta (s) t'(s)\big (k\eta (t(s))-C\delta (t(s))+\kappa _\delta (t(s)) N_\delta (t(s))\big )\nonumber \\{} & {} +\;\cos \theta (s) \big (t'(s)X_{tz}(t(s),z(s))+ z'(s)X_{zz}(t(s),z(s))\big ).\nonumber \\ \end{aligned}$$

(23)

Bearing in mind that $\{ T_\delta ,\,X_z,\,N,\,\frac{1}{R} \gamma \}$ is an orthonormal frame of $\mathbb {R}^{4}_\nu$ along $\gamma$, we have

$$\begin{aligned} \delta (t(s))&= \textstyle f\big (\frac{z(s)}{R}\big )\gamma (s)-R g\big (\frac{z(s)}{R}\big ) X_z(t(s),z(s)),\\ N_\delta (t(s))&= \textstyle -\sin (a) N(t(s),z(s))+\cos (a)\Big (f\big (\frac{z(s)}{R}\big )X_z(t(s),z(s))+ \frac{\varepsilon }{R}g\big (\frac{z(s)}{R}\big )\gamma (s)\Big ),\\ \eta (t(s))&= \textstyle \cos (a) N(t(s),z(s))+\sin (a)\Big (f\big (\frac{z(s)}{R}\big )X_z(t(s),z(s))+ \frac{\varepsilon }{R}g\big (\frac{z(s)}{R}\big )\gamma (s)\Big ),\\ X_{tz}(t(s),z(s))&= \textstyle -\Big (\frac{\varepsilon }{R}g\big (\frac{z(s)}{R}\big )+ f\big (\frac{z(s)}{R}\big )\big (\sin (a)k+\cos (a)\kappa _\delta (t(s))\big )\Big )T_\delta (t(s)), \\ X_{zz}(t(s),z(s))&= -C\gamma (s). \end{aligned}$$

From these equations, jointly with (23) and the fact that $\gamma$ is a geodesic in M (and so $N_\gamma (s)=N(\gamma (s))$; the case $N_\gamma (s)=-N(\gamma (s))$ is similar), we deduce

$$\begin{aligned} \theta '(s)= & {} \textstyle t'(s)\Big (CR g\big (\frac{z(s)}{R}\big )+ f\big (\frac{z(s)}{R}\big )\big (\sin (a)k+\cos (a)\kappa _\delta (t(s))\big )\Big ), \end{aligned}$$

(24)

$$\begin{aligned} \kappa _\gamma (s)= & {} \sin \theta (s) t'(s)\big (k\cos (a)-\sin (a)\kappa _\delta (t(s))\big ). \end{aligned}$$

(25)

On the other hand, it is easy to see that $B_\gamma (s)=\cos \theta (s) T_\delta (t(s))-\sin \theta (s) X_z(t(s),z(s))$ and by taking derivative here, we have

$$\begin{aligned} \tau _\gamma (s)=\cos \theta (s) t'(s)\big (-\cos (a)k+\sin (a)\kappa _\delta (t(s))\big ). \end{aligned}$$

(26)

Hence, we have shown the following result:

Proposition 11

Let M be a nontrivial concircular surface in $\mathbb {M}^3(C)$, locally parametrized by (11). An arclength parametrized curve $\gamma (s)=X\big (t(s),z(s)\big )$, with $\kappa _\gamma >0$, is a geodesic if and only if there is a differentiable function $\theta (s)$ such that Eqs. (21), (22) and (24) are satisfied. Moreover, the curvature and torsion of $\gamma$ are given by (25) and (26), respectively.

We finish this section with the following characterization of concircular helices in $\mathbb {M}^3(C)$.

Theorem 12

Let $\gamma (s)$ be an arclength parametrized curve in $\mathbb {M}^3(C)$, $\kappa _\gamma >0$. Then $\gamma$ is a proper concircular helix if and only if $\gamma$ is (congruent to) a geodesic of a proper concircular surface.

Proof

We need only prove the direct implication. Let $\gamma (s)$ be a arclength parametrized proper concircular helix in $\mathbb {M}^3(C)$ with axis $V=p_0-C\left\langle p_0,\phi \right\rangle \phi$, such that $\left\langle N_\gamma ,V \right\rangle =\lambda$ is constant along $\gamma$. Let M be the ruled surface with base curve $\gamma$ and director curve $D_\gamma$ (the unit Darboux vector field of $\gamma$), which can be parametrized as follows:

$$\begin{aligned} X(s,z)=f\Big ({\frac{z}{R}}\Big )\gamma (s)+R\,g\Big ({\frac{z}{R}}\Big )\, \left( \frac{\rho (s)}{\sqrt{1+\rho (s)^2}}T_\gamma (s)+\frac{1}{\sqrt{1+\rho (s)^2}}B_\gamma (s)\right) . \end{aligned}$$

(27)

Since $X_s\in \text {span}\{T_\gamma ,B_\gamma \}$ and $X_z\in \text {span}\{\gamma ,\,T_\gamma ,\,B_\gamma \}$, we obtain that the unit normal vector field N is collinear with the principal normal vector field $N_\gamma$. From here, we conclude that M is a concircular surface in $\mathbb {M}^3(C)$ with axis V, and that $\gamma$ is a geodesic in M. This concludes the proof. $\square$