Concircular Helices

• Concircular helices are unit-speed curves in constant curvature 3-manifolds with a principal normal that projects constantly on a concircular field, generalizing classical helices.
• They are characterized by a closed system of ODEs linking curvature, torsion, and the concircular factor, enabling precise analytic descriptions in various space forms.
• These curves serve as geodesics on concircular surfaces, unifying generalized, slant, and rectifying helices and offering insights into differential geometric applications.

A concircular helix is a unit-speed curve in a Riemannian 3-manifold of constant sectional curvature for which the principal normal projects constantly onto a specific type of vector field, called a concircular field. These curves generalize classical helices and include them—as well as planar and rectifying curves—as special cases. Concircular helices admit a complete analytic characterization in terms of their curvature and torsion, and in all space forms are precisely the geodesics of a special family of ruled hypersurfaces called concircular surfaces (Lucas et al., 25 Jan 2026, Lucas et al., 27 Jan 2026).

1. Definitions and Characterizations

Let $\,^3(C)$ denote the simply-connected, complete Riemannian 3-manifold of constant sectional curvature $C$. This space is $S^3$ when $C>0$, $\mathbb{R}^3$ for $C=0$, and $H^3$ for $C<0$. The Levi-Civita connection is denoted $\nabla$. A vector field $V$ on $\,^3(C)$ is concircular if

$\nabla_X V = \mu X$

for every tangent $X$, with $\mu : \,^3(C) \to \mathbb{R}$ a smooth function, called the concircular factor. In Euclidean space, the canonical form is $Y(p) = \mu p + v$ for $v \in \mathbb{R}^3$.

Given a unit-speed curve $\gamma: I \to \,^3(C)$ with Frenet frame $\{T,N,B\}$, curvature $\kappa(s)>0$, and torsion $\tau(s)$, the curve $\gamma$ is called a concircular helix if there exists a concircular field $V$ (with factor $\alpha(s) := \mu(s)$) such that the principal normal $N(s)$ has constant projection onto $V$:

$$\langle N(s), V(\gamma(s)) \rangle = \lambda \text{ (constant)}.$$

The decomposition along the curve reads

$V|_\gamma = t(s) T(s) + \lambda N(s) + z(s) B(s),$

with $t(s)$ and $z(s)$ smooth.

2. Fundamental ODE Systems

For concircular helices in space forms, the construction leads to a system of ODEs connecting the Frenet invariants and concircular factor: \begin{align*} t' - \lambda \kappa &= \alpha, \ t\kappa - z \tau &= 0, \ z' + \lambda \tau &= 0, \ \alpha' + C t &= 0. \end{align*} Setting $z = \frac{t\kappa}{\tau}$, differentiating, and eliminating auxiliary variables, one obtains the central pair of linear ODEs $[2601.18003,\,\text{Theorem~5.4}]$: \begin{align*} \alpha'' + C\alpha + C\lambda\kappa &= 0, \ \left( \frac{\alpha'}{\tau/\kappa} \right)' &= C\lambda \tau. \end{align*} Thus, the curvature $\kappa$, torsion $\tau$, and the concircular factor $\alpha$ are linked by a closed system of linear ODEs, providing a precise analytic framework for these curves in any space form.

In Euclidean space ($C=0$), the system reduces to a scalar ODE for the curvature ratio $\rho = \tau/\kappa$ $[2601.19252]$:

$$\left( \frac{\rho'}{\kappa (1+\rho^2)^{3/2}} \right)' = m\,\frac{\rho''}{\kappa^2(1+\rho^2)^{5/2}}, \quad m = -\frac{\mu}{\lambda} \neq 0.$$

A first integral yields

$$\kappa(s) = \frac{m \rho''(s) + n}{\rho'(s)\, (1 + \rho(s)^2)},$$

with torsion $\tau(s) = \rho(s)\kappa(s)$.

3. Classification and Explicit Examples

Three principal cases arise depending on parameter values:

• $\lambda=0$ (Planar and Rectifying Curves): These include planar geodesics ($\tau \equiv 0$) and rectifying curves ($t=0$ or $\kappa=0$).
• $\lambda \ne 0$ with Constant Rectifying Slope: Here, $\tau/\kappa = \rho = \mathrm{const}$, and $\kappa(s) = m \alpha(s)$. The ODE becomes $\alpha'' + C\,\frac{\rho^2}{1+\rho^2}\,\alpha=0$, with solutions corresponding to:
  • Ordinary circular helices in $\mathbb{R}^3$ ($C=0$)
  • Spherical helices in $S^3$ ($C>0$)
  • Hyperbolic helices in $H^3$ ($C<0$)
• General Case ($\lambda \ne 0$, Nonconstant Slope): The pair of ODEs must both be solved, defining a two-parameter family of concircular helices.

Representative examples include the classical circular helix in $\mathbb{R}^3$,

$$\gamma(s) = (a \cos(s / \sqrt{a^2 + b^2}),\, a\sin(s/\sqrt{a^2 + b^2}),\, b s/\sqrt{a^2 + b^2}),$$

and the Hopf helix in $S^3$,

$$\gamma(s) = (\cos\theta\,e^{i s\cos\theta}, \sin\theta\,e^{j s\sin\theta})\in\mathbb{C}^2\simeq\mathbb{R}^4,$$

as well as explicit non-circular concircular helices defined by nonconstant $\rho(s)$, e.g.,

$$\kappa(s) = \frac{1}{1+s^2},\quad \tau(s) = \frac{s}{1+s^2}$$

for $\rho(s) = s$, with $m=1$, $n=1$.

4. Geometric Relations: Geodesics on Concircular Surfaces

A central geometric result is that concircular helices are precisely the geodesics of the concircular surfaces—special ruled hypersurfaces determined by an axis vector field $V$ or $Y$ with the property that the unit normal of the surface has constant projection onto the concircular field:

$\langle N_M, V \rangle = \mathrm{const}.$

Ruled concircular surfaces in $\mathbb{R}^3$ include generalized cylinders, conical surfaces, and the tangent surfaces to rectifying curves ((Lucas et al., 27 Jan 2026), Theorem 5.1). In space forms, the ruled parametrization

$X(p, z) = \exp_p(z W(p))$

allows constructing all such surfaces, and every concircular helix locally arises as a geodesic on one of them ((Lucas et al., 25 Jan 2026), Theorem 6.3).

5. Integrability and Global Properties

Existence and uniqueness for the underlying ODEs guarantee a local $4$-parameter family of concircular helices in any space form. For Euclidean and hyperbolic space, generic solutions yield globally defined curves. In the spherical case, closure conditions depend on rationality constraints between the ODE frequency and geometric rotation—only certain parameter choices yield closed curves.

Notably, the construction of concircular helices is globally analytical: any smooth curvature ratio function $\rho(s)$ with $\rho' \ne 0$ generates a proper concircular helix via the corresponding formulas for curvature and torsion, subject to the initial conditions and prescribed axis data.

6. Connections and Special Curve Types

Concircular helices generalize several noteworthy curve classes:

• Generalized helices: Given by $\tau/\kappa = \mathrm{const}$, which corresponds to the degenerate case $Y$ constant, $\mu = 0$.
• Slant helices: Satisfy $$(\rho') / [\kappa(1+\rho^2)^{3/2}] = \mathrm{const}$$, corresponding to constant angle between principal normal and a fixed direction.
• Rectifying curves: Realized when $Y$ is the position vector; characterized by $\tau/\kappa$ linear in arc-length.

The analytic framework for concircular helices in both Euclidean and general constant curvature settings unifies classical differential geometry with the broader theory of ruled hypersurfaces and has been formalized in recent works by Lucas and Ortega-Yagües (Lucas et al., 25 Jan 2026, Lucas et al., 27 Jan 2026).