Concircular Hypersurfaces in Geometry

• Concircular hypersurfaces are submanifolds in Riemannian geometry characterized by a constant scalar product with a concircular vector field, integrating geodesic and umbilical properties.
• They are classified as ruled hypersurfaces with one vanishing principal curvature, where the integral curves of the principal tangent field are ambient geodesics.
• Parametric constructions in Euclidean and space forms use concircular vector field conditions, leading to differential equations that characterize concircular helices and related geometric structures.

A concircular hypersurface is a distinguished class of submanifolds in Riemannian geometry, defined by the existence of a vector field whose covariant derivative is proportional to the identity and whose scalar product with the unit normal is constant. These hypersurfaces are characterized by intricate geometric structures, connecting the theory of vector fields with the ruled and umbilical geometry of submanifolds. In both Euclidean and general space forms, they admit a classification as certain ruled hypersurfaces intimately related to geodesic structures and totally umbilical submanifolds.

1. Concircular Vector Fields and Hypersurface Definition

A vector field $Y$ on a Riemannian manifold $M$ with Levi–Civita connection $\nabla$ is concircular if it satisfies

$\nabla_X Y = \mu X$

for all tangent vectors $X$, where $\mu\in C^\infty(M)$ is the concircular factor. In Euclidean space $\mathbb{R}^n$, every concircular vector field takes the form $Y(p)=\mu p+v$ for some constant $\mu\in\mathbb{R}$ and $v\in\mathbb{R}^n$.

A submanifold $M\subset\mathbb{R}^n$ with normal $n$ is called concircular (with axis $Y$) if there exists $Y\in\mathrm{Con}(\mathbb{R}^n)$ such that $\langle n(p),Y(p)\rangle = \lambda$ is constant along $M$ (Lucas et al., 27 Jan 2026).

For general $n$-dimensional space forms ${}^n(C)$ of curvature $C\neq0$, a vector field $V$ is concircular if

$\nabla_X V = \mu X$

and $V$ is the tangential part of a constant vector $p_0$ under the standard immersion, explicitly

$$V = p_0 + \mu \varphi,\qquad \mu = -C \langle p_0,\varphi\rangle$$

where $\varphi:{}^n(C)\to\mathbb{R}^{n+1}_\nu$ is the canonical immersion (Lucas et al., 25 Jan 2026).

A hypersurface $M^{n-1}\subset{}^n(C)$ with unit normal $N$ is concircular if there exists a concircular $V$ on ${}^n(C)$ such that $\lambda = \langle N,V \rangle$ is constant on $M$.

2. Shape Operator, Tangent Distributions, and Ruled Structure

A key property of concircular hypersurfaces is the vanishing of one principal curvature. Let $A$ denote the shape operator associated to the normal $N$, with second fundamental form $\sigma(X,Y)=\langle A X,Y\rangle$. Given the concircularity condition $\nabla_X V = \mu X$ and the decomposition $V|_M = \alpha T + \lambda N$ with a unit tangent field $T$, it follows that (Lucas et al., 25 Jan 2026): $$\sigma(X,T) = 0\quad\text{for all tangent } X,\qquad \text{i.e., } A T = 0$$ Thus, $T$ defines a principal direction with zero principal curvature. The corresponding integral curves of $T$ are necessarily ambient geodesics, and the orthogonal complement $T^\perp$ integrates to a family of $(n-2)$-dimensional submanifolds. This leads to the ruled surface (in $\mathbb{R}^3$) or hypersurface structure described below.

3. Parametric Classification and Construction

Every nontrivial concircular hypersurface in a space form is a ruled hypersurface generated by a family of ambient geodesics ("rulings") orthogonal to an $(n-2)$-dimensional "directrix" in a totally umbilical submanifold. In Euclidean space $\mathbb{R}^3$, any nontrivial concircular surface $M$ with axis $Y$ admits a local parametrization (Lucas et al., 27 Jan 2026): $$X(t,z) = \beta(t) + z\big(\cos\varphi\,N_{\beta}(t) + \sin\varphi\,\eta(t)\big),\qquad \varphi\in(0,\pi/2]$$ where:

• $S$ is a totally umbilical surface (plane or sphere) with normal $\eta$ everywhere parallel to $Y$
• $\beta:I\to S$ is a unit-speed curve on $S$ with Darboux frame $\{T_\beta,N_\beta,\eta\}$
• $z\mapsto X(t,z)$ traces the ruling through $\beta(t)$

In general space forms, the local parametrization of a concircular hypersurface is given by (Lucas et al., 25 Jan 2026): $$\Psi_a(p,z) = \exp_p(z W_a(p)) = f(z/R)\,p + R\,g(z/R)\,W_a(p)$$ where $R = 1/\sqrt{|C|}$, $f(t)$ and $g(t)$ are trigonometric or hyperbolic functions depending on $C$, and

$$W_a(p) = \cos a\cdot \eta_1(p) + \sin a\cdot \eta_2(p)$$

with $\eta_1,\eta_2$ the normals to the directrix and ambient totally umbilical submanifold, respectively.

4. Special Cases: Proper Concircular Surfaces and Hypersurfaces

Proper concircular hypersurfaces (where the concircular factor is nonzero) in $\mathbb{R}^3$ fall into two congruence types (Lucas et al., 27 Jan 2026):

1. Parallel to a Cone: Shifted (paralleled) rulings of a cone over a fixed vertex. Parametrized as $X(t,z) = r\,\beta(t) + z\,N_\beta(t)$ with $\beta\subset S^2(r)$.
2. Normal Surface to a Spherical Curve: The union of all normal lines to a curve $\delta(u)\subset S^2(R)$, parametrized as $X(u,v) = \delta(u) + v\,N_\delta(u)$.

In space forms, these cases correspond to the construction with the angle parameter $a$ describing the orientation of the rulings relative to the totally umbilical directrix (Lucas et al., 25 Jan 2026).

5. Differential Equations for Concircular Helices

A concircular helix is a curve with the property that its normal vector makes constant angle with a concircular vector field. For a unit-speed curve $\gamma:I\to\mathbb{R}^3$ with curvature $\kappa>0$ and torsion $\tau$ and Frenet frame $\{T,N,B\}$, define $\rho(s) = \tau(s)/\kappa(s)$. The concircularity condition leads to the third-order ordinary differential equation (Lucas et al., 27 Jan 2026): $$\left(\frac{\rho'}{\kappa(1+\rho^2)^{3/2}}\right)' = m\,\frac{\rho''}{\kappa^2(1+\rho^2)^{5/2}},\qquad m=-\mu/\lambda\neq 0$$ This ODE characterizes proper concircular helices and is equivalent to the constancy of $\langle N, Y \rangle$ where $Y$ is concircular and $\nabla Y = \mu I$.

6. Geodesic Structure and Relationship with Ruled Surfaces

Geodesics on concircular ruled surfaces are precisely the concircular helices. On ruled surfaces parametrized as in Section 3, a curve $\gamma(s)=(t(s), z(s))$ is a geodesic if and only if it satisfies a specific system involving the Darboux frame of the base curve, with curvature and torsion tied to the parameters of the ruled surface (Lucas et al., 27 Jan 2026). The same correspondence holds in higher-dimensional space forms (Lucas et al., 25 Jan 2026).

For tangent-normal surfaces to spherical curves, geodesics are characterized via a separate system, again yielding curvature and torsion that satisfy the above ODE.

7. Examples and Classification in Different Space Forms

Trivial concircular hypersurfaces are totally umbilical (hyperplanes, spheres in $\mathbb{R}^n$; geodesic spheres or horospheres in $S^n$, $H^n$). Non-trivial examples include:

• Cylinders in $\mathbb{R}^n$
• Generalized "latitude" ruled hypersurfaces in $S^n$
• Equidistant hypersurfaces ruing by geodesics at constant angle in $H^n$

These constructions fully characterize concircular hypersurfaces in both Euclidean and constant curvature (space form) settings (Lucas et al., 27 Jan 2026, Lucas et al., 25 Jan 2026).