Möbius Semi-Parallel Submanifolds

• Möbius semi-parallel submanifolds are umbilic-free immersions with a conformally rescaled metric and a trace-free Möbius second fundamental form satisfying a vanishing curvature operator condition.
• The theory employs structural equations like the Gauss–Codazzi–Ricci relations and utilizes diagonalization techniques to classify models in both Euclidean and spherical geometries.
• Key classifications include cone, rotational, and product submanifolds, where eigenvalue relations and curvature symmetries dictate geometric integrability and parallelism.

A Möbius semi-parallel submanifold is an umbilic-free immersion $f: M^n \rightarrow \mathbb{R}^m$ or $f: M^n \rightarrow S^m$ whose Möbius second fundamental form commutes, under an appropriate curvature action, with the induced Möbius metric. The fundamental invariants are the Möbius metric $\langle X,Y \rangle^* = \rho^2 \langle X,Y \rangle$, where $\rho^2 = \frac{n}{n-1}(\|\alpha\|^2 - n\, \|H\|^2)$ detects the umbilical locus, and the Möbius second fundamental form $$\beta(X,Y) = \rho [\alpha(X,Y) - \langle X,Y \rangle H]$$, where $\alpha$ is the classical second fundamental form and $H$ is the mean curvature vector. Möbius semi-parallelism is characterized by the vanishing of a curvature operator applied to the Möbius second fundamental form: $\bar{R} \cdot \beta = 0$ (the van der Waerden–Bortolotti curvature action), or equivalently by the symmetry of the second covariant derivative of $\beta$ in the first two arguments. This property is fundamentally Möbius-invariant, unifying parallelism conditions, integrability of curvature distributions, and flatness conditions under the Möbius (conformal) group (Antas et al., 29 Oct 2025, Antas, 24 Dec 2025, Burstall et al., 2010).

1. Möbius-Geometric Invariants and Semi-Parallelity

Given an umbilic-free immersion $f: M^n \rightarrow \mathbb{R}^m$, the Möbius metric $$\langle X, Y \rangle^* = \rho^2 \langle X, Y \rangle$$ is a conformal re-scaling that is well-defined away from the umbilical locus ($\rho > 0$). The Möbius second fundamental form

$$\beta(X, Y) = \rho [\alpha(X, Y) - H \langle X, Y \rangle]$$

is trace-free with respect to the Möbius metric. Diagonalizing $\beta$ leads to Möbius principal normals $\bar{\eta}_i$ and Blaschke tensor eigenvalues $\theta_i$. The semi-parallel condition, written as

$$(\bar{R} \cdot \beta)(X, Y, Z, W) = R^\perp(X, Y)[\beta(Z, W)] - \beta(R^*(X, Y) Z, W) - \beta(Z, R^*(X, Y) W) = 0,$$

requires the vanishing of the Möbius curvature operator acting on $\beta$ (Antas, 24 Dec 2025, Antas et al., 29 Oct 2025). An equivalent formulation uses the commutation relation in the Ricci equation of the conformal Cartan framework: $R^\perp \cdot \beta = 0,$ or, in hypersurface cases, $B \wedge R^* = 0$ for the trace-free Möbius shape operator $B$ (Antas et al., 29 Oct 2025, Burstall et al., 2010).

2. Structural Equations and Conformal Invariance

The geometric structure is governed by the conformal Gauss–Codazzi–Ricci equations for the Möbius geometry. Let $\nabla^*, R^*$ denote the Levi–Civita connection and curvature with respect to the Möbius metric, $\hat{\psi}$ the Blaschke tensor, and $\omega$ the Möbius form. The equations take the form (Antas et al., 29 Oct 2025):

• Gauss:

$$R^*(X, Y) = B X \wedge^* B Y + \hat{\psi} X \wedge^* Y + X \wedge^* \hat{\psi} Y$$

• Codazzi:

$$(\nabla^*_X B)Y - (\nabla^*_Y B)X = \omega(X) Y - \omega(Y) X$$

• Ricci:

$$d\omega(X, Y) = \langle [\hat{\psi}, B] X, Y \rangle^*$$

Under the semi-parallel hypothesis, $\omega$ is closed, principal curvature distributions are umbilical and integrable, and the number of distinct Möbius principal curvatures is bounded above by three. The Cartan connection and tractor calculus from conformal submanifold geometry provide a conformally invariant recasting of these equations and clarify the distinction between curvature and torsion in this context (Burstall et al., 2010).

3. Classification of Möbius Semi-Parallel Submanifolds

The main classification results follow from the Möbius semi-parallel condition and structural equations. In the hypersurface case ($f: M^n \rightarrow \mathbb{R}^{n+1}$, $n \geq 4$) with three distinct principal curvatures, there are two core cases (Antas et al., 29 Oct 2025):

• Case A: $(m_1=1, m_2=1, m_3 \geq 2)$. Locally, up to Möbius transformation, $f$ is equivalent to:
  • (i) A cone over a Clifford torus in $S^3$;
  • (ii) A rotational hypersurface over a hyperbolic cylinder in $\mathbb{H}^3$.
  • The semi-parallel relation takes the form $$\bar{\lambda}_i \bar{\lambda}_j + \theta_i + \theta_j = 0$$, with $\bar{\lambda}_i$ Möbius principal curvatures, $\theta_i$ Blaschke tensor eigenvalues.
• Case B: $(m_1 \geq 1, m_2 \geq 2, m_3 \geq 2)$, where the Möbius metric splits as a Riemannian product and $B$ is parallel, resulting in a Möbius-parallel (isoparametric) hypersurface.

For isometric immersions $f: M^n \rightarrow \mathbb{R}^{n+p}$ with flat normal bundle, the two-normal case ($k$ and $n-k$) is classified as Möbius-equivalent to cylinders, cones, tori, or rotational submanifolds over curves with explicitly prescribed curvatures. In more than two principal normal directions ($k \geq 3$), semi-parallelity typically implies parallelism, with constancy or vanishing of Möbius scalar curvature (Antas, 24 Dec 2025).

Table: Local Models for Möbius Semi-Parallel Hypersurfaces ($n \geq 4$, three distinct curvature principal directions)

Case	Local Model (up to Möbius transformation)	Curvature Condition
A (1,1,$\geq$2)	Cone over Clifford torus or rotational hypersurface over hyperbolic cylinder	$$\bar{\lambda}_i\bar{\lambda}_j + \theta_i+\theta_j=0$$
B ($\geq$1,$\geq$2,$\geq$2)	Product metric, parallel $B$	$B$ parallel

4. Relation to Möbius-Flat and Isothermic Submanifolds

Möbius semi-parallelism generalizes several classical conformal submanifold theories. Möbius-flat submanifolds are characterized by vanishing normal tractor curvature and either conformal flatness ($m \geq 3$) or the existence of commuting holomorphic quadratic differentials ($m=2$) (Burstall et al., 2010). For surfaces:

• Möbius semi-parallelity includes isothermic and Guichard surfaces, channel surfaces (vanishing trace-free part of $B$), and conformally flat hypersurfaces.
• Spectral deformation and T-transform methods yield further isothermic and constrained Willmore surfaces within this framework.

In higher dimensions ($n \geq 4$), Möbius-flat hypersurfaces coincide with channel hypersurfaces, while Möbius semi-parallel hypersurfaces are either channel-type or Möbius-parallel. This theory unites classical submanifold geometries (parallel second fundamental form, integrable curvature distributions) under Möbius-invariant criteria.

5. Technological Frameworks and Proof Strategies

The structural and classification results rely on diagonalization of $B$ and the Blaschke tensor in local Möbius-orthonormal frames, symmetry conditions forced by the vanishing of curvature actions, and decomposition results for twisted and warped product metrics. The Dajczer–Florit–Tojeiro splitting applies in the presence of extrinsic umbilical foliations, leading to reduction to standard product, cone, or warped rotational constructions (Antas, 24 Dec 2025). Under closedness of $\omega$ and flatness of the normal bundle, the ODE arising from the curvature relation $$\langle \bar{\eta}_i, \bar{\eta}_j \rangle + \theta_i + \theta_j = 0$$ prescribes curve geometries for the rotational and cone factors. For high multiplicities in principal curvatures, de Rham decomposition yields global metric splitting and Möbius-parallel submanifolds.

6. Unified Möbius-Invariant Theory and Connections to Spherical Geometry

Research by Hu–Xie–Zhai completed the classification for Möbius semi-parallel submanifolds in the sphere $S^{n+p}$, establishing the maximal number of distinct Möbius principal curvatures and precise conditions for classification (Antas et al., 29 Oct 2025). The Euclidean and spherical results together illustrate that every umbilic-free Möbius semi-parallel hypersurface (with three distinct principal curvatures) is Möbius-equivalent to a cone over a Clifford torus, a rotational hypersurface over a hyperbolic cylinder, or an isoparametric product, depending on the multiplicity structure. This outcome links the theory of Möbius-invariant submanifolds across both Euclidean and spherical spaces, presenting two intrinsically matched sides of the same Möbius-invariant geometric classification (Antas et al., 29 Oct 2025, Antas, 24 Dec 2025).

A plausible implication is that further study of the spectral deformations, integrability conditions, and flatness criteria in Möbius geometry will yield refined characterizations for broader classes of constrained Willmore, isothermic, or channel submanifolds in arbitrary codimension.