Primitive Totally Geodesic Immersed Surfaces

• Primitive totally geodesic immersed surfaces are isometric immersions whose images are maximally embedded with geodesics preserved in ambient manifolds.
• They play a pivotal role in arithmeticity and rigidity, linking geometric structures in complex hyperbolic and metric Lie group settings.
• Their classification informs superrigidity results and arithmetic criteria in locally symmetric spaces, with concrete parametrizations and nonexistence phenomena.

A primitive totally geodesic immersed surface is an isometric immersion of a two-dimensional surface into a (generally higher-dimensional) ambient manifold in such a way that: (1) the image is totally geodesic—that is, all geodesics on the surface (with respect to the induced metric) are also geodesics in the ambient space, and (2) the immersion is primitive (equivalently, maximal): it does not factor, up to finite covers, through a proper inclusion into a totally geodesic submanifold of higher dimension. This concept plays a critical role in the arithmetic, rigidity, and geometry of locally symmetric spaces, particularly in complex and real hyperbolic manifolds and 3-dimensional metric Lie groups, where it connects with superrigidity and arithmeticity phenomena (Bader et al., 2020, Castro et al., 22 Jul 2025).

1. Definitions and Fundamental Properties

A properly immersed totally geodesic subspace $N \to M$ of a (finite volume) complex hyperbolic $n$-manifold $M = K \backslash G / \Gamma$, with $G = \operatorname{SU}(n,1)$, is called a geodesic submanifold. Equivalently, such submanifolds are the projections to $M$ of closed orbits of noncompact, connected, almost-simple subgroups $S < G$ acting on $G/\Gamma$ (Bader et al., 2020).

A geodesic submanifold is maximal if it is not strictly contained in any larger geodesic submanifold; i.e., the corresponding subgroup $S < G$ is not properly contained in a larger closed almost-simple subgroup of $G$. An immersion is primitive if it does not factor (up to finite covers) through any inclusion into a higher-dimensional geodesic submanifold. Primitivity coincides with maximality: every primitive totally geodesic immersion is maximal, and conversely, maximality forces primitivity (Bader et al., 2020).

In 3-dimensional unimodular metric Lie groups (both Riemannian and Lorentzian, with Milnor operator diagonalizable), a totally geodesic immersion $\phi: \Sigma \to G$ of a nondegenerate surface (with unit normal $N$) is primitive if its image does not lie in any larger totally geodesic submanifold. The shape operator $S$ vanishes identically ($S \equiv 0$), which is equivalent to the left-invariant Gauss map $(\nu_1, \nu_2, \nu_3)$ satisfying the algebraic condition $\sum_{i=1}^3 c_i \nu_i^2 = 0$ (Castro et al., 22 Jul 2025).

2. Classification Results in Complex Hyperbolic Manifolds

The principal classification theorems connect the existence and abundance of primitive (maximal) totally geodesic immersed surfaces to strong arithmeticity properties. Specifically:

• Arithmeticity Theorem: Any finite-volume complex hyperbolic $n$-manifold ($n \geq 2$) containing infinitely many maximal properly immersed totally geodesic submanifolds of real dimension at least $2$ must be arithmetic. Explicitly, if $M = K \backslash G / \Gamma$ with $G = \operatorname{SU}(n,1)$ and $\Gamma$ a lattice, and $\ell$ the adjoint-trace field of $\Gamma$, then $\Gamma$ is commensurable with an arithmetic lattice. The only real place $v_0$ of $\ell$ for which $$\mathbf{G}(\ell_{v_0}) \cong \operatorname{PU}(n,1)$$ is that giving the original embedding, while for $v \neq v_0$, $\Gamma$ projects to a compact subgroup [(Bader et al., 2020), Theorem 1.1].
• For nonarithmetic complex hyperbolic manifolds, only finitely many maximal primitive totally geodesic surfaces can exist [(Bader et al., 2020), Corollary 1.2].

In the 2-dimensional case, totally geodesic curves in a complex hyperbolic surface $M = X \setminus D$ (where $X$ is a smooth projective toroidal compactification, $D$ is the compactifying divisor) can be characterized using Hirzebruch–Höfer proportionality. For any projective curve $C \subset X$, $$3 C \cdot C + 3 \deg(D \cap C) \geq -K_X \cdot C + 2 D \cdot C$$, with equality precisely if $C$ is the image of a totally geodesic complex curve in $M$ [(Bader et al., 2020), Theorem 1.15].

3. Superrigidity and Homogeneous Dynamics

The existence and structure of primitive totally geodesic immersions in locally symmetric spaces and the associated lattices are deeply constrained by superrigidity phenomena:

• Superrigidity Theorem: Given a lattice $\Gamma \subset G = \operatorname{SU}(n,1)$, any homomorphism $\rho: \Gamma \to H(k)$ (with $H$ a connected adjoint $k$-algebraic group and $k$ a local field) whose image is unbounded and Zariski dense in $H$ extends uniquely to a continuous homomorphism $\bar{\rho}: G \to H(k)$ under suitable compatibility or real-structure hypotheses. This is established via construction of equivariant boundary maps and use of the Bader–Furman algebraic gates mechanism. In the real case, Pozzetti’s chain rigidity theorem is essential for rationality arguments [(Bader et al., 2020), Theorem 1.6].

Superrigidity underpins the arithmeticity criterion, since infinitely many ergodic homogeneous measures (arising from many maximal geodesic submanifolds) force the associated lattice to be arithmetic via measure rigidity theorems (Ratner–Mozes–Shah equidistribution) [(Bader et al., 2020), Proposition 3.5, Proposition 6.4].

4. Explicit Examples and Nonexistence Phenomena

Elementary examples and classification data include:

• Simple-type lattices: Manifolds modeled on quotients by arithmetic lattices in $$\operatorname{SL}_{n+1}(\mathbb{Z}[i]) \cap \operatorname{SU}(n,1)$$ admit totally geodesic complex submanifolds in each lower dimension $1 \leq m \leq n$ (via coordinate hyperplanes) and real geodesic submanifolds for $2 \leq m \leq n$ (via real coordinate subspaces) [(Bader et al., 2020), Example 6.1].
• Division-algebra lattices: For manifolds derived from division algebra lattices, no positive-dimensional complex geodesic submanifolds exist, and the only geodesic immersions are closed geodesics; neither complex nor real primitive totally geodesic surfaces of dimension $\geq 2$ occur [(Bader et al., 2020), Example 6.2].
• Nonexistence for hyperbolic $3$-manifolds: Many finite-volume real hyperbolic $3$-manifolds (e.g., nonarithmetic knot complements with nonintegral trace field) cannot embed as primitive totally geodesic surfaces in any complex hyperbolic manifold, as their adjoint-trace fields are not compatible with those of any complex hyperbolic lattice [(Bader et al., 2020), Theorem 1.17, Example 6.3].

5. Structure and Parametrization in Metric Lie Groups

In the framework of simply connected 3-dimensional unimodular metric Lie groups $G$ (including Riemannian and Lorentzian cases with diagonalizable Milnor operator), the following holds (Castro et al., 22 Jul 2025):

• The primitive totally geodesic surfaces are classified by the vanishing of the shape operator $S \equiv 0$, which is equivalent to the algebraic Gauss map condition $\sum_{i=1}^3 c_i \nu_i^2 = 0$ for the structure constants $(c_1, c_2, c_3)$ of $G$ (where $[E_1, E_2]=c_3 E_3$, etc.).
• In $\mathbb{R}^3$ and $\mathbb{S}^3$, every affine plane and every great sphere, respectively, is totally geodesic, corresponding to cosets of 2-dimensional subgroups.
• For non-constant curvature, sum-type groups ($c_3 = c_1 + c_2$) admit a 2-parameter family of primitive totally geodesic cylinders, with the normal vector and the integrating distribution explicitly given.
• The only nontrivial totally geodesic surfaces in these groups arise in the sum-type case; for other parameter combinations, there are no such surfaces.

Ambient Group	Algebraic Condition	Primitive Totally Geodesic Surfaces
$\mathbb{R}^3$	$c_i \equiv 0$	All affine planes
$\mathbb{S}^3$	$c_1 = c_2 = c_3 \neq 0$	All great 2-spheres
Sum-type	$c_3 = c_1 + c_2$	Flat (or timelike) cylinders with explicit normal and ruling
Other types	None	None

Primitive totally geodesic immersions in these ambient spaces connect deeply to the representation theory of their isometry groups and the structure of their left-invariant metrics.

6. Moduli, Invariants, and Parametrizations

In complex hyperbolic surfaces, the set of primitive totally geodesic curves is parametrized as follows:

• Each primitive geodesic curve corresponds to a conjugacy class of standard subgroups, either $\operatorname{SU}(1,1)$ (complex geodesics) or $\operatorname{SO}_0(2,1)$ (real geodesics), in $\operatorname{SU}(2,1)$, corresponding to 2-planes in $\mathbb{C}^3$ of signature $(1,1)$ defined over the adjoint-trace field $\ell$.
• Up to $\operatorname{SU}(2,1)$-conjugacy, totally geodesic complex curves are parametrized by $\ell$-rational points on the Hermitian Grassmannian of signature $(1,1)$ 2-planes. Their isometry groups are $\operatorname{SU}(1,1)$-lattices defined over $\ell$ [(Bader et al., 2020), Section 5.4].
• The collection of such curves on an arithmetic surface forms an arithmetic family, with arithmeticity detected by intersection pairings in $H^2(X)$.

A plausible implication is that the presence or absence of primitive totally geodesic surfaces provides a topological and arithmetic invariant distinguishing arithmetic manifolds from nonarithmetic ones, as reflected by the intersection structures on cohomology and the dynamics of the isometry group.

7. Significance and Broader Context

Primitive totally geodesic immersed surfaces are critical to the understanding of rigidity, arithmetic, and topological characterization of locally symmetric spaces, particularly in complex hyperbolic and 3-dimensional metric Lie group geometries. Their classification is directly implicated in superrigidity theory, measure rigidity, and arithmeticity, and their moduli provide concrete links between geometric, arithmetic, and group-theoretic invariants. Their scarcity or abundance demarcates the fundamental dichotomy between arithmetic and nonarithmetic lattices, with finiteness or infinitude of such surfaces serving as a geometric criterion for arithmeticity (Bader et al., 2020, Castro et al., 22 Jul 2025).