Totally Geodesic Almost Complex Surfaces

• Totally geodesic almost complex surfaces are two-dimensional, J-invariant submanifolds with vanishing second fundamental form in (pseudo-)Riemannian manifolds.
• Their classification in homogeneous nearly Kähler 6-manifolds reveals discrete curvature values, explicit model immersions, and links between submanifold rigidity and ambient symmetry.
• These surfaces are pivotal in constructing associative 3-folds in G₂-cones, advancing the understanding of calibrated geometry and submanifold theory.

Totally geodesic almost complex surfaces are two-dimensional submanifolds of a (pseudo-)Riemannian manifold endowed with an almost complex structure, whose tangent bundles are invariant under the almost complex structure and have vanishing second fundamental form. Their study is central in the theory of strictly nearly Kähler 6-manifolds, both in the compact Riemannian and in the pseudo-Riemannian, noncompact settings. These surfaces provide canonical, highly rigid submanifolds, and their classification in homogeneous nearly Kähler spaces elucidates the geometric and topological structure of such ambient manifolds.

1. Definitions and Basic Constructions

A nearly Kähler manifold $(M^6, g, J)$ is a 6-dimensional (pseudo-)Riemannian manifold equipped with an almost complex structure $J$ satisfying $J^2 = -\mathrm{Id}$, $g(JX, JY) = g(X, Y)$ for all $X, Y \in TM$, and $(\nabla_X J) X = 0$ for all $X$, where $\nabla$ denotes the Levi-Civita connection. The structure is ‘strict’ if $(\nabla J) \ne 0$ everywhere. Homogeneous nearly Kähler 6-manifolds include compact Riemannian spaces such as $S^6$, $\mathbb{C}P^3$, the flag manifold $F_{1,2}(\mathbb{C}^3)$, $S^3 \times S^3$, and certain pseudo-Riemannian analogues, e.g., $SL(2, \mathbb{R}) \times SL(2, \mathbb{R})$, $SU(2,1)/(U(1)\times U(1))$, and $SL(3,\mathbb{R})/(\mathbb{R} \times SO(2))$ (Lorenzo-Naveiro et al., 2024, Cwilinski et al., 2021, Bolton et al., 2014, Anarella et al., 6 Jan 2026, Ghandour et al., 2020).

An almost complex surface $M^2 \subset M^6$ is an immersed surface whose tangent bundle is $J$-invariant: $J(T_p M) = T_p M$ for all $p \in M^2$. Such a surface is totally geodesic if its second fundamental form $\mathrm{II}$ vanishes identically, i.e., all geodesics of $M^2$ remain geodesics of $M^6$. Equivalently, for all $X, Y \in TM^2$, the ambient covariant derivative satisfies $\tilde{\nabla}_X Y \in TM^2$.

2. Structural Criteria in Homogeneous Nearly Kähler Manifolds

For a surface $M^2$ to be both almost complex and totally geodesic in a homogeneous nearly Kähler 6-manifold, two main compatibility conditions must be met:

• The tangent planes are preserved by the almost complex structure: $J(T_p M) = T_p M$ for every $p$.
• The second fundamental form vanishes: for all $X, Y \in T_p M$, the projection of $\tilde{\nabla}_X Y$ onto the normal bundle is zero.

In the presence of additional structure, such as an almost product tensor $P$ (notably in $S^3 \times S^3$ and its analogues), the problem reduces to analyzing $P$-invariant and $P$-orthogonal $J$-invariant 2-planes, with curvature implications provided by the Gauss equation (Bolton et al., 2014, Bolton et al., 2012, Ghandour et al., 2020).

3. Classification in Homogeneous Nearly Kähler 6-Manifolds

Up to isometry and ambient symmetry, totally geodesic almost complex surfaces in homogeneous nearly Kähler 6-manifolds are rigidly classified. The following table summarizes the types and their key properties in the primary examples:

Ambient Manifold	Type of Surface	Curvature $K$
$S^3 \times S^3$ (Bolton et al., 2014, Lorenzo-Naveiro et al., 2024, Bolton et al., 2012)	Flat torus ($P$-invariant)	$0$
	Round 2-sphere ($P$-orthogonal)	$\frac{2}{3}$
$SL(2,\mathbb{R}) \times SL(2,\mathbb{R})$ (Ghandour et al., 2020)	Flat surfaces (Lorentzian or Riemannian)	$0$
	Hyperbolic plane ($P$ interchanges $T,N$)	$-3$
Flag $F_{1,2}(\mathbb{C}^3)$ (Cwilinski et al., 2021, Lorenzo-Naveiro et al., 2024)	Spheres in integrable, mixed, or full directions (various orbits)	$4, 1, 0$
$SL(3,\mathbb{R})/(\mathbb{R} \times SO(2))$ (Anarella et al., 6 Jan 2026)	Compact/Noncompact Spheres, Planes	$1, 4, 0$ (sign depends on type)
$S^6$ (Lorenzo-Naveiro et al., 2024)	G$_2$-conjugate 2-sphere	$1$

Totally geodesic almost complex surfaces correspond to:

• Orbits of certain subgroups (e.g., tori, SU(2), SO(3), SO$^+(2,1)$, SL(2,$\mathbb{R}$)) (Anarella et al., 6 Jan 2026, Lorenzo-Naveiro et al., 2024, Cwilinski et al., 2021).
• Explicit immersions parametrized by exponential maps in group coordinates (Cwilinski et al., 2021, Bolton et al., 2014, Ghandour et al., 2020).
• The intersection of canonical J-invariant planes in the tangent Lie algebra with the almost complex and curvature constraints.

4. Curvature Values and Model Examples

In all such homogeneous settings, the possible values for the intrinsic Gaussian curvature $K$ of a totally geodesic almost complex surface are discrete and determined by the ambient structure:

• In $S^3 \times S^3$, $K=0$ (flat torus) or $K=\frac{2}{3}$ (round 2-sphere) (Bolton et al., 2014, Bolton et al., 2012).
• In $SL(2,\mathbb{R}) \times SL(2,\mathbb{R})$, $K=0$ (flat) or $K=-3$ (hyperbolic) (Ghandour et al., 2020).
• In flag manifolds, $K \in \{4,1,0\}$, with $K = 4$ appearing for spheres tangent to an integrable summand, $K=0$ for flat tori (Cwilinski et al., 2021, Lorenzo-Naveiro et al., 2024).
• The manifold $SL(3,\mathbb{R})/(\mathbb{R} \times SO(2))$ exhibits five classes: two spheres of positive definite or Lorentzian signature, a hyperbolic plane, a flat plane (neutral signature), and a degenerate lightlike plane (Anarella et al., 6 Jan 2026).

Explicit model immersions for each of these types are provided in the literature. For instance, the $S^3 \times S^3$ round 2-sphere is given by $\psi(x) = (1-\sqrt{3}x, 1+\sqrt{3}x)$ for $x \in S^2 \subset \mathbb{R}^3$ (Bolton et al., 2014, Bolton et al., 2012).

5. Rigidity, Uniqueness, and Parallel Second Fundamental Form

Totally geodesic almost complex surfaces are maximally rigid: there are no continuous families except those given by the ambient symmetry. Notably:

• Any almost complex surface with parallel second fundamental form (i.e., $\nabla^{\perp} \mathrm{II} = 0$) is necessarily totally geodesic and, up to congruence, is one of the classified types (Bolton et al., 2014, Bolton et al., 2012, Ghandour et al., 2020).
• For compact surfaces of nonnegative curvature, rigidity implies that only the totally geodesic examples exist, such as the flat torus or round 2-sphere in $S^3 \times S^3$ (Bolton et al., 2012).
• In $F_{1,2}(\mathbb{C}^3)$ and its pseudo-Riemannian analogues, all totally geodesic almost complex surfaces are homogeneous orbits, and any parallel-II almost complex surface is totally geodesic (Cwilinski et al., 2021).

6. Consequences, Extensions, and Connections

Totally geodesic almost complex surfaces in homogeneous nearly Kähler 6-manifolds serve as building blocks for submanifold theory, calibrated geometry, and the study of associative 3-folds in $G_2$-cones. Each such surface $\Sigma$ lifts canonically to an associative 3-fold $\mathrm{Cone}(\Sigma)$ in the $G_2$-cone over $M^6$ (Lorenzo-Naveiro et al., 2024). These analyses complete the classification of cohomogeneity-one associative submanifolds in $G_2$-cones over homogeneous strictly nearly Kähler 6-manifolds.

No further non-trivial (i.e., non-congruent, non-orbit) totally geodesic almost complex surfaces exist in these settings. The ambient $J$-holomorphic sectional curvatures admit only a finite discrete spectrum, and all possible examples arise from explicit group-theoretic constructions (Cwilinski et al., 2021, Lorenzo-Naveiro et al., 2024, Bolton et al., 2014, Anarella et al., 6 Jan 2026).

Key developments, classifications, and explicit models are detailed in the following works:

• S. Dioos, J. Fastenakels, J. M. Lorenzo-Naveiro, K. Van der Veken, L. Vrancken, "Almost complex totally geodesic surfaces in the nearly Kähler $$\frac{\text{SL}(3,\mathbb R)}{\mathbb R\times \text{SO}(2)}$$" (Anarella et al., 6 Jan 2026).
• J. M. Lorenzo-Naveiro & A. Rodríguez-Vázquez, "Totally geodesic submanifolds of the homogeneous nearly Kähler 6-manifolds and their G2-cones" (Lorenzo-Naveiro et al., 2024).
• J. M. Lorenzo-Naveiro & A. Rodríguez-Vázquez, "Almost complex surfaces in the nearly Kaehler flag manifold" (Cwilinski et al., 2021).
• J. C. Díaz-Ramos et al., "Almost complex surfaces in the nearly Kahler SL(2,R)xSL(2,R)" (Ghandour et al., 2020).
• J. Bolton, F. Dillen, B. Dioos, L. Vrancken, "On almost complex surfaces in the nearly Kähler $S^3 \times S^3$" (Bolton et al., 2014), and "Almost complex surfaces in the nearly Kähler $S^3\times S^3$" (Bolton et al., 2012).

These results collectively provide a comprehensive understanding of the structure, classification, and rigidity of totally geodesic almost complex surfaces in the landscape of homogeneous nearly Kähler geometry.