Nonzero Parallel Vector Fields

• A nonzero parallel vector field is a smooth, nowhere-vanishing field that remains parallel under the Levi-Civita connection, ensuring constant length and reduced holonomy.
• Its existence forces geometric decompositions such as Riemannian products or mapping tori, with notable topological constraints like specific Betti number patterns.
• In Lorentzian and pseudo-Riemannian settings, these fields underpin pp-wave metrics and affect curvature, influencing both physical models and geometric classifications.

A nonzero parallel vector field is a globally defined, nowhere-vanishing smooth vector field $V$ on a (pseudo-)Riemannian manifold $(M,g)$ satisfying $\nabla V = 0$, where $\nabla$ is the Levi-Civita connection of $g$. The geometric and topological implications of the existence of such vector fields vary dramatically with the dimension, the signature of the metric, and the additional geometric structure present on $M$. Their paper constrains the holonomy group, impacts the topology (e.g., Betti numbers, fibration structure), and interacts with curvature and the classification of manifolds in both Riemannian and Lorentzian settings.

1. Fundamental Properties and Constraints

The condition $\nabla V = 0$ implies that $V$ is both Killing and has constant length, i.e., $g(V,V)$ is constant on $M$. In the Riemannian case (positive-definite metric), $V$ is either everywhere spacelike or, if $g(V,V)=0$, must be zero. In Lorentzian signature, $V$ can be timelike, spacelike, or null (lightlike). The existence of such a vector field imposes a severe restriction: the holonomy group $\mathrm{Hol}(g)$ reduces, and in many cases, $M$ admits a global splitting or even a bundle structure over $S^1$ or $\mathbb R$.

For compact Riemannian manifolds, the presence of a nonzero parallel vector field guarantees that $M$ geometrically splits as a Riemannian product $M' \times S^1$ (de Rham theorem). However, this decomposition is refined in special geometric contexts, e.g., Kähler, locally conformally Kähler (lcK), or in the presence of additional symmetries (e.g., contact or Sasakian structures).

2. Topological and Cohomological Consequences

The topology of manifolds admitting a nonzero parallel vector field is tightly constrained. In dimension three, the question of existence was completely answered: a closed, orientable 3–manifold admits a nonzero parallel vector field with respect to some Riemannian metric if and only if it is a Kähler mapping torus, i.e., a fiber bundle over $S^1$ with fiber a compact Kähler surface and Kähler holonomy (Gnandi et al., 11 Nov 2025).

In this context, each such $3$-manifold $M$ is diffeomorphic to the mapping torus $M_\varphi$ of a Kähler automorphism $\varphi$: $$M_{\varphi} = (\Sigma \times [0,1])/(x,1) \sim (\varphi(x),0),$$ where $\Sigma$ is a compact Kähler surface and $\varphi$ is a Kähler automorphism. This bundle structure guarantees the existence of a parallel field in the $S^1$ direction after equipping $M$ with a Riemannian metric locally isometric to $dt^2 + g_\Sigma$. Consequently, the holonomy reduces from $SO(3)$ to $SO(2)$, and $X = \partial/\partial t$ is parallel.

A striking consequence is that, on any such $3$-manifold, all Betti numbers $b_k(M)$ are necessarily odd, as follows from the cohomology Leray–Serre decomposition and classical results in co–Kähler geometry. In particular: $$b_0(M)=1,\quad b_1(M)=\mathrm{even}+1,\quad b_2(M)=\mathrm{odd}+\mathrm{even},\quad b_3(M)=1.$$ These results preclude, for instance, spherical $3$-manifolds from admitting such fields. The classification for $3$-manifolds is thus exhaustive: only mapping tori with Kähler fibers arise.

3. Nonzero Parallel Vector Fields in Complex and Conformal Geometry

Locally conformally Kähler (lcK) geometry provides a rich setting for nonzero parallel vector fields. For a compact lcK manifold $(M, g, J)$ with $\nabla V = 0$ and $\dim_\mathbb{R} M \geq 6$, the structure theorem divides all cases into two distinct types (Moroianu, 2015):

(a) Vaisman Case: If the Lee form $\theta$ is a nonzero constant multiple of $V^\sharp$ (the metric dual of $V$), i.e., $\theta = \lambda V^\sharp$, $M$ is a Vaisman manifold. Here, the Kähler form and complex structure are preserved up to a globally conformal change, and the Lee form is parallel. These manifolds are known to be mapping tori of Sasakian manifolds, reflecting a bundle structure over $S^1$.

(b) Globally Conformally Kähler Case: When $M$ is globally conformally Kähler but not Kähler, up to finite cover, $M = \mathbb R^2 \times N$ for a simply-connected complete Kähler manifold $N$. The metric and Kähler form take the form: $$g = ds^2 + dt^2 + e^{2c(t)}g_N,\qquad \Omega = ds \wedge dt + e^{2c(t)}\Omega_N,$$ where $c(t)$ is a smooth function. Both $V = \partial/\partial s$ and $J V = \partial/\partial t$ are parallel for the globally conformal Kähler metric. The Lee form is exact: $\theta = 2c'(t)\,dt$. The splitting

$TM = \mathrm{Span}\{V, JV\} \oplus D$

with both summands $\nabla$-parallel and integrable, reflects the global geometric decomposition enforced by the existence of the parallel field.

4. Parallel Vector Fields in Lorentz and Pseudo-Riemannian Geometry

In Lorentzian and related pseudo-Riemannian geometries, the causal character of $V$ (timelike, spacelike, null/lightlike) distinguishes the geometric consequences. In Lorentzian signature, the existence of a nonzero timelike or spacelike parallel field forces the metric to be a Riemannian or Lorentzian product, but the null (lightlike) case is exceptional.

For Lorentzian manifolds, particularly those with a parallel lightlike vector field, the geometry is governed by the Brinkmann or pp-wave metric ansatz (Pelegrín et al., 2016, Mahara et al., 2021): $$g = 2 du\,dv + H(u, x^i)\,du^2 + \sum_{i=1}^{n-1} (dx^i)^2,$$ with $V = \partial_v$ parallel and null ($g(V,V) = 0,\,\nabla V = 0$). The only nonzero curvature components are

$$R(\partial_i, \partial_u, \partial_u, \partial_j) = -\tfrac{1}{2} \frac{\partial^2 H}{\partial x^i \partial x^j},$$

and the Ricci tensor only has a $R_{uu}$ component, tied to the Laplacian of $H$ in the transverse directions.

Imposing the timelike convergence condition (TCC), i.e., $\mathrm{Ric}(Z,Z)\ge0$ for all timelike $Z$, leads to $\Delta_x H(u,x^i)\le0$. Every spacelike hypersurface of constant mean curvature in such a spacetime is highly constrained: compact examples must be totally geodesic if maximal, and no compact examples exist with nonzero constant mean curvature. This demonstrates the strong rigidity imposed by parallel null directions in Lorentzian geometry—a result generalizing the Calabi–Bernstein theorem.

In Einstein–Lorentzian settings, such as Ricci-flat metrics, every nontrivial solution with a parallel null vector field is a pp-wave, locally characterized by $$g=2du\,dv+H(u, x^2, x^3)\,du^2-\delta_{ij}dx^i dx^j$$ with $H$ harmonic in the transverse plane. The possible holonomy is further reduced, and the class of (non-flat) gravitational wave solutions with parallel rays is entirely classified (Mahara et al., 2021).

5. Nonzero Parallel Fields in Special Geometries: Sasakian and Hypersurfaces

For a noninvariant hypersurface $M$ of a Sasakian manifold $\widetilde M$, equipped with the induced $(\phi, g, u, v, \lambda)$-structure, the existence of a nonzero parallel vector field $V$ is highly restrictive (Srivastava et al., 2012). The fundamental formula

$\nabla_X V = \phi X + \lambda H X$

implies, by skew-symmetry and symmetry considerations, that $M$ must be totally geodesic if such a $V$ exists. Thus, unless the second fundamental form vanishes identically ($M$ totally geodesic within $\widetilde M$), a nonzero parallel vector field tangent to $M$ cannot exist. The only nontrivial instances are essentially flat, e.g., hyperplanes in a flat Sasakian model.

This rigidity result shows that the presence of a nonzero parallel vector field is generically obstructed by curvature of the hypersurface or its embedding, and only appears in totally geodesic circumstances.

6. Physical and Geometric Applications

Nonzero parallel vector fields are central in both classical geometry and mathematical physics. In quantum condensed matter theory, the concept underpins the response of electronic states to external vector potentials. For instance, in thin topological insulator (TI) films, the response to parallel magnetic fields is analyzed via the surface Dirac Hamiltonian, where a static in-plane vector potential may be gauged away for a single surface, resulting in no orbital diamagnetism. However, when top and bottom TI surfaces are hybridized, this gauge invariance is broken, introducing a nontrivial field dependence and allowing for a quantum phase transition driven by the parallel field. The character of the field-dependent Hamiltonian and its resulting phases trace directly to the interplay of hybridization and the non-gaugeability of the effective vector potential (Zyuzin et al., 2011).

In Riemannian and complex geometry, parallel fields induce or result from holonomy reduction, fiber bundle structures (e.g., mapping tori), and have implications for the topology (Betti numbers) and symmetry of the manifold.

In Lorentzian geometry and general relativity, parallel null vector fields are the key to constructing and classifying all vacuum gravitational pp-wave solutions, whose spacetime curvature, causal structure, and physical interpretation (e.g., as gravitational waves with globally parallel rays) are governed by the existence and properties of such fields.

7. Classification, Obstructions, and Uniqueness

The classification theorems for nonzero parallel vector fields are now comprehensive in several major settings:

• In dimension three for closed, orientable manifolds: parallel fields correspond precisely to Kähler mapping tori (Gnandi et al., 11 Nov 2025).
• For lcK manifolds (real dimension $\geq 6$): only Vaisman or globally conformally Kähler metrics admit nonzero parallel vector fields (Moroianu, 2015).
• For hypersurfaces of Sasakian manifolds: only totally geodesic, noninvariant hypersurfaces allow such fields (Srivastava et al., 2012).
• In Lorentzian geometry, the only non-flat spacetimes admitting a parallel (non-spacelike) vector field are those with a parallel null direction—namely, Brinkmann or pp-wave metrics (Pelegrín et al., 2016, Mahara et al., 2021).

A plausible implication is that the existence of a nonzero parallel vector field almost always enforces a geometric or topological product structure, drastically reduces holonomy, and places severe restrictions on curvature and cohomology: only in low-curvature, highly symmetric, or bundle-type situations can such fields exist. Conversely, in generic curved settings, curvature, holonomy, or topological obstructions preclude their existence. This structure theory provides a unifying thread connecting geometry, topology, and mathematical physics.