Invariant Lightlike Submanifolds

Updated 18 April 2026

Invariant lightlike submanifolds are degenerate metric spaces that remain invariant under ambient structure tensors and quarter-symmetric metric connections.
They inherit induced connections with torsion and specialized curvature identities that modify classical Gauss and Codazzi equations.
Explicit models in Sasakian, para-Sasakian, and quaternionic Kähler manifolds validate curvature inequalities and integrability conditions for these submanifolds.

An invariant lightlike submanifold is a geometric object arising in the context of semi-Riemannian geometric structures—specifically, those equipped with additional structure tensors (almost contact, complex, or paracontact structures) and non-standard affine connections, notably quarter-symmetric metric connections. In such ambient geometries, these submanifolds exhibit special invariance properties with respect to both the lightlike (degenerate) metric and the structural tensors, which has significant consequences for their intrinsic and extrinsic geometry, as well as for curvature and foliation phenomena.

1. Quarter-Symmetric Metric Connections and Ambient Structures

A quarter-symmetric metric connection $∇^Q$ on a (pseudo-)Riemannian manifold $(\tilde N, \tilde g)$ is a linear connection whose torsion tensor $T^Q$ is controlled by a $1$-form $\eta$ (often the structure $1$-form of an almost contact, complex, or paracontact structure) and a $(1,1)$ -tensor $\phi$ . Explicitly,

$T^Q(X, Y) = ∇^Q_X Y - ∇^Q_Y X - [X,Y] = \eta(Y)\phi X - \eta(X)\phi Y,$

with $∇^Q$ required to satisfy $(\tilde N, \tilde g)$ 0. This structure adapts to various settings:

Lorentzian or indefinite Sasakian manifolds $(\tilde N, \tilde g)$ 1,
$(\tilde N, \tilde g)$ 2-cosymplectic, para-Sasakian, almost Hermitian, or quaternionic Kähler backgrounds.

Metric-compatibility and the prescribed torsion ensure that $(\tilde N, \tilde g)$ 3 interpolates between the Levi-Civita connection and more general metric connections with torsion determined by the geometric structure of the ambient manifold (Hui et al., 2017, V. et al., 2019, Gupta et al., 14 Dec 2025).

2. Definition and Structure of Invariant Lightlike Submanifolds

A lightlike submanifold $(\tilde N, \tilde g)$ 4 of a semi-Riemannian manifold is characterized by a degenerate induced metric, where at each $(\tilde N, \tilde g)$ 5 the radical (null) distribution $(\tilde N, \tilde g)$ 6 is nontrivial. For $(\tilde N, \tilde g)$ 7 to be invariant in an ambient manifold with structural tensor $(\tilde N, \tilde g)$ 8 and characteristic vector field $(\tilde N, \tilde g)$ 9, the invariance conditions are:

$T^Q$ 0 is tangent to $T^Q$ 1 everywhere,
For all $T^Q$ 2, $T^Q$ 3.

This ensures the structure is compatible with the degenerate metric of $T^Q$ 4 and that the tangent bundle $T^Q$ 5 is $T^Q$ 6-invariant (Hui et al., 2017). More generally, in the context of screen generic lightlike (SGL) submanifolds in indefinite Sasakian statistical manifolds, invariant properties manifest in the interplay of the screen, radical, and co-screen distributions under the ambient structures (Gupta et al., 14 Dec 2025).

3. Induced Connections, Torsion, and Curvature Identities

The quarter-symmetric metric connection induces a compatible connection on $T^Q$ 7, preserving the degenerate metric. For invariant lightlike submanifolds, the induced connection $T^Q$ 8 on $T^Q$ 9 inherits a torsion tensor $1$0 of form

$1$1

paralleling the ambient form, with induced second fundamental forms $1$2, $1$3 encoding, respectively, the transversal and screen geometry (Gupta et al., 14 Dec 2025).

Curvature identities adapt the curvature tensor of $1$4:

In Lorentzian concircular structures or para-Sasakian settings: $1$5 where $1$6 is Levi-Civita curvature, with correction terms as detailed explicitly for $1$7-structures and para-Sasakian geometries (Hui et al., 2017, V. et al., 2019).
In almost contact or $1$8-cosymplectic manifolds, the Ricci tensor gains an additional $1$9 component under quarter-symmetric deformation, while scalar curvature remains invariant (Roy et al., 2021).

In the setting of SGL submanifolds, torsion and curvature propagate to the induced structure and modify Gauss, Codazzi, and Ricci equations, introducing corrections that reflect both the torsion and the underlying degeneracy of the metric (Gupta et al., 14 Dec 2025).

4. Integrability, Foliations, and Geodesicity

Integrability and foliation theory for invariant lightlike submanifolds is highly nuanced due to both the degenerate metric and the influence of the ambient quarter-symmetric connection. Typical features include:

Integrability criteria: Distributions such as $\eta$ 0 (screen), $\eta$ 1, $\eta$ 2, and $\eta$ 3 are integrable (defining foliations) precisely when certain torsion- and structure-determined conditions are met, e.g.,

$\eta$ 4

Totally geodesic foliations: A foliation is totally geodesic if the associated second fundamental forms vanish on the relevant distributions and the distributions are parallel under the induced connection (Gupta et al., 14 Dec 2025).
Mixed geodesic submanifold: $\eta$ 5 is mixed-geodesic if for $\eta$ 6, $\eta$ 7,

$\eta$ 8

with further compatibility conditions governing the interaction with the screen distribution.

These conditions are expressed via Gauss-Weingarten type formulas adapted to the ambient quarter-symmetric geometry, and are essential for the qualitative understanding of the submanifold geometry.

5. Curvature Inequalities and Geometric Consequences

Invariant lightlike submanifolds—particularly in the presence of quarter-symmetric metric connections—exhibit nontrivial geometric inequalities. In the context of submanifolds in quaternionic Kähler manifolds with Ricci quarter-symmetric metric connection, Chen and Casorati-type curvature inequalities acquire additional torsion-dependent terms:

Chen inequality: $\eta$ 9 where $1$0 is the scalar curvature at $1$1, $1$2 the mean curvature, $1$3 the constant of space form, and $1$4 projections of quaternionic structures (Wani et al., 2020).
Generalized δ-Casorati curvature inequalities: The presence of torsion influences both lower and upper bounds on normalized Casorati curvatures, with extremal cases corresponding to quasi-umbilical, trivial normal connection scenarios.

Such inequalities generalize classical results and underscore the geometric richness and constraints imposed by the ambient connection and structure.

6. Explicit Models and Illustrative Examples

Concrete examples demonstrate the theory:

The explicit construction of invariant submanifolds in $1$5 manifolds with explicit frame, metric, and structure, confirming theoretical expectations (equality of mean curvature with respect to Levi-Civita and quarter-symmetric connections, total geodesicity, etc.) (Hui et al., 2017).
Contact SGL submanifolds of indefinite Sasakian statistical manifolds, where structure tensors, signature of the metric, and the statistical deformation are specified, and all integrability, foliation, and geodesicity conditions are checked explicitly (Gupta et al., 14 Dec 2025).
3- and 5-dimensional para-Sasakian manifolds admitting quarter-symmetric metric connections, verifying possibilities for pseudosymmetry or Einstein property relative to the adapted connection (V. et al., 2019).

These models validate the general theory and illustrate the subtle interplay between invariance, degenerate geometry, and connection-induced torsion.

The geometric analysis of invariant lightlike submanifolds under quarter-symmetric metric connections fits into a broader context of non-Riemannian connections, lightlike geometry, and structure-rich geometric flows:

The extension to almost Hermitian and Kähler geometry via quarter-symmetric $1$6-connections creates connections to projective and holomorphic curvature invariants, with the emergence of structure-independent combinations such as the Weyl and holomorphic projective tensors (Zlatanović et al., 2022).
Research in cosymplectic, $1$7-cosymplectic, and paracontact settings analyzes how the torsion modifies Ricci and scalar curvature, further constraining curvature identities and soliton equations under the deformed connection (Roy et al., 2021, V. et al., 2019).
The integrated perspective provided by quarter-symmetric connections enables the systematic study of geometric invariants, curvature pinching, and the regularity of special foliations and submanifold types.

This suggests a fertile landscape for further exploring rigidity and classification phenomena, particularly where lightlike geometry, structure invariance, and non-symmetric connections coalesce.