Indefinite Sasakian Statistical Manifold

• Indefinite Sasakian statistical manifolds are odd-dimensional semi-Riemannian spaces with a Sasakian structure and dual torsion-free affine connections, ensuring precise metric compatibility.
• The incorporation of a quarter-symmetric metric connection enriches the analysis by adapting statistical and contact structures, affecting the curvature and integrability of submanifolds.
• Screen generic lightlike submanifolds within these manifolds are characterized by specific decompositions and structure equations that govern geodesic foliations and invariant lightlike properties.

An indefinite Sasakian statistical manifold is an odd-dimensional semi-Riemannian manifold equipped with both a Sasakian structure (with indefinite signature) and a pair of torsion-free affine connections that are dual with respect to the metric. Of particular current interest are lightlike and, more generally, screen generic lightlike (SGL) submanifolds in such ambient spaces, especially under additional connection structures such as quarter-symmetric metric connections. These geometric objects offer a rich intersection of statistical, contact, and lightlike geometry, characterized by intricate compatibility conditions among metric, connection, and tensor structures (Gupta et al., 14 Dec 2025, Bahadır, 2020).

1. Foundational Definitions

Let $(\tilde N, \tilde\rho)$ be a semi-Riemannian manifold with indefinite signature. An indefinite Sasakian statistical structure comprises:

• A (torsion-free) affine connection $\bar\nabla$ and its dual $\bar\nabla^*$, satisfying for all vector fields $X, Y, Z$:

$$\bar\nabla_XY - \bar\nabla_Y X = [X, Y] \quad\text{and}\quad X \tilde\rho(Y, Z) = \tilde\rho(\bar\nabla_X Y, Z) + \tilde\rho(Y, \bar\nabla^*_X Z).$$

• A triple $(\phi, \nu, \eta)$ on $\tilde N$ such that

$$\phi^2 = -\mathrm{Id} + \eta \otimes \nu, \quad \eta(\nu) = 1, \quad \tilde\rho(\phi X, \phi Y) = \tilde\rho(X, Y) - \eta(X) \eta(Y),$$

and $d\eta(X,Y) = \tilde\rho(X, \phi Y)$.

• The Levi-Civita connection $\bar\nabla^\circ$ satisfies

$$\bar\nabla^\circ_X \nu = -\phi X, \quad (\bar\nabla^\circ_X \phi)Y = \tilde\rho(X, Y) \nu - \eta(Y) X.$$

Compatibility between the affine connection and the Sasakian structure is ensured by the condition on the symmetric difference tensor $K(X, Y) = \bar\nabla_X Y - \bar\nabla^\circ_X Y$: $K(X, \phi Y) + \phi(K(X, Y)) = 0.$ This implies

$$\bar\nabla_X \phi\,Y - \phi(\bar\nabla^*_X Y) = \tilde\rho(X, Y)\,\nu - \eta(Y)\,X,\qquad \bar\nabla_X \nu = -\phi X + \tilde\rho(\bar\nabla_X\nu, \nu)\nu.$$

(Gupta et al., 14 Dec 2025, Bahadır, 2020).

2. Quarter-Symmetric Metric Connection

A quarter-symmetric connection $\tilde D$, in the sense of Golab, is defined by the torsion tensor

$$\tilde T(X, Y) = \tilde D_XY - \tilde D_YX - [X, Y] = \eta(Y)\phi X - \eta(X)\phi Y.$$

It is called metric if $\tilde D \tilde\rho = 0$. Within the indefinite Sasakian statistical framework, the quarter-symmetric metric connection and its dual are given by

$$\tilde D_X Y = \bar\nabla_X Y - K(X, Y) - \eta(X)\phi Y, \qquad \tilde D^*_X Y = \bar\nabla^*_X Y + K(X, Y) - \eta(X)\phi Y.$$

The curvature tensor of $\tilde D$, for $X, Y, Z \in T\tilde N$, is

$$\tilde R(X, Y)Z = \tilde D_X \tilde D_Y Z - \tilde D_Y \tilde D_X Z - \tilde D_{[X, Y]} Z,$$

which extends the Levi-Civita curvature by terms involving $K$ and $\phi$. The quarter-symmetric metric connection adapts the statistical and contact structures to the torsion, encoding additional geometric data relevant for submanifold analysis (Gupta et al., 14 Dec 2025).

3. Screen Generic Lightlike Submanifolds

Let $N \subset \tilde N$ be an $m$-dimensional lightlike submanifold, where the induced metric is degenerate. The fundamental distributions in this context include:

• The radical (null) distribution $\Rad(TN)$,
• The screen distribution $S(TN)$, which is a nondegenerate complement: $TN = \Rad(TN) \oplus S(TN)$,
• The lightlike transversal bundle $ltr(TN)$ and the screen transversal bundle $S(TN^\perp)$.

A submanifold $N$ is called a screen generic lightlike (SGL) submanifold if:

1. $\phi(\Rad(TN)) = \Rad(TN)$,
2. There exists a nondegenerate subbundle $E_\circ = \phi(S(TN)) \cap S(TN)$ within $S(TN)$, permitting the splitting

$$S(TN) = E_\circ \oplus E' \oplus \langle \nu \rangle.$$

Any vector field $X \in TN$ can be decomposed as

$$X = P_\circ X + P_1 X + Q X + \eta(X) \nu = P X + Q X + \eta(X) \nu,$$

where $P X \in E_\circ \oplus \Rad(TN)$ and $Q X \in E'$.

For $X \in TN$,

$\phi X = T X + w X,$

with $T X$ tangential and $w X$ transversal. Similar decompositions hold for normal bundle elements, facilitating detailed study of induced geometry (Gupta et al., 14 Dec 2025).

4. Structure Equations and Integrability

The Gauss and Weingarten formulas for SGL submanifolds, with respect to the quarter-symmetric metric connection $\tilde D$, are: $$\tilde D_X Y = D_X Y + \tilde h^l(X, Y) + \tilde h^s(X, Y),$$ where

$D_X Y = \nabla_X Y - \eta(X) T Y - K(X, Y),$

$$\tilde h^l(X, Y) = h^l(X, Y), \qquad \tilde h^s(X, Y) = h^s(X, Y) - \eta(X) w Y,$$

and the induced torsion is

$T^D(X, Y) = \eta(Y) T X - \eta(X) T Y.$

The integrability of the subbundles is governed by precise conditions. For instance:

• $E_\circ$ is integrable if and only if

$$2\,\tilde\rho(Y, \phi X) = \eta(\tilde D_X \nu) \eta(Y) - \eta(\tilde D_Y \nu) \eta(X),\quad \forall X, Y \in E_\circ.$$

• $E_\circ \oplus \langle\nu\rangle$ is integrable if and only if, for $X, Y \in E_\circ, Z \in E', N \in ltr(TN)$:

$$\tilde\rho(D_X' \phi Y - D_Y' \phi X, T Z) = \tilde\rho(\tilde h^s(Y, \phi X) - \tilde h^s(X, \phi Y), w Z),$$

$$\tilde\rho(\tilde h'(X, \phi Y), \phi N) = \tilde\rho(\tilde h'(Y, \phi X), \phi N).$$

Parallelism of distributions, total geodesicity, and mixed geodesicity are formulated in terms of the vanishing of specific second fundamental forms or derived conditions involving the induced connections and tensor fields (Gupta et al., 14 Dec 2025).

5. Characterization of Geodesic Foliations and Mixed Geodesic Submanifolds

A submanifold $N$ is said to be $E$-geodesic if both the lightlike and screen second fundamental forms vanish on $E$, i.e.,

$$\tilde h^l(X, Y) = 0, \quad \tilde h^s(X, Y) = 0, \quad \forall X, Y \in E.$$

Mixed geodesicity requires $\tilde h(X, Y) = 0$ for all $X \in E$, $Y \in E' \oplus \langle \nu \rangle$.

Key results include:

• $E \oplus \langle\nu\rangle$ defines a totally geodesic foliation if and only if $N$ is $(E \oplus \nu)$-geodesic and this distribution is parallel under $D$.
• $N$ is mixed geodesic if and only if, for all $X \in E$, $Z \in E' \oplus \nu$,

$\tilde h^l(X, T Z) = -D^l(X, w Z),$

$$\tilde\rho(\tilde A_{wZ}X - D_X(T Z), B W) = \tilde\rho(\tilde h^s(X, T Z) + \tilde\nabla^s_X(w Z), C W).$$

These structural equations provide precise criteria for the geometric behavior of SGL submanifolds, relevant for curvature and foliation theory in indefinite metric settings (Gupta et al., 14 Dec 2025).

6. Invariant Lightlike Submanifolds and Inheritance

A lightlike hypersurface $M$ of an indefinite Sasakian statistical manifold $\widetilde M$ is called invariant if

$\phi(S(TM)) \subset S(TM),\qquad \phi(\Rad(TM)) \subset \Rad(TM).$

If the structure vector field $\xi$ is tangent to $M$, then the induced objects $(M, g, D, D^*, \phi, \xi, \eta)$ constitute an indefinite Sasakian statistical structure on $M$ (Bahadır, 2020). Explicit local models illustrate these constructions, such as lightlike hyperplanes in $\mathbb{R}^5$ or more complex SGL submanifolds in higher dimensions (Gupta et al., 14 Dec 2025).

7. Examples and Model Spaces

A canonical example is given in $\mathbb{R}^{13}_6$ with coordinates $(x_1, y_1, \dots, x_6, y_6, z)$ and a semi-Euclidean metric of signature $(-,-,-,+,+,+,-,-,-,+,+,+,+)$. The standard Sasakian structure is specified, and a statistical connection of the form $\bar\nabla = \bar\nabla^\circ + K$, $K$ satisfying $K(X, \phi Y) + \phi K(X, Y) = 0$, is used. An explicit parameterization for a submanifold $N$ yields an SGL submanifold with all the defining properties confirmed:

• $\Rad(TN)$ is $\phi$-invariant,
• The screen bundle decomposes $$S(TN) = E_\circ \oplus E' \oplus \langle \nu \rangle$$,
• The Gauss–Weingarten and integrability conditions for the induced connection and distributions are satisfied.

Similar flat models are constructed for lightlike hypersurfaces in lower-dimensional semi-Euclidean spaces, corroborating the general theory (Gupta et al., 14 Dec 2025, Bahadır, 2020).

• "The geometric characteristics of SGL submanifolds in an indefinite Sasakian statistical manifold equipped with a quarter symmetric metric connection," (Gupta et al., 14 Dec 2025).
• "On Lightlike Geometry of Indefinite Sasakian Statistical Manifolds," (Bahadır, 2020).