Further characterizations of the $m$-weak group inverse of a complex matrix

1.   Introduction

The concept of lightlike submanifolds in geometry was initially established and expounded upon in a work produced by Duggal and Bejancu ^[1]. A nondegenerate screen distribution was employed in order to produce a nonintersecting lightlike transversal vector bundle of the tangent bundle. They defined the CR-lightlike submanifold as a generalization of lightlike real hypersurfaces of indefinite Kaehler manifolds and showed that CR-lightlike submanifolds do not contain invariant and totally real lightlike submanifolds. Further, they defined and studied GCR-lightlike submanifolds of Kaehler manifolds as an umbrella of invariant submanifolds, screen real submanifolds, and CR-lightlike and SCR-lightlike submanifolds in ^[2,3], respectively. Subsequently, B. Sahin and R. Gunes investigated geodesic property of CR-lightlike submanifolds ^[4] and the integrability of distributions in CR-lightlike submanifolds ^[5]. In the year 2010, Duggal and Sahin published a book ^[6]pertaining to the field of differential geometry, specifically focusing on the study of lightlike submanifolds. This book provides a comprehensive examination of recent advancements in lightlike geometry, encompassing novel geometric findings, accompanied by rigorous proofs, and exploring their practical implications in the field of mathematical physics. The investigation of the geometric properties of lightlike hypersurfaces and lightlike submanifolds has been the subject of research in several studies (see ^{[7,8,9,10,11,12,13,14]}).

Crasmareanu and Hretcanu^[15] created a special example of polynomial structure ^[16] on a differentiable manifold, and it is known as the golden structure $(\overline{M}, g)$. Hretcanu C. E. ^[17] explored Riemannian submanifolds with the golden structure. M. Ahmad and M. A. Qayyoom studied geometrical properties of Riemannian submanifolds with golden structure ^{[18,19,20,21]} and metallic structure ^[22,23]. The integrability of golden structures was examined by A. Gizer et al. ^[24]. Lightlike hypersurfaces of a golden semi-Riemannian manifold was investigated by N. Poyraz and E. Yasar ^[25]. The golden structure was also explored in the studies ^{[26,27,28,29]}.

In this research, we investigate the CR-lightlike submanifolds of a golden semi-Riemannian manifold, drawing inspiration from the aforementioned studies. This paper has the following outlines: Some preliminaries of CR-lightlike submanifolds are defined in Section 2. We establish a number of properties of CR-lightlike submanifolds on golden semi-Riemannian manifolds in Section 3. In Section 4, we look into several CR-lightlike submanifolds characteristics that are totally umbilical. We provide a complex illustration of CR-lightlike submanifolds of a golden semi-Riemannian manifold in the final section.

2.   Opening statements

Assume that $(\overline{\aleph}, g)$ is a semi-Riemannian manifold with $(k + j)$-dimension, $k, j \ge 1$, and $g$ as a semi-Riemannian metric on $\overline{\aleph}.$ We suppose that $\overline{\aleph}$ is not a Riemannian manifold and the symbol $q$ stands for the constant index of $g$.

^[15] Let $\overline{\aleph}$ be endowed with a tensor field $\psi$ of type $(1, 1)$ such that

where $I$ represents the identity transformation on $\Gamma (\Upsilon \overline{\aleph})$. The structure $\psi$ is referred to as a golden structure. A metric $g$ is considered $\psi$-compatible if

for all $\gamma$, $\zeta$ vector fields on $\Gamma (\Upsilon \overline{\aleph})$, then $(\overline{\aleph}, g, \psi)$ is called a golden Riemannian manifold. If we substitute $\psi \gamma$ into $\gamma$ in (2.2), then from (2.1) we have

for any $\gamma, \zeta \in \Gamma (\Upsilon \overline{\aleph}).$

If $(\overline{\aleph}, g, \psi)$ is a golden Riemannian manifold and $\psi$ is parallel with regard to the Levi-Civita connection $\overline{\nabla}$ on $\overline{\aleph}:$

then $(\overline{\aleph}, g, \psi)$ is referred to as a semi-Riemannian manifold with locally golden properties.

The golden structure is the particular case of metallic structure ^[22,23] with $p = 1, \ q = 1$ defined by

where $p$ and $q$ are positive integers.

^[1] Consider the case where $\aleph$ is a lightlike submanifold of $k$ of $\overline{\aleph}.$ There is the radical distribution, or $Rad(\Upsilon\aleph)$, on $\aleph$ that applies to this situation such that $Rad(\Upsilon\aleph) = \Upsilon\aleph \cap \Upsilon\aleph^{\bot},$ $\forall$ $p \in \aleph.$ Since $Rad\Upsilon\aleph$ has rank $r \ge 0,$ $\aleph$ is referred to as an $r$-lightlike submanifold of $\overline{\aleph}.$ Assume that $\aleph$ is a submanifold of $\aleph$ that is $r$-lightlike. A screen distribution is what we refer to as the complementary distribution of a Rad distribution on $\Upsilon \aleph$, then

As $S(\Upsilon\aleph)$ is a nondegenerate vector sub-bundle of $\Upsilon\overline{\aleph}|_{\aleph},$ we have

where $S(\Upsilon\aleph)^{\bot}$ consists of the orthogonal vector sub-bundle that is complementary to $S(\Upsilon\aleph)$ in $\Upsilon\overline{\aleph}|_{\aleph}.$ $S(\Upsilon\aleph), S(\Upsilon\aleph^{\bot})$ is an orthogonal direct decomposition, and they are nondegenerate.

Let the vector bundle

Thus,

Assume that the Levi-Civita connection is $\overline{\nabla}$ on $\overline{\aleph}.$ We have

and

where $\{ \nabla_{\gamma}\zeta, A_{h}\gamma \}$ and $\{ h(\gamma, \zeta), \nabla_{\gamma}^{\bot}h \}$ belongs to $\Gamma (\Upsilon\aleph)$ and $\Gamma (tr(\Upsilon\aleph)),$ respectively.

Using projection $L : tr(\Upsilon\aleph) \rightarrow ltr(\Upsilon\aleph),$ and $S: tr(\Upsilon\aleph) \rightarrow S(\Upsilon\aleph^{\bot}),$ we have

and

for any $\gamma, \zeta \in \Gamma (\Upsilon\aleph), \; \aleph \in \Gamma (ltr(\Upsilon\aleph)),$ and $\chi \in \Gamma (S(\Upsilon\aleph^{\bot})),$ where $h^{l}(\gamma, \zeta) = Lh(\gamma, \zeta), \; h^{s}(\gamma, \zeta) = Sh(\gamma, \zeta), \; \nabla_{\gamma}^{l}\aleph, \; \lambda^{l}(\gamma, \chi) \in \Gamma (ltr(\mathcal{T}\aleph)), \; \nabla_{\gamma}^{s}\; \lambda^{s}(\gamma, \aleph) \in \Gamma (S(\Upsilon\aleph^{\bot})),$ and $\nabla_{\gamma}\zeta, \; A_{\aleph}\gamma, \; A_{\chi}\gamma \in \Gamma (\Upsilon\aleph).$

The projection morphism of $\Upsilon\aleph$ on the screen is represented by $P$, and we take the distribution into consideration.

where $\gamma, \zeta \in \Gamma(\Upsilon\aleph), \; \xi \in \Gamma(Rad(\Upsilon\aleph)).$

Thus, we have the subsequent equation.

Consider that $\overline{\nabla}$ is a metric connection. We get

Using the characteristics of a linear connection, we can obtain

Based on the description of a CR-lightlike submanifold in ^[4], we have

where $\lambda = Rad(\Upsilon\aleph) \bot {\psi} Rad(\Upsilon\aleph) \bot \lambda_{0}.$

$S$ and $Q$ stand for the projection on $\lambda$ and $\lambda'$, respectively, then

for $\gamma, \zeta \in \Gamma (\Upsilon\aleph),$ where $f\gamma = {\psi}S\gamma$ and $w\gamma = {\psi}Q\gamma.$

On the other hand, we have

for any $\zeta \in \Gamma (tr(\Upsilon\aleph)),$ $B\zeta \in \Gamma (\Upsilon\aleph)$ and $C\zeta \in \Gamma (tr(\Upsilon\aleph)),$ unless $\aleph_{1}$ and $\aleph_{2}$ are denoted as ${\psi}L_{1}$ and ${\psi}L_{2},$ respectively.

Lemma 2.1. Assume that the screen distribution is totally geodesic and that $\aleph$ is a CR-lightlike submanifold of the golden semi-Riemannian manifold, then $\nabla_{\gamma}\zeta \in \Gamma (S(\Upsilon N)),$ where $\gamma, \zeta \in \Gamma (S(\Upsilon \aleph)).$

Proof. For $\gamma, \zeta \in \Gamma (S(\Upsilon\aleph)),$

Using (2.8),

Using (2.11),

Since screen distribution is totally geodesic, $h^{*}(\gamma, \zeta) = 0,$

Using Lemma 1.2 in ^[1] p.g. 142, we have

where $\gamma, \zeta \in \Gamma (S(\Upsilon \aleph)).$

Theorem 2.2. Assume that $\aleph$ is a locally golden semi-Riemannian manifold $\overline{\aleph}$ with CR-lightlike properties, then $\nabla_{\gamma}\psi\gamma = \psi\nabla_{\gamma}\gamma$ for $\gamma \in \Gamma (\lambda_{0}).$

Proof. Assume that $\gamma, \zeta \in \Gamma (\lambda_{0}).$ Using (2.5), we have

Nondegeneracy of $\lambda_{0}$ implies

where $\gamma \in \Gamma (\lambda_{0}).$

3.   Geodesic CR-lightlike submanifolds

Definition 3.1. ^[4] A CR-lightlike submanifold of a golden semi-Riemannian manifold is mixed geodesic if $h$ satisfies

where $h$ stands for second fundamental form, $\gamma \in \Gamma (\lambda),$ and $\alpha \in \Gamma (\lambda').$

For $\gamma, \zeta \in \Gamma (\lambda)$ and $\alpha, \beta \in \Gamma (\lambda')$ if

and

then it is known as $\lambda$-geodesic and $\lambda'$-geodesic, respectively.

Theorem 3.2. Assume $\aleph$ is a CR-lightlike submanifold of $\overline{\aleph}$, which is a golden semi-Riemannian manifold. $\aleph$ is totally geodesic if

and

for $\alpha, \beta \in \Gamma (\Upsilon\aleph), \; \xi \in \Gamma (Rad(\Upsilon\aleph)) \;$, and $\; \chi \in \Gamma (S(\Upsilon\aleph^{\bot})).$

Proof. Since $\aleph$ is totally geodesic, then

for $\gamma, \zeta \in \Gamma (\Upsilon\aleph).$

We know that $h(\gamma, \zeta) = 0$ if

and

Since ${g}(h(\gamma, \zeta), \xi) = 0,$ we have

Similarly,

Since ${g}(h(\gamma, \zeta), \chi) = 0,$ we get

for $\chi \in \Gamma (S(\Upsilon\aleph^{\bot})).$

Lemma 3.3. Assume that $\overline{\aleph}$ is a golden semi-Riemannian manifold whose submanifold $\aleph$ is CR-lightlike, then

for $\gamma \in \Gamma (\lambda), \; \zeta \in \Gamma (\lambda') \;$ and $\; \chi \in \Gamma (S(\Upsilon\aleph^{\bot})).$

Proof. Using (2.5), we get

From (2.9), it follows that

where $\gamma \in \Gamma (\lambda), \; \zeta \in \Gamma (\lambda') \; , \; \chi \in \Gamma (S(\Upsilon\aleph^{\bot})).$

Theorem 3.4. Assume that $\aleph$ is a CR-lightlike submanifold of the golden semi-Riemannian manifold and $\overline{\aleph}$ is mixed geodesic if

and

for $\gamma \in \Gamma (\lambda), \; \xi \in \Gamma (Rad(\Upsilon\aleph)), \;$ and $\; \chi \in \Gamma (S(\Upsilon\aleph^{\bot})).$

Proof. For $\gamma \in \Gamma (\lambda), \; \zeta \in \Gamma (\lambda'), \;$ and $\; \chi \in \Gamma (S(\Upsilon\aleph^{\bot})),$ we get

Using (2.5),

Again using (2.5), we obtain

Using (2.10), we have

Since the CR-lightlike submanifold $\aleph$ is mixed geodesic, we have

where $\gamma \in \Gamma (\lambda), \zeta \in \Gamma (\lambda').$

From (2.5), we get

From (2.9), we get

Since, $\aleph$ is mixed geodesic, then ${g}(h(\gamma, \zeta), \chi) = 0$

Theorem 3.5. Suppose that $\aleph$ is a CR-lightlike submanifold of a golden semi-Riemannian manifold $\overline{\aleph},$ then $\aleph$ is $\lambda'$-geodesic if $A_{\chi}\eta$ and $A_{\xi}^{*}\eta$ have no component in $\aleph_{2}\bot {\psi}Rad(\Upsilon\aleph)$ for $\eta \in \Gamma (\lambda'), \; \xi \in \Gamma (Rad(\Upsilon\aleph)), \; \;$ and $\; \chi \in \Gamma (S(\Upsilon\aleph^{\bot})).$

Proof. From (2.5), we obtain

where $\chi, \beta \in \Gamma (\lambda').$

Using (2.9), we have

Since $\aleph$ is $\lambda'$-geodesic, then ${g}(h(\eta, \beta), \chi) = 0.$

From (3.1), we get

Now,

From (2.10), we get

Since $\aleph$ is $\lambda'$- geodesic, then

Thus, $A_{\chi}\eta$ and $A^{*}_{\xi}\eta$ have no component in $M_{2}\bot \psi Rad(\Upsilon\aleph).$

Lemma 3.6. Assume that $\overline{\aleph}$ is a golden semi-Riemannian manifold that has a CR-lightlike submanifold $\aleph$. Due to the distribution's integrability, the following criteria hold.

(ⅰ) $\psi{g}(\lambda^{l}({\psi}\gamma, \chi), \zeta) - {g}(\lambda^{l}(\gamma, \chi), {\psi}\zeta) = {g}(A_{\chi}{\psi}\gamma, \zeta) - {g}(A_{\chi}\gamma, {\psi}\zeta),$

(ⅱ) ${g}(\lambda^{l}({\psi}\gamma), \xi) = {g}(A_{\chi}\gamma, {\psi} \xi),$

(ⅲ) ${g}(\lambda^{l}(\gamma, \chi), \xi) = {g}(A_{\chi}{\psi}\gamma, \psi\xi) - {g}(A_{\chi}\gamma, \psi\xi),$

where $\gamma, \zeta \in \Gamma (\Upsilon\aleph), \; \xi \in \Gamma (Rad(\Upsilon\aleph)),$ and $\chi \in \Gamma (S(\Upsilon\aleph^{\bot})).$

Proof. From Eq (2.9), we obtain

Using (2.5), we get

Again, using (2.5), we get

Using (2.9), we have

(ⅱ) Using (2.9), we have

Using (2.10), we get

(ⅲ) Replacing $\zeta$ by $\psi\xi$ in (ⅰ), we have

Using Definition 2.1 in ^[18] p.g. 9, we get

4.   Totally umbilical CR-lightlike submanifolds

Definition 4.1. ^[12] A CR-lightlike submanifold of a golden semi-Riemannian manifold is totally umbilical if there is a smooth transversal vector field $H \in tr\ \Gamma (\Upsilon \aleph)$ that satisfies

where $h$ is stands for second fundamental form and $\chi, \ \eta \in \ \Gamma (\Upsilon \aleph).$

Theorem 4.2. Assume that the screen distribution is totally geodesic and that $\aleph$ is a totally umbilical CR-lightlike submanifold of the golden semi-Riemannian manifold $\overline{\aleph}$, then

Proof. Given that $\overline{\aleph}$ is a golden semi-Riemannian manifold,

Using (2.5) and (2.6), we have

Interchanging $\eta$ and $\chi$, we obtain

Subtracting Eqs (4.1) and (4.2), we get

Taking the inner product with $\gamma \in \Gamma (\lambda_{0})$ in (4.3), we have

Now,

Since $\psi$ is parallel to $\overline{\nabla},$ i.e., $\overline{\nabla}_{\gamma}\psi = 0,$

Using (2.7), we have

Since $\aleph$ is a totally umbilical CR-lightlike submanifold and the screen distribution is totally geodesic,

and

where $\chi \in \Gamma (\lambda')$ and $\gamma \in \Gamma (\lambda_{0}).$

From (4.5), we have

From Lemma 2.1, we get

Similarly,

Using (4.4), we have

Since $\lambda_{0}$ is nondegenerate,

Theorem 4.3. Let $\aleph$ be the totally umbilical CR-lightlike submanifold of the golden semi-Riemannian manifold $\overline{\aleph}$. Consequently, $\aleph$'s sectional curvature, which is CR-lightlike, vanishes, resulting in $\overline{K}(\pi) = 0,$ for the entire CR-lightlike section $\pi.$

Proof. We know that $\aleph$ is a totally umbilical CR-lightlike submanifold of $\overline{\aleph},$ then from (2.13) and (2.14),

for a CR-lightlike section $\pi = \gamma\wedge \omega, \; \gamma \in \Gamma (\lambda_{0}), \omega \in \Gamma (\lambda').$

From (2.12), we have $(\nabla_{U}g)(\zeta, \omega) = 0.$ Therefore, from (4.6) and (4.7), we get

Now, from (4.8) and (4.9), we get

For any $\beta \in \Gamma (tr(\Upsilon\aleph))$, from Equation (4.10), we get

For any unit vectors $\gamma \in \Gamma (\lambda)$ and $\omega \in \Gamma (\lambda'),$ we have

We have

where

or

i.e.,

for all CR-sections $\pi.$

5.   Example

Example 5.1. We consider a semi-Riemannian manifold $R_{2}^{6}$ and a submanifold $\aleph$ of co-dimension $2$ in $R_{2}^{6},$ given by equations

where $\alpha \in R - \{ \frac{\pi}{2} + k\pi; \ k \in z \}.$ The structure on $R^{6}_{2}$ is defined by

Now,

It follows that $(R_{2}^{6}, \psi)$ is a golden semi-Reimannian manifold.

The tangent bundle $\Upsilon\aleph$ is spanned by

Thus, $\aleph$ is a 1-lightlike submanifold of $R_{2}^{6}$ with $Rad\Upsilon\aleph = \mathrm{Span}\{X_{0}\}$. Using golden structure of $R_{2}^{6},$ we obtain that $X_{1} = \psi(X_{0}).$ Thus, $\psi(Rad\Upsilon\aleph)$ is a distribution on $\aleph.$ Hence, the $\aleph$ is a CR-lightlike submanifold.

6.   Conclusions

In general relativity, particularly in the context of the black hole theory, lightlike geometry finds its uses. An investigation is made into the geometry of the $\aleph$ golden semi-Riemannian manifolds that are CR-lightlike in nature. There are many intriguing findings on completely umbilical and completely geodesic CR-lightlike submanifolds that are examined. We present a required condition for a CR-lightlike submanifold to be completely geodesic. Moreover, it is demonstrated that the sectional curvature $K$ of an entirely umbilical CR-lightlike submanifold $\aleph$ of a golden semi-Riemannian manifold $\overline{\aleph}$ disappears.

Use of AI tools declaration

The authors declare they have not used Artificial Intelligence (AI) tools in the creation of this article.

Acknowledgments

The present work (manuscript number IU/R$\&$D/2022-MCN0001708) received financial assistance from Integral University in Lucknow, India as a part of the seed money project IUL/IIRC/SMP/2021/010. All of the authors would like to express their gratitude to the university for this support. The authors are highly grateful to editors and referees for their valuable comments and suggestions for improving the paper. The present manuscript represents the corrected version of preprint 10.48550/arXiv.2210.10445. The revised version incorporates the identities of all those who have made contributions, taking into account their respective skills and understanding.

Conflict of interest

Authors have no conflict of interests.