De Spinadel [1] developed the theory of metallic means family, in which bronze ratio has an important location in studying topics such as dynamical system and quasicrystals. The geometry of bronze structures has an important relation with pure Riemannian metrics according to the corresponding structure. Pandey [2] introduced the concept of bronze structure on Riemannian manifold which was further studied by Aknipar [3]. Moreover, Duggal and Bejancu [4] initiated the theory of lightlike submanifolds of semi-Riemannian manifolds which provides an outstanding framework for geometric characteristics and has versatile applications, particularly in general relativity.

In view of this, Jin [5] developed generic lightlike submanifolds for an indefinite cosymplectic manifold which was further explored by [6, 5] for indefinite Sasakian manifold and indefinite Kaehler manifold, respectively. Several new connections such that semi-symmetric non-metric connection [7], non-metric ϕ- symmetric connection [8] etc. applied to the SG-lightlike submanifolds came into existence. the generalization of screen Cauchy Riemann(SCR) and generic lightlike submanifolds, Dogan et al. [9] introduced the notion of SG-lightlike submanifold for indefinite Kaehler manifold. Then, a few more similar classes namely, contact totally umbilical SG-lightlike and minimal SG-lightlike submanifolds of indefinite Sasakian manifolds were researched by Gupta [10]. Recently, the theory of SGlightlike submanifolds was investigated for golden semi-Riemannian manifold [11], cosympletic manifold [12] and semi-Riemannian product manifold [13].

Jin[14] introduced the notion of non-symmetric and non-metric -type connection on semi-Riemannian manifold as follows: A linear connection Ω on a semi-Riemannian manifold^¯ is called an type connection if its torsion tensor satisfies

where l and m are smooth functions, is a tensor field of and is a 1-form associated with a smooth unit spacelike vector field which is called the characteristic vector field, by . We set and denote by and , the smooth vector fields on . Jin et al. [15] further developed the geometry of generic lightlike submanifolds for an indefinite Kaehler manifold equipped with an connection. However, the concept of this connection for a locally bronze semi-Riemannian manifold having SG-lightlike submanifolds is yet to be investigated.

In this paper, we introduce the bronze structure for semi-Riemannian manifold and analyze the geometry of SG-lightlike submanifolds of a locally bronze semi-Riemannian manifolds equipped with an connection. The integrability, parallelism and geodesicity of the distributions are characterized. We prove the non-existence of coisotropic, isotropic or totally proper SG-lightlike submanifold of a locally bronze semi-Riemannian manifold. Totally umbilical proper SG-lightlike submanifolds for locally bronze semi-Riemannian manifolds are also worked upon. Additionally, the structure of a minimal SG-lightlike submanifolds has been substantiated with an example.

Let be an -dimensional semi-Riemannian manifold with semi Riemannian metric of constant index q such that Let be an m-dimensional lightlike submanifold of . There exists a smooth distribution on of rank 0 \end{document} ]]> , known as radical distribution on such that RadT where and are degenerate orthogonal spaces but not complementary. Then is called an r-lightlike submanifold of

Consider complementary distribution of radical distribution in , called screen distribution, denoted by and , called screen transversal vector bundle which is a complementary vector subbundle to in ⊥

Since S(TM^◦) is a non degenerate vector subbundle of , we have

where is the complementary orthogonal vector subbundle of in

Let and be complementary vector bundles to in and to RadTM^◦ in . Then

Following four cases exist for the lightlike submanifold :

(1) is r-lightlike if

(2) is co-isotropic i

(3) is isotropic if and

(4) is totally lightlike if

Theorem 2.1.[4] be a r-lightlike submanifold of a semi-Riemannian manifold . Then there exists a complementary vector bundle , called a lightlike transversal bundle of RadTM^◦ in , and a basis of consisting of smooth sections of , where is a coordinate neighborhood of , such that

where is a lightlike basis of

Let be the Levi-Civita connection on . Then, the corresponding Gauss and Weingarten formulae are as follows:

where and belong to and , respectively. and are linear connections on and on the vector bundle tr , respectively.

Considering the projection morphisms L and S of on and respectively, we have

and . Then, considering (3)-(5) and the fact that is a metric connection, the following holds:

Let be a projection of on . Then, we have

for any and ), where and belong to and , respectively. By using the above equations, we obtain

In general, the connection on is not metric connection. Since is a metric connection, from (3), we derive

for any . Here, is a metric connection on

Definition 2.2.A polynomial structure on a semi-Riemannian manifold is known as bronze structure if it is determined by such that

where is a tensor field of type (1, 1) on

Definition 2.3.If a semi-Riemannian metric satisfies the equation

which yields

then is called ^¯J^′-compatible.

Definition 2.4.”Ifsemi-Riemannian metric is compatible with the bronze structure , then the pair is called a bronze semi-Riemannian structure and a bronze semi-Riemannian manifold.”

Definition 2.5. is a bronze semi-Riemannian manifold and

then is termed as a locally bronze semiRiemannian manifold.

Let be a locally bronze semi-Riemannian manifold, where is a semi-Riemannian metric and is a bronze structure. Then, For each tangent to can be written as follows

where and w are the tangential and the transversal parts of and are the projections on and ), respectively. In addition, for any can be written as

where and are the tangential and the transversal parts of , respectively.

Definition 3.1.[9] A real r-lightlike submanifold M^◦ oflocally bronze semi-Riemannian manifold is said to be a SG-lightlike submanifold of if the following conditions are satisfied:

(a) Rad(TM^◦) is invariant with respect to , that is

(b) There exist a subbundle such that

where is a non-degenerate distribution on

The above definition implies that there exist a complementary non-degenerate distribution in such that

Let and Q be the projection morphisms on and respectively. Then for all

where , B is invariant and It is clear that

Now for a vector field and using (17), we have

where and w ).

is said to be a proper SG-lightlike submanifold of a locally bronze semi-Riemannian manifold if and

Proposition 3.2. Let be an SG-lightlike submanifold of a locally bronze semi-Riemannian manifold . Then, the distribution is invariant with respect to

For Levi-Civita connection on the locally bronze semi-Riemannian manifold , we set

for any . Since is torsion free and metric connection, therefore we obtain

for any , where is a torsion tensor of the connection where l and m are smooth functions and θ is a 1-form associated with a smooth unit spacelike vector field termed as the characteristic vector field, by . Setting , Ω is called an^¯ -type connection. Since admits a tensor field of type (1, 1), therefore for any , we have

Let be a SG-lightlike submanifold of a locally bronze semi-Riemannian manifold with an type connection Ω. Let Ω be the^¯ induced linear connection on from Ω. Therefore the Gauss formula inherent in^¯ its structure is as follows:

for any , where , and are lightlike second fundamental form and the screen second fundamental form of , respectively. Now from (3), (17), (23) and (26), we obtain

Moreover using, (17), (22), (24) and (26), we get,

for any , where is torsion tensor of the induced connection Ω on . Hence, the following result holds:

Theorem 4.1. The induced connection Ω on the SG-lightlike submanifold of a locally bronze semi-Riemannian manifold with an -type connection Ω^¯, is also an (l, m)-type connection.

Suppose that vanishes identically on . Therefore

follows from (30).

Consequently, we attain the following result:

Theorem 4.2.For a SG-lightlike submanifold ofa locally bronze semi-Riemannian manifold with an -type connection , the induced connection Ω on is also an (l, m)-type connection if and only vanishes identically on and the characterstic vector field such that

Corresponding to an pe connection Ω, the Weingarten formulae are^¯ given by:

and , where 2 and and are linear connections on and , respectively. Both and are linear operators on

Now, employing equations (4), (5), (23), (32) and (33), we attain

Now, using equations (6), (7), (29), (35) and (37), we obtain

Let be the projection of on , then any can be written as 2 where is a basis for Rad . The subsequent decomposition w.r.t Ω is

for any , where and {h^¯◦∗(X^◦, J^′Y^◦), Ω^¯∗t◦ξ^◦} ∈ Γ(RadTM^◦).

From (8), (9), (17), (39) and (40), we obtain

Further, using (10), (11), (12), (41) and (42), we derive

Definition 5.1."Let be a lightlike submanifold of a locally bronze semi-Riemannian manifold . If

then is an invariant lightlike submanifold of a locally bronze semi-Riemannian manifold

Proposition 5.2. A screen Cauchy Riemann (SCR) lightlike submanifold of a locally bronze semi-Riemannian manifold is a SG-lightlike submanifold such that distribution B is totally anti-invariant , that is,

where is the normal component of bronze semi-Riemannian structure and is a non-degenerate invariant distribution.

Proposition 5.3.There is a non-existence of coisotropic , isotropic or totally proper SG-lightlike submanifold of a locally bronze semi-Riemannian manifold. If M^◦ is SG-isotropic, coisotropic or totally lightlike submanifold, then it is invariant lightlike submanifold.

Proof. Let be a proper SG-lightlike submanifold of a locally bronze semi-Riemannian manifold . Let is isotropic, which implies that and Therefore, Hence, which shows that is invariant submanifold with respect to . If is coisotropic, then . Therefore, using (44), we obtain and . Also, ⊕ and is invariant with respect to . Further, if is totally lightlike, then and . Thus, we obtain which proves is invariant.

This proves the assertion.

□

Example 5.4.Consider a locally bronze semi-Riemannian manifold of signature with respect to the basis

Let be a submanifold defined as

Here is spanned by , where

Therefore, is submanifold with such that and . This shows that Rad(TM^◦) is invariant with respect to . Since and . Therefore, is invariant with respect to

Furthermore, is spanned by , where

We derive, and , which means , is not invariant with respect to and and

Therefore, is spanned by

which is invariant with respect to ， which shows that , where is invariant with respect to .

Hence, become a proper SG-2-lightlike submanifold of

Definition 5.5. Let be a SG-lightlike submanifold of a locally bronze semi-Riemannian manifold with an connection . Then the distribution B is said to be integrable if and only if for any

Remark 5.6.The above definition implies that the distribution B is said to be integrable and only if for any and

Theorem 5.7. Let be a SG-lightlike submanifold of a locally bronze semi-Riemannian manifold with an connection Ω^¯. Then, is integrable if and only if the following conditions hold:

for any

Proof. The distribution is integrable if and only if

Therefore, from equations (23), (13), (21),(14),(25), (26) and (39), we obtain

Similarly, on applying (13), (14), (20), (22), (26) and (39), we obtain

The required assertion is obtained.

Theorem 5.8.The distribution of a submanifold in a locally bronze semi-Riemannian manifold with an connection Ω^¯ is integrable if and only if

has no component in Γ(B ),

for any

Proof. is integrable if and only if Since, Ω is an^¯ -type connection, therefore applying equations (14), (15), (25), (26), (32) and (22), we obtain

Now, using equation (39), we get

Similarly,

Then, from the equations (20) and (21), we get the desired result.

Theorem 5.9.For a SG-lightlike submanifold ofa locally bronze semi-Riemannian manifold with an connection , the distribution B is integrable if and only if

for any and ).

Proof. Since is a locally bronze semi-Riemannian manifold equipped with an (l, m)-type connection , therefore

Now, using (22) and (26), we obtain

Since B is integrable, therefore . Also, is SG-lightlike submanifold.

Thus,

□

Theorem 5.10.Let be a SG-lightlike submanifold of a locally bronze semi-Riemannian manifold with an - type connection Ω^¯. Then, the distribution is parallel if and only if the following conditions hold:

for all and N^◦ ∈ Γ(ltr(TM^◦)).

Proof. Since the distribution is parallel, therefore for all and ^◦)),

The concept of locally bronze semi-Riemannian manifold leads to

Employing equations (14) and (25), we obtain

Equations (39) and (45) leads to

Similarly, from equations (13), (14), (22) and (25), we get

From (26), (39) and (45), we attain

Theorem 5.11.For a locally bronze semi-Riemannian manifold with an type connection , the distribution of a SG-lightlike submanifold is parallel if and only if

for any

Proof. Let the distribution is parallel. Therefore, for all and

Now, using equations (13) and (25), we obtain

Employing (14), (22), (26) and (33), we attain

Further, from equation (39), we get

Similarly,

Equations (26),(33) and (39) leads to

Definition 5.12.[11] is a screen B-geodesic SG-lightlike submanifold if its second fundamental form satisfies

Remark 5.13.M^◦ is B-geodesic SG-lightlike submanifold of endowed with an -type connection if

Theorem 5.14.The distribution B ofa SG-lightlike submanifold ofa locally bronze semi-Riemannian manifold with an -type connection Ω^¯ defines a totally geodesic foliation in if and only if is B-geodesic and B is parallel with respect to on

Proof. If B defines totally geodesic foliation in , then for all and 0, for all and . T (9c) (99) d (20)

Using equations (26), (28) and (29), we get

Since is SG-lightlike submanifold of . Therefore, the distribution B is invariant with respect to bronze structure . Hence, and vanishes.

Thus, , which implies that is B-geodesic and B is parallel with respect to Ω on

Conversely, let be B-geodesic and parallel with respect to Ω on . Therefore, , which implies that

Definition 5.15.[11] " is a mixed geodesic SG-lightlike submanifold of a locally bronze semi-Riemannian manifold if its second fundamental form satisfies

Remark 5.16.If its second fundamental forms satisfies the conditions

then is said to be mixed geodesic submanifold with an connection.

Theorem 5.17. Let be a SG-lightlike submanifold of a locally bronze semi-Riemannian manifold equipped with an -type connection Ω^¯. Then is mixed geodesic if and only if

for all and

Proof. Since is mixed geodesic, therefore

, for all and

Using Gauss formula, we derive

From equations (19), (22) and (25), we infer

Similarly, using equations (13), (14), (22) and (25), we get

Employing equations (26) and (33), we attain

Theorem 5.18.Let be a SG-lightlike submanifold of a locally bronze semi-Riemannian manifold equipped with an -type connection Ω^¯. Then, for any ，

Proof. Using (13), (22) and (25), we get

From (17), (18), (26) and (33), we obtain

Comparing tangential parts in the above equation,

Theorem 5.19.For a locally bronze semi-Riemannian manifold with an (l, m)-type connection Ω^¯, a SG-lightlike submanifold is mixed geodesic and only if the following conditions hold

for any

Proof. Let be a mixed geodesic SG-lightlike submanifold.

From (1), (13), (22) and (23), we get

Employing (17), (18), (26) and (33) on above equation and comparing the transversal parts, we obtain

Definition 6.1.[16] ”Consider a lightlike submanifold ofa semi-Riemannian manifold is said to be totally umbilical in there exist a smooth transversal vector field on called the transversal curvature vector field of , such that, for all ，

The Gauss and Weingarten equations for implies that is totally umbilical if and only if on each coordinate neighborhood , there exist smooth vector fields ) and ), such that

for all

Lemma 6.2.Let be a totally umbilical proper SG-lightlike submanifold of a locally bronze semi-Riemannian manifold with an (l, m)-type connection Ω^¯. Then,

for any

Proof. Since is a locally bronze semi-Riemannian manifold, it implies that ,

Following equation (23), we obtain

Using equations (13), (17) and (18), we get

From equations (26), (32), (33) and (51), we attain

Thus the required result is obtained by comparing the tangential and normal parts of above equation. □

Theorem 6.3.If is a totally umbilical proper SG-lightlike submanifold of a locally bronze semi-Riemannian manifold with an connection Ω^¯, then

Proof. Let be a totally umbilical proper SG-lightlike submanifold of a locally bronze semi-Riemannian manifold, then

Now, using equations (13), (17), (18), (23), (26) and comparing transversal parts, we obtain

From the concept of totally umbilical submanifold of , we get

Since is a SG-lightlike submanifold of , it is inferred that the distribution B is invariant with respect to . Also, if we take , then

Hence,

Theorem 6.4.Let be a totally umbilical proper SG-lightlike submanifold of a locally bronze semi-Riemannian manifold with an -type connection Ω^¯. Then the necessary and suficient condition for induced connection to be metric is

Proof. Lemma (6.2) implies

Since is a totally umbillical proper SG-lightlike submanifold, therefore

Since the distribution is invariant with respect to bronze structure , we have . Also, from equations (50) and (51), we attain

, which leads to

If we take , then , so equation (51) implies that

Employing equation (30), we have

Definition 7.1.[17] A lightlike submanifold , of a semi-Riemannian manifold , is said to be minimal :

(i) on and

(ii) trace , where trace is written with respect to restricted to

The above definition is independent of and ), but is dependent on .

Example 7.2.let be a semi-Riemannian space of signature , with canonical basis

Let be a submanifold given by

Consider the bronze structure ^¯J^′ defined by

Then, is spanned by , where

where and . It is proved that , which shows that Rad(TM^◦) is invariant under . Since, . Therefore, , is invariant under ^¯J^′.

Upon calculation, we derive that is spanned by

such that , which implies that is invariant under is spanned by

Since, , we see that is invariant under . Now, such that and , which shows that and . Therefore. M° is a proper SG-1-lightlike submanifold of

Also, we derive upon calculation

Hence, for all

That is on and traceh 0.

Therefore, M^◦ is minimal proper SG-lightlike submanifold of

Theorem 7.3.The distribution in a SG-lightlike submanifold of a locally bronze semi-Riemannian manifold with an -type connection Ω^¯ is minimal if and only if

for any and .

Proof. is minimal if and only if

for all

Since is a SG-lightlike submanifold of . Therefore, the distribution is invariant w.r.t . Also using equations (27), (3), (16), (14), (5) and (36), we get

and

for any and

Also, since the shape operator is symmetric on , we obtain

for any and

Employing (52) on above equation, we attain

Theorem 7.4.A SG-lightlike submanifold ofa locally bronze semi-Riemannian manifold with an connection Ω^¯ is minimal if and only if

, where

dim , dim , dim is a basis of and for and

Proof. being minimal submanifold implies that

and

If we choose an orthonormal basis of as , then we get

where is an orthonormal basis of

Since, k , we derive

traceh^¯◦|S(TM◦) = traceA^¯∗ξ◦k |Bo⊕B′ + traceA^¯W◦j |Bo⊕B′ ,

Therefore, trace and trace

On the other hand, using concept of minimal SG-lightlike submanifold with equation (38), we obtain

∀ , which proof is completed. □

This research work has initiated the lightlike geometry of screen generic lightlike submanifolds in the bronze semi-Riemannian manifold equipped with an (l, m)- type connection.

The bronze Riemannian manifolds and bronze means are the impoprtant classes of metallic means family, are instrumental as the basis of proportion to design sculptures. These manifolds are of great use to physicists to analyze the behaviour of nonlinear dynamic systems in the transition from periodicity to semiperiodicity. The study undertaken suggests that there is a potential for further investigation of various metric or non-metric connections derived from (l, m)-type connection equipped with bronze semi-Riemannian manifold. Also, the geometry of screen generic lightlike submanifolds can be examined for various classes of metallic semi-Riemannian manifold equipped with distinct connections inherent to their structure. Also, the bronze semi-Riemannian manifolds, being intrinsically related to the theoretical explanation of behavior in quantum physics, can motivate the geometers to delve into the applications of this noteworthy area of study.