Abstract
This study attempts to establish new upper bounds on the mean curvature and constant sectional curvature of the first positive eigenvalue of the ψ − Laplacian operator on Riemannian manifolds. Various approaches are being used to find the first eigenvalue for the ψ − Laplacian operator on closed oriented bi-slant submanifolds in a Sasakian space form. We extend different Reilly-like inequalities to the ψ − Laplacian on bi-slant submanifolds in a unit sphere depending on our results for the Laplacian operator. The conclusion of this study considers some special cases as well.
1 Introduction
It is one of the most significant aspects of Riemannian geometry to determine the bounds of the Laplacian on a given manifold. One of the major objectives is to find the eigenvalue that arises as a solution of the Dirichlet or Neumann boundary value problems for curvature functions. Because different boundary conditions exist on a manifold, one can adopt a theoretical perspective to the Dirichlet boundary condition using the upper bound for the eigenvalue as a technique of analysis for the Laplacian’s appropriate bound on a given manifold. Assessing the eigenvalue for the Laplacian and ψ − Laplacian operators has been progressively well-known over a long time. The generalization of the usual Laplacian operator, which is an anisotropic mean curvature, was studied in [17]. Let K denote a complete noncompact Riemannian manifold and B signify the compact domain within K. Let λ1(B) > 0 be the first eigenvalue of the Dirichlet boundary value problem.where Δ represents the Laplacian operator on the Riemannian manifold Km. The Reilly’s formula deals exclusively with the fundamental geometrical characteristics of a given manifold. This is generally acknowledged by the following statement. Let (Km, g) be a compact m − dimensional Riemannian manifold and λ1 denote the first nonzero eigenvalue of the Neumann problem.where η is the outward normal on ∂Km.
As a result of Reilly [24], we have the following inequality for a manifold Km immersed in a Euclidean space with ∂Km = 0where H is the mean curvature vector of immersion Km into Rn, signifies the first nonzero eigenvalue of the Laplacian on Km, and dV represents the volume element of Km.
Zeng and He computed the upper bounds for the ψ − Laplace operator as it relates to the first eigenvalue for Finsher submanifolds in Minkowski space. The first eigenvalue of the Laplace operator on a closed manifold was described by Seto and Wei . Nevertheless, Du et al. [16] derived the generalized Reilly inequality and calculated the first nonzero eigenvalue of the ψ − Laplace operator. By adopting a very similar strategy, Blacker and Seto [3] demonstrated a Lichnero-type lower limit for the first nonzero eigenvalue of the ψ − Laplacian for Neumann and Dirichlet boundary conditions.
The studies [14, 15] illustrate the first nonnull Laplacian eigenvalue, which is considered an extension of Reilly’s work . The results of the distinct classes of Riemannian submanifolds for diverse ambient spaces show that the results of both first nonzero eigenvalues portray similar inequality and have the same upper bounds [13, 14]. In the case of the ambient manifold, it is known from past research that Laplace and ψ − Laplace operators on Riemannian manifolds played a vital role in accomplishing different achievements in Riemannian geometry (see [2, 5, 10, 11, 17, 22, 23,]).
The ψ − Laplacian on a m − dimensional Riemannian manifold Km is defined aswhere ψ > 1 and if ψ = 2; then, the abovementioned formula becomes the usual Laplacian operator.
The eigenvalue of Δh, on the other hand, is Laplacian-like. If a function h ≠ 0 meets the following equation with Dirchilet boundary condition or Neumann boundary condition as discussed earlierwhere λ is a real number called the Dirichlet eigenvalue. In the same way, the previous requirements apply to the Neumann boundary condition.
Looking at Riemannian manifolds without boundaries, the Reilly-type inequality for the first nonzero eigenvalue λ1,ψ for ψ − Laplacian was computed in .
On the other hand, Chen was the first to propose the geometry of slant immersions as a logical extension of both holomorphic and totally real immersions. In addition, Lotta introduced the notion of slant submanifolds within the context of almost contact metric manifolds, and Cabrerizo et al. [9] delved more into these submanifolds. More precisely, Cabrerizo et al. explored slant submanifolds in the setting of Sasakian manifolds. However, Cabrerizo et al. introduced another generalization of slant and contact CR-submanifolds; that is, they proposed the idea of bi-slant and semi-slant submanifolds in the almost contact metric manifolds and provided several examples of these submanifolds.
After examining the literature, a logical question arises: can the Reilly-type inequalities for submanifolds of spheres be obtained using almost contact metric manifolds, as described in [1, 14, 15]? To answer this question, we explore the Reilly-type inequalities for bi-slant submanifolds isometrically immersed in a Sasakian space form (odd dimensional sphere). To this end, our aim is to compute the bound for the first nonzero eigenvalues viaψ − Laplacian. The present study is led by the application of the Gauss equation and studies carried out in [13, 14, 16].
2 Preliminaries
A (2n + 1) − dimensional C∞ − manifold is said to have an almost contact structure, if on , there exists a tensor field ϕ of type (1, 1) and a vector field ξ and a 1-form η satisfying the following properties:
The manifold with the structure (ϕ, ξ, η) is called almost contact manifold. There exists a Riemannian metric g on an almost contact metric manifold , satisfying the following relationfor all , where is the tangent bundle of .
An almost contact metric manifold is said to be Sasakian manifold if it satisfies the following relation .for any , where denotes the Riemannian connection of the metric g.
A Sasakian manifold is said to be a Sasakian space form if it has constant ϕ-holomorphic sectional curvature κ and is denoted by . The curvature tensor of the Sasakian space form is given by [4].for all vector fields e1, e2, e3 on .
K is assumed to be a submanifold of an almost contact metric manifold with the induced metric g. The Riemannian connection of induces canonically the connections ∇ and ∇⊥ on the tangent bundle TK and the normal bundle T⊥K of K respectively, and then the Gauss and Weingarten formulas are governed byfor each e1, e2 ∈ TK and v ∈ T⊥K, where σ and Av are the second fundamental form and the shape operator, respectively, for the immersion of K into ; they are related aswhere g is the Riemannian metric on and the induced metric on K.
If Te1 and Ne1 represent the tangential and normal part of ϕe1, respectively, for any e1 ∈ TK, we can write
Similarly, for any v ∈ T⊥K, we writewhere tv and nv are the tangential and normal parts of ϕv, respectively. Thus, T (resp. N) is 1-1 tensor field on TK (resp. T⊥K) and t (resp. n) is a tangential (resp. normal) valued 1-form on T⊥K (resp. TK).
The notion of slant submanifolds in contact geometry was first defined by A. Lotta . Later, these submanifolds were studied by Cabrerizo et al. [9]. Now, we have the following definition of slant submanifolds:
Definition
A submanifold K of an almost contact metric manifold is said to be slant submanifold if for any x ∈ K and X ∈ TxK − ⟨ξ⟩, where ⟨ξ⟩ is the distribution spanned by the vector field ξ, the angle between X and ϕX is constant. The constant angle α ∈ [0, π/2] is then called the slant angle of K in . If α = 0, the submanifold is invariant submanifold, and if α = π/2, then it is an anti-invariant submanifold. If α ≠ 0, π/2, it is a proper slant submanifold.
Moreover, Cabrerizo et al. [9] proved the characterizing equation for the slant submanifold. More precisely, they proved that a submanifold Nm is said to be a slant submanifold if ∃ a constant τ ∈ [0, 1] and a (1, 1) tensor field T, which satisfies the following relation:where τ = − cos2α.
From (2.10), it is easy to conclude the following:
Now, we define the bi-slant submanifold, which was introduced by Cabrerizo et al. .
A submanifold
Kof an almost contact metric manifold
is said to be bi-slant submanifold if there exist two orthogonal complementary distributions
and
such that.
1) .
2) The distribution is slant with the slant angle α1 ≠ 0, π/2.
3) The distribution is slant with the slant angle α2 ≠ 0, π/2.
If α1 = 0 and α2 = π/2, then the bi-slant submanifold is a semi-invariant submanifold. Now, we have the following example of a bi-slant submanifold:
Example.
Considering the 5-dimensional submanifold in R9 with the usual Sasakian structure, such thatfor any α1, α2 ∈ (0, π/2), then it is easy to see that this is an example of a bi-slant submanifold M in R9 with slant angles α1 and α2. Moreover, it can be observed thatform a local orthonormal frame of TK, in which and , where and are the slant distributions with slant angles α1 and α2, respectively.
It is assumed that Kd=2p+2q+1 is a bi-slant submanifold of dimension d in which 2p and 2q are the dimensions of the slant distributions and respectively. Moreover, let {u1, u2, … , u2p, u2p+1 = v1, u2p+2 = v2, … , ud−1 = v2q, ud = v2q+1 = ξ} be an orthonormal frame of vectors which form a basis for the submanifold K2p+2q+1, such that {u1, u2 = sec α1Tu1, u3, u4 = sec α1Tu3, … , u2p = sec α1Tu2p−1} is tangential to the distribution , and the set {v1, v2 = sec α2Tv1, v3, v4 = sec α2Tv3, … v2q = sec α2Tv2q−1} is tangential to . By Eq. 2.4, the curvature tensor for the bi-slant submanifold N2p+2q+1 is given by the formula:
The dimension of the bi-slant submanifold Kd can be decomposed as d = 2p + 2q + 1; then, using the formula (2.10) for slant distributions, we haveand
Then
The relation (2.12) implies that
From the relation (2.13) and Gauss equation we haveor
In the study [1], Ali et al. studied the effect of conformal transformation on the curvature and second fundamental form. More precisely, it is assumed that together with a conformal metric , where . Then, stands for the dual coframe of and represents the orthogonal frame of . Moreover, we havewhere ρa is the component of the covariant derivative of ρ along the vector ea, that is, dρ = ∑aρaea.
Applying the pullback property in (2.15) to Kmvia the point x, we getwhere and are the components of the second fundamental form and mean curvature vector.
The following significant relation was proved in [1].
3 Main Results
Initially, some basic results and formulas will be discussed which are compatible with the studies ([1, 22]).
It is well-known that a simply connected Sasakian space form is a (2t + 1)-sphere S2t+1 and Euclidean space R2t+1 with constant sectional curvature κ = 1 and κ = −3, respectively.
Now, we have the following result, which is based on the preceding arguments:
Lemma 3.1[1]LetKdbe a slant submanifold of a Sasakian space formwhich is closed and oriented with dimension. Ifis embedding fromKdto, then there is a standard conformal mapsuch that the embedding Ω = x°f = (Ω1, … , Ω2t+2) satisfiesforψ > 1.Remark: The Lemma 3.1 is also true for the bi-slant submanifolds and can be proved on the same lines as derived in [1].In the next result, we obtain a result which is analogous to Lemma 2.7 of [22]. Indeed, in Lemma 3.1 by the application of test function, we obtain the higher bound for λ1,ψ in terms of conformal function.
Proposition 3.2LetKdbe ad − dimensional bi-slant submanifold which is closed orientable isometrically immersed in a Sasakian space form. Then we havewherexis the conformal map used inLemma 3.1, andψ > 1. The standard metric is identified byLc,, and we considerx*L1 = e2pLc.Proof: Considering Ωa as a test function, along with Lemma 3.1, we haveObserving that and then |Ωa| ≤ 1, we getOn using 1 < ψ ≤ 2, we concludeBy the application of Holder’s inequality together with (3.2).–.(3.4), we getwhich is (3.1). On the other hand, if we assume ψ ≥ 2, then by Holder inequalityAs a result, we getThe Minkowski inequality providesBy the application of 3.2, 3.7, and .3.8, it is easy to get (3.1).In the next theorem, we are going to provide a sharp estimate for the first eigenvalue of the ψ − Laplace operator on the bi-slant submanifold of the Sasakian space form .
LetKdbe ad − dimensional bi-slant submanifold of a Sasakian space form, then
1. The first nonnull eigenvalueλ1,ψof theψ − Laplacian satisfiesfor 1 < ψ ≤ 2 andfor , where 2pand 2q are the dimensions of the invariant and slant distributions, respectively.
2. The equality is satisfied in (3.9) and (3.10) if ψ = 2 andKd are minimally immersed in a geodesic sphere of radius rκof with the following relations
Proof . Proposition 3.2, together with Holder inequality, provides
We can calculate e2ρ with the help of conformal relations and the Gauss equation. Let , , and . From (2.14), the Gauss equation for the embedding f and the bi-slant embedding Ω = x◦f, we have
On tracing (2.16), we have
Using 3.12, 3.13, and 3.14, we get
The abovementioned relation implies that
Furthermore, on simplification, we get
On integrating along dV, it is easy to see thatwhich is equivalent to (3.9). If ψ > 2, then it is not possible to apply Holder inequality to govern by using . Now, multiplying both sides of Eq. 3.18 by e(ψ−2)ρ and integrating on Kd,
From the assumption, it is evident that d ≥ 2ψ − 2. On applying Young’s inequality, we arrive at
From Eqs 3.20, 3.21, we conclude the following:
Substituting (3.22) in (3.1), we obtain (3.10). For the bi-slant submanifolds, the equality case holds true in (3.9), and the equality cases of (3.2) and 3.4 imply thatfor a = 1, … , 2t + 2. For 1 < ψ < 2, we have |Ωa| = 0 or 1. Therefore, there exists only one a for which |Ωa| = 1 and λ1,ψ = 0, which is not possible since the eigenvalue λi,ψ ≠ 0. This leads to using the value of ψ equal to 2, so we can apply Theorem 1.5 of [15].
For ψ > 2, the equality in (3.10) still holds; this indicates that equalities in (3.7) and (3.8) are satisfied, and this leads toand there exists a such that |∇Ωa| = 0. It shows that Ωa is a constant and λ1,ψ = 0; this again contradicts the fact that λ1,ψ ≠ 0, which completes the proof.
Note 3.1 If ψ = 2, then the ψ − Laplacian operator becomes the Laplacian operator. Therefore, we have the following corollary.
Corollary 3.4LetKdbe ad − dimensional bi-slant submanifold of a Sasakian space form, then the first nonnull eigenvalue of the Laplacian satisfiesBy the application of Theorem 3.3 for 1 < ψ ≤ 2, we have the following result.
LetKdbe ad − dimensional bi-slant submanifold of a Sasakian space form, then the first nonnull eigenvalue λ1,ψof theψ − Laplacian satisfiesfor 1 < ψ ≤ 2.
Proof: If 1 < ψ ≤ 2, we have , and then the Holder inequality provides
On combining (3.9) and (3.25), we get the required inequality. This completes the proof.
Note 3.2 If κ = 1, then simply the connected Sasakian space form becomes an odd dimensional sphere, B2t+1(1). Furthermore, if κ = −3, then changes to (2t + 1) − dimensional Euclidean space.
As a result of the abovementioned arguments, we conclude
Corollary 3.6 Let Kd be a d − dimensional bi-slant submanifold of a Sasakian space form B2t+1(1) (odd dimensional sphere), then
1. The first nonnull eigenvalueλ1,ψof theψ − Laplacian satisfiesfor 1 < ψ ≤ 2 andfor, where 2p and 2q are the dimensions of the anti-invariant and slant distributions, respectively.
Note 3.3 If α1 = 0 and α2 = π/2, then the bi-slant submanifolds become the semi-invariant submanifolds.
With the application of the abovementioned findings, we can deduce the following results for semi-invariant submanifolds in the setting of Sasakian manifolds.
Corollary 3.7 Let Kd be a d − dimensional semi-invariant submanifold of a Sasakian space form , then
1. The first nonnull eigenvalueλ1,ψof theψ − Laplacian satisfiesfor 1 < ψ ≤ 2 andfor , where 2p and 2q are the dimensions of the anti-invariant and slant distributions, respectively.
2. The equality is satisfied in (3.28) and (3.29) if ψ = 2 andKd are minimally immersed in a geodesic sphere of radiusrcof with the following relation
Furthermore, by Corollary 3.4 and Note 3.1, we deduce the following.
Corollary 3.8 Let Kd be a d − dimensional semi-invariant submanifold of a Sasakian space form , then the first nonnull eigenvalue of the Laplacian satisfies
In addition, we also have the following corollary, which can be derived from Theorem 3.5.
Corollary 3.9 Let Kd be a d − dimensional semi-invariant submanifold of a Sasakian space form , then the first nonnull eigenvalue λ1,ψ of the ψ − Laplacian satisfiesfor 1 < ψ ≤ 2.
Statements
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The original contributions presented in the study are included in the article/Supplementary Material, further inquiries can be directed to the corresponding author.
Author contributions
All the authors listed have made a substantial, direct, and intellectual contribution to the work and approved it for publication.
Funding
The first author extends his appreciation to the Deanship of Scientific Research at King Khalid University for funding this work through research groups program under grant number R.G.P.1/206/42.
Conflict of interest
The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.
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Summary
Keywords
eigenvalues, Laplacian, bi-slant submanifolds, Sasakian space form, Reilly-like inequalities
Citation
Alkhaldi AH, Khan MA, Aquib M and Alqahtani LS (2022) Estimation of Eigenvalues for the ψ-Laplace Operator on Bi-Slant Submanifolds of Sasakian Space Forms. Front. Phys. 10:870119. doi: 10.3389/fphy.2022.870119
Received
05 February 2022
Accepted
10 March 2022
Published
18 May 2022
Volume
10 - 2022
Edited by
Josef Mikes, Palacký University, Czechia
Reviewed by
Anouar Ben Mabrouk, University of Kairouan, Tunisia
Aligadzhi Rustanov, Moscow State Pedagogical University, Russia
Updates
Copyright
© 2022 Alkhaldi, Khan, Aquib and Alqahtani.
This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) and the copyright owner(s) are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.
*Correspondence: Ali H. Alkhaldi, ahalkhaldi@kku.edu.sa; Meraj Ali Khan, meraj79@gmail.com
This article was submitted to Statistical and Computational Physics, a section of the journal Frontiers in Physics
Disclaimer
All claims expressed in this article are solely those of the authors and do not necessarily represent those of their affiliated organizations, or those of the publisher, the editors and the reviewers. Any product that may be evaluated in this article or claim that may be made by its manufacturer is not guaranteed or endorsed by the publisher.