Home Riemannian invariants for warped product submanifolds in Q ε m × R and their applications
Article Open Access

Riemannian invariants for warped product submanifolds in Q ε m × R and their applications

  • Yanlin Li , Norah Alshehri and Akram Ali EMAIL logo
Published/Copyright: October 30, 2024
Download Article (PDF)
Open Mathematics
From the journal Open Mathematics Volume 22 Issue 1

Abstract

This article investigates the geometric and topologic of warped product submanifolds in Riemannian warped product Q ε m × R . In this respect, we obtain the first Chen inequality that involves extrinsic invariants like the length of the warping functions and the mean curvature. This inequality involves two intrinsic invariants (sectional curvature and δ -invariant). In addition, an integral bound is provided for the Bochner operator formula of compact warped product submanifolds in terms of the Ricci curvature gradient. We aim to apply this theory to many structures and obtain Dirichlet eigenvalues for problem applications. Some new results regarding the vanishing mean curvature are presented as a partial solution, and this can be considered for the well-known problem given by Chern.

Keywords: warped products; Riemannian product manifolds; delta-invariant; gradient Ricci curvature; Dirichlet eigenvalues
MSC 2010: 53C25; 53C21; 53D15

1 Introduction and motivations

As the warping function determines the properties of the warped product manifold, their geometry and physics are rich and varied. Studying these objects requires a fundamental understanding of how this function behaves. Recently, there has been a surge of interest in warped product manifolds, driven partly by their wide range of applications and their relationship with other mathematical fields. Thus, warped product manifolds can be applied in many fields of geometry and physics. For example, certain types of black hole spacetime can be modeled with warped product manifolds in general relativity. The study of vector bundles on algebraic varieties causes them to arise in algebraic geometry. The use of such manifolds has been used in topology for the construction of examples of exotic manifolds without smooth structures [1].

Many intrinsic invariants exist for (sub)manifolds in contemporary research. A significant physical and geometrical aspect of product manifolds extends beyond Hermitian geometry. According to Einstein’s general relativity, their respective metrics determine the topology of three-dimensional space and one-dimensional time. In addition to Kaluza-Klein, brane, and gauge theories, product manifolds can be used to solve complex problems. It is also well-known that many researchers have investigated submanifolds and product manifolds. We need to give the new forms of Riemannian invariants that are distinct from “classical” invariants to address these issues. Furthermore, the new intrinsic invariants for submanifolds should be connected to the essential extrinsic invariants in a general optimum way. This is why Chen [2] introduced the notion of delta-invariants on Riemannian manifolds. It was the first author, in the 1990s, who introduced a new family of curvature functions on submanifolds. It is roughly described by an isometric immersion of a Riemannian manifold into a real space form that creates the least tension within the space around each point. According to Chen, the ideal submanifold is one that meets the equality condition, and numerous inequalities are derived as invariants of this submanifold. The ideal submanifolds of real and complex space forms have been described by Chen [2], Chen et al. [3]. Furthermore, Petrovic, Dillen, Verstraelen, Tripathi, and Ozgur studied conformally flat, semisymmetric, and Ricci-semisymmetric submanifolds obeying Chen’s inequality in real space forms [4–6]. As a result, we discovered the following problems posed by Chen [4]:

Problem 1.1

Suppose that ℬ 1 × f ℬ 2 is an arbitrary warped product isometrically immersed in E m (or in Q m ( c ) ) as Riemannian submanifold. What are the connections between the warping function and the extrinsic structures ℬ 1 × f ℬ 2 ?

Problem 1.2

Given a warped product ℬ 1 × f ℬ 2 , what are the necessary conditions for the warped product to admit minimal isometric immersion in a Euclidean m -space E m (or in Q m ( c ) )?

Aside from Chen’s optimal inequality for CR-warped products in complex space [7], there have not been many studies on δ -invilliant for warped product structure. It has recently been reported that Mustafa et al. [8] and other authors in [9–11] have constructed the first Chen invariant for warped product submanifolds on real space forms and documented the minimality conditions for them. Conversely, product spaces S n × R and H n × R have been intensively researched in recent years [12–16]. For example, n = 2 , they are two of the eight models proposed by Thurston in his geometrization conjecture. A generalized Hopf differential was presented by Abresch and Rosenberg in [17] and used to categorize constant mean curvature surfaces in S 2 × R and H 2 × R . Since then, numerous studies have been published on this topic about the geometry of submanifolds in those product spaces, the majority of them are focused on submanifolds with constant mean curvature or minimal submanifolds. The product spaces S n × R and H n × R demonstrate greater submanifold geometrically characteristics richer than the space forms S n , R n , and H n . In this context, the Ricci and Codazzi and Gauss equations are essential tools in the study of submanifolds. The Codazzi and Gauss equations are defined intrinsically on submanifolds when the ambient space is a space form. However, the vertical vector ∂ ∂ t concerning the R direction is implicated in the preceding product spaces. We shall provide the basic equations for an n -dimensional manifold ℬ n isometrically embedded into Q ε m × R in Section 2 such that Q ε m stands for a real space forms with constant sectional curvature ε . The basic equations in such a scenario consist of only the Ricci and Codazzi and Gauss equations but also two additional equations generated by ∂ ∂ t ’s parallelism. Let T represent ∂ ∂ t ’s projection on ℬ n . Because T appears in both the basic equations and the fundamental theorems for isometric immersions in Q ε m × R , the first fundamental form is a symmetric 2-tensor that measures the scalar curvature of the submanifold, while the second fundamental form is the normal curvature tensor that measures the curvature of the submanifold in the normal direction. This invariant T is important because it is used to calculate the length of curves on the submanifold and to define the submanifold’s area.

Here, we present a new approach for constructing δ -invariant curvature inequalities for warped product submanifolds immersed in Riemannian product manifolds of the type Q ε m × R , which has been discussed in [14]. Several inequalities for Euclidean spaces and hyperbolic spaces and spheres were also generalized based on the main results.

2 Preliminaries

Suppose that ( ℬ , g ) represents a Riemannian manifold of dimension m . In Riemannian space form, a Riemannian manifold is described by the following relation for Riemannian curvature tensor ℛ :

(2.1) ℛ ( Z 1 , Z 2 , Z 3 , Z 4 ) = ε { g ( Z 1 , Z 4 ) g ( Z 2 , Z 3 ) − g ( Z 2 , Z 4 ) g ( Z 1 , Z 3 ) } ,

for any Z 1 , Z 2 , Z 3 , Z 4 ∈ X ( ℬ ) and it is denoted by ℬ m ( ε ) .

If ∇ and ∇ ⊥ are generated connections on the tangent bundle T ℬ n and normal bundle T ⊥ ℬ n of ℬ n n-dimensional Riemannian submanifold of an m -dimensional Riemannian manifold ℬ ˜ m with generated metric g , respectively. Following are the Gauss and Weingarten formulas

(2.2) ( i ) ∇ ˜ Z 1 Z 2 = ∇ Z 1 Z 2 + Π ( Z 1 , Z 2 ) , ( i i ) ∇ ˜ Z 1 N = − A N Z 1 + ∇ Z 1 ⊥ N ,

for each Z 1 , Z 2 ∈ X ( ℬ n ) and N ∈ X ⊥ ( ℬ n ) , where ∇ ˜ is the connection on ℬ ˜ m , and Π and A N are the second fundamental form and shape operator (corresponding to the normal vector field N ), respectively. Their relationship is as follows:

(2.3) g ( Π ( Z 1 , Z 2 ) , N ) = g ( A N Z 1 , Z 2 ) .

As well, Gauss’ and Codazzi’s equations are as follows:

(2.4) R ( Z 1 , Z 2 , Z 3 , Z 4 ) = R ˜ ( Z 1 , Z 2 , Z 3 , Z 4 ) + g ( Π ( Z 1 , Z 4 ) , Π ( Z 2 , Z 3 ) ) − g ( Π ( Z 1 , Z 3 ) , Π ( Z 2 , Z 4 ) ) .

(2.5) ( R ˜ ( Z 1 , Z 2 ) Z 3 ) ⊥ = ( ∇ ˜ Z 1 Π ) ( Z 2 , Z 3 ) − ( ∇ ˜ Z 2 Π ) ( Z 1 , Z 3 ) .

for all Z 1 , Z 2 , Z 3 , Z 4 ∈ X ( ℬ ˜ m ) , where ℛ ˜ and ℛ are the curvature tensor of ℬ ˜ m and ℬ n , respectively. This is the mean curvature of Riemannian submanifolds ℋ

(2.6) ℋ = 1 n trace ( Π ) .

A submanifold ℬ n of Riemannian manifold ℬ ˜ m is said to be totally umbilical and totally geodesic if Π ( Z 1 , Z 2 ) = g ( Z 1 , Z 1 ) ℋ and Π ( Z 1 , Z 2 ) = 0 , for any Z 1 , Z 2 ∈ X ( ℬ n ) , respectively. Moreover, if ℋ = 0 , then ℬ n is called minimal in ℬ ˜ m . Furthermore, the kernel or the null space of the second fundamental form is defined as follows:

(2.7) ℬ x = { Z ∈ T x ℬ n : Π ( Z , Z 1 ) = 0 , for all Z 1 ∈ T x ( ℬ n ) } .

In this context, if we have { e 1 , … , e m } be an orthonormal basis of the tangent space T x ℬ m ˜ at some x in ℬ ˜ m , we give the definition of another important Riemannian intrinsic invariant called the scalar curvature of ℬ ˜ m , and denoted at τ ˜ ( x ) , which is given by

(2.8) τ ˜ ( x ) = ∑ 1 ≤ i < j ≤ m K ˜ i j ,

K ˜ i j = K ˜ ( e i ∧ e j ) stand for the sectional curvature of space form ℬ ˜ m ( c ) . In a later proof, it will be frequently used to illustrate that first equality (2.8) is congruent with the following equation:

(2.9) 2 τ ˜ ( x ) = ∑ 1 ≤ i ≠ j ≤ m K ˜ i j ,

and the scalar curvature τ ˜ ( L ) of r -dimensional subspace of T x ℬ ˜ m is as follows:

(2.10) τ ˜ ( L ) = ∑ 1 ≤ i < j ≤ r K ˜ i j .

Suppose that { e 1 , … , e n } is an orthonormal basis of the tangent space T x ℬ n and if e r ∈ { e n + 1 , … , e m } an orthonormal basis of the normal space T ⊥ ℬ n , then we obtain

(2.11) Π i j r = g ( Π ( e i , e j ) , e r ) and ∣ ∣ Π ∣ ∣ 2 = ∑ r = n + 1 m ∑ i , j = 1 n ( Π i j r ) 2 .

Suppose that K i j denotes the sectional curvatures of the plane section spanned by { e i , e j } in the submanifold ℬ n . Therefore, K ˜ i j and K i j are the extrinsic and intrinsic sectional curvature of the span { e i , e j } at x ; therefore, from the Gauss equation (2.4), we obtain

(2.12) K i j = K ˜ i j + ∑ r = n + 1 m ( Π i i r Π j j r − ( Π i j r ) 2 ) .

A second invariant is known as the Chen first invariant, and the definition is given as x ∈ ℬ ˜ m :

(2.13) δ ℬ ˜ m ( x ) = τ ˜ ( x ) − inf { K ˜ ( π ) : π ⊂ T x ℬ ˜ m , dim π = 2 } .

Assume that ( ℬ 1 d 1 , g 1 ) and ( ℬ 2 d 2 , g 2 ) are two Riemannian manifolds. Suppose that f is a smooth function defined on ℬ 1 d 1 . Then, the warped product manifold ℬ n = ℬ 1 d 1 × f ℬ 2 d 2 of the manifold ℬ 1 d 1 × ℬ 2 d 2 is furnished by the Riemannian metric g = g 1 + f 2 g 2 . Then, for any Z 1 ∈ X ( ℬ d 1 ) and Z 2 ∈ X ( ℬ d 2 ) , we obtain

(2.14) ∇ Z 2 Z 1 = ∇ Z 1 Z 2 = ( Z 1 ln f ) Z 2 .

According to units, vector fields Z 1 and Z 2 are tangent to ℬ 1 d 1 and ℬ 2 d 2 , respectively, and thus derive

(2.15) K ( Z 1 ∧ Z 2 ) = g ( R ( Z 1 , Z 2 ) Z 1 , Z 2 ) = ( ∇ Z 1 Z 1 ) ln f g ( Z 2 , Z 2 ) − g ( ∇ Z 1 ( ( Z 1 ln f ) Z 2 ) , Z 2 ) = ( ∇ Z 1 Z 1 ) ln f g ( Z 2 , Z 2 ) − g ( ∇ Z 1 ( Z 1 ln f ) Z 2 + ( Z 1 ln f ) ∇ Z 1 Z 2 , Z 2 ) = ( ∇ Z 1 Z 1 ) ln f − ( Z 1 ln f ) 2 − Z 1 ( Z 1 ln f ) .

Suppose that { e 1 , … , e n } is an orthonormal frame for ℬ n , then summing up over the vector fields such that

∑ j = d 1 + 1 n ∑ i = 1 d 1 K ( e i ∧ e j ) = ∑ j = d 1 + 1 n ∑ i = 1 d 1 ( ( ∇ e i e i ) ln f − e i ( e i ln f ) − ( e i ln f ) 2 ) ,

which implies that

(2.16) ∑ j = d 1 + 1 n ∑ i = 1 d 1 K ( e i ∧ e j ) = d 2 ( Δ ( ln f ) − ∣ ∣ ∇ ( ln f ) ∣ ∣ 2 ) , n = d 1 + d 2 .

Nevertheless, for arbitrary warped product submanifolds, it has been shown in [4] that

(2.17) ∑ j = d 1 + 1 n ∑ i = 1 d 1 K ( e i ∧ e j ) = d 2 Δ f f .

Thus, from (2.16) and (2.17), we obtain

(2.18) Δ f f = Δ ( ln f ) − ∣ ∣ ∇ ( ln f ) ∣ ∣ 2 .

Warped product submanifolds tend to result in the following remarks:

Remark 2.1

A warped product manifold ℬ n = ℬ 1 d 1 × f ℬ 2 d 2 is said to be trivial if the warping function f is constant or simply a Riemannian product manifold.

Remark 2.2

If ℬ n = ℬ 1 d 1 × f ℬ 2 d 2 is a warped product manifold, then ℬ 1 d 1 is totally geodesic and ℬ 2 d 2 is totally umbilical submanifold of ℬ n .

A key algebraic figure can be found in the following lemma:

Lemma 2.1

Let l 1 , l 2 , … , l n , s be ( n + 1 ) ( n ≥ 2 ) real numbers such that

(2.19) ∑ i = 1 n ( t i ) 2 = ( d 1 ) ∑ i = 1 n t i 2 + s .

Then, 2 l 1 l 2 ≥ s , with equality holds if and only if l 1 + l 2 = l 3 = … = l n .

Let ℬ n be an n -dimensional submanifold of Q ε m × R with codimension p that m + 1 = n + p . Assume that the ℛ stands for curvature operator for ℬ n and ℛ ˜ for Q ε m × R , respectively. Let A is the second fundamental form of ℬ n such that Π : X ( ℬ n ) ⊙ X ( ℬ n ) ⟶ X ⊥ ( ℬ n ) as g ( A μ Z 1 , Z 2 ) = g ( Π ( Z 1 , Z 2 ) , μ ) , for all Z 1 , Z 2 ∈ X ( ℬ n ) and μ ∈ X ⊥ ( ℬ n ) . If the squared norm of A and Π are equal, then ‖ A ‖ 2 = ‖ Π ‖ 2 , where Π is also a second fundamental form of ℬ n . If { e 1 , e 2 , … , e n } be a local orthonormal frame of tangent bundle and { ξ 1 , ξ 2 , … , ξ p } be an orthonormal frame for normal bundle, then we have

(2.20) ∂ ∂ t = T + ∑ α ν α ξ α = ∑ i T i e i + ∑ α ν α ξ α ,

where T i and ν α stand for smooth functions for all 1 ≤ i ≤ n and 1 ≤ α ≤ p . The following formulas are given as fundamental equations for submanifold ℬ n isometrically immersed into Q ε m × R [14]:

(2.21) g ( ℛ ( Z 1 , Z 2 ) Z 3 , Z 4 ) = ε { g ( Z 2 , Z 3 ) g ( Z 1 , Z 4 ) + g ( Z 1 , T ) g ( Z 1 , Z 3 ) g ( Z 4 , T ) + g ( Z 1 , T ) g ( Z 3 , T ) g ( Z 2 , Z 4 ) − g ( Z 1 , Z 3 ) g ( Z 2 , Z 4 ) − g ( Z 1 , T ) g ( Z 2 , Z 3 ) g ( Z 4 , T ) − g ( Z 2 , T ) g ( Z 3 , T ) g ( Z 1 , Z 4 ) } + g ( Π ( Z 2 , Z 3 ) , Π ( Z 1 , Z 4 ) ) − g ( Π ( Z 1 , Z 3 ) , Π ( Z 2 , Z 4 ) ) ,

(2.22) g ( ( ∇ Z 1 Π ) ( Z 2 , Z 3 ) , ξ α ) − g ( ( ∇ Z 2 Π ) ( Z 1 , Z 3 ) , ξ α ) = ε ν α ( g ( Z 1 , T ) Z 2 − g ( T , Z 2 ) Z 1 ) ,

(2.23) g ( ℛ ⊥ ( Z 1 , Z 2 ) ξ , η ) = g ( [ A ξ , A η ] Z 1 , Z 2 ) ,

(2.24) ∇ Z 1 T = ∑ j ν j A j ( Z 1 ) ,

(2.25) d ν α ( Z 1 ) = ∑ j ν α ω α j ( Z 1 ) − g ( A α ( Z 1 ) , T ) ,

for all Z 1 , Z 2 , Z 3 , Z 4 ∈ X ( ℬ n ) , and ξ , η ∈ X ⊥ ( ℬ n ) . As a result of a straightforward calculation, we obtain the Gauss equation as follows:

(2.26) ℛ i j k l = ε ( ( δ i k δ j l − δ i l δ j k ) + T j T k δ i l + T i T l δ j k − T i T k δ j l − T j T l δ i k ) + ∑ r ( Π i k r Π j l r − Π i l r Π k k r ) .

We are giving an example of a minimal surface in product spaces.

Example 2.1

[18] The horizontal sphere S 2 × R and vertical cylinders S 1 × R are examples of product spaces and helicoids most famous examples of minimal surfaces in S 2 × R . Let us assume that ℳ β is a helicoid such that β ≠ 0 . We define conformal immersion as

ϕ ( u 1 , v 1 ) = sin ϕ ( u 1 ) cos β v 1 sin ϕ ( u 1 ) sin β v 1 cos ϕ ( u 1 ) v 1

such that ϕ satisfied the relation ϕ ′ ( u 1 ) 2 = 1 + β 2 sin 2 ϕ ( u 1 ) , ϕ ″ ( u 1 ) 2 = β 2 sin ϕ ( u 1 ) cos ϕ ( u 1 ) , if we consider that ϕ ( 0 ) = 0 and ϕ ′ ( u 1 ) > 0 . We can classify that ℳ β is a right helicoid if β > 0 and left helicoid if β < 0 . The normal to S 2 × R in R 4 is given by

N ˜ ( u 1 , v 2 ) = sin ϕ ( u 1 ) cos β v 1 sin ϕ ( u 1 ) sin β v 1 cos ϕ ( u 1 ) 0 .

The normal to ℳ β in S 2 × R is given by

N ˜ ( u 1 , v 2 ) = 1 ϕ ′ ( u 1 ) sin β v 1 cos β v 1 0 β sin ϕ ( u 1 ) .

Now, we calculate that as

g ∂ 2 x ∂ u 1 2 , N = g ∂ 2 x ∂ v 1 2 , N = 0 , g ∂ 2 x ∂ u 1 ∂ v 1 , N = − β cos ϕ ( u 1 ) .

Therefore, g ( S Z 1 , Z 2 ) = g ( d Z 2 ( Z 1 ) , N ) , we derive the matrix of S in the frame ∂ ∂ u 1 , ∂ ∂ u 2 is − β cos ϕ ( u 1 ) ϕ ′ ( u 1 ) 0 1 1 0 . For umbilical point such that cos ϕ ( u 1 ) = 0 . We find that T = − 1 ϕ ′ ( u 1 ) 2 ∂ ∂ v 1 and ν = β sin ϕ ( u 1 ) ϕ ′ ( u 1 ) . If β = 0 , the formula defines a vertical cylinder S 2 × R and if β ⟶ ∞ , the surface reduces to horizontal sphere S 2 × { t } .

3 Main results

The results presented in this section provide solutions to Problems 1.1 and 1.2.

Theorem 3.1

Assuming that ℬ n = ℬ 1 d 1 × f ℬ 2 d 2 is an isometric immersion into a product manifold Q ε m × R . Then, for each point x ∈ ℬ n and each plane section π i ⊂ T x ℬ i d i , for i = 1 , 2 , we obtain

  1. Let π 1 ⊂ T x ℬ 1 d 1 , then

    (3.1) δ ℬ 1 d 1 ( x ) ≤ n 2 2 ∣ ∣ ℋ ∣ ∣ 2 + d 2 ∣ ∣ ∇ ( ln f ) ∣ ∣ 2 − d 2 Δ ( ln f ) + d 1 2 ( d 1 + 2 d 2 − 1 ) − ( d 1 − 1 ) ‖ T ‖ 2 − 1 ε .

    It is only possible to compute the equality of the above inequality at x ∈ ℬ n if and only if there exists an orthonormal basis { e 1 , … , e n } of T x ℬ n and an orthonormal basis { e n + 1 , … , e m } of T x ⊥ ℬ n such that

    1. π 1 = Span { e 1 , e 2 } and

    2. Shape operators can be expressed as follows:

      ( i ) A e n + 1 = μ 1 Π 12 n + 1 0 ⋯ 0 1 d 1 0 1 d 1 + 1 ⋯ 0 1 n Π 12 n + 1 μ 2 0 ⋯ ⋮ ⋮ ⋯ ⋮ 0 0 μ ⋯ ⋮ ⋮ ⋯ ⋮ ⋮ ⋮ ⋮ ⋱ ⋮ ⋮ ⋯ ⋮ 0 d 1 1 0 0 ⋯ μ 0 d 1 d 1 + 1 ⋯ 0 d 1 n 0 d 1 + 11 ⋯ ⋯ ⋯ 0 d 1 + 1 d 1 Π d 1 + 1 d 1 + 1 n + 1 ⋯ Π d 1 + 1 n n + 1 ⋮ ⋱ ⋱ ⋱ ⋮ ⋮ ⋱ ⋮ 0 n 1 ⋯ ⋯ ⋯ 0 n d 1 Π n d 1 + 1 n + 1 ⋯ Π n n n + 1 ,

      where μ = μ 1 + μ 2 . If r ∈ { n + 2 , … , m } , then it calculates the matrix

      ( i i ) A e r = Π 11 r Π 12 r 0 ⋯ 0 1 d 1 0 1 d 1 + 1 ⋯ 0 1 n Π 21 r − Π 11 r 0 ⋯ ⋮ ⋮ ⋯ ⋮ 0 0 0 33 ⋯ ⋮ ⋮ ⋯ ⋮ ⋮ ⋮ ⋮ ⋱ ⋮ ⋮ ⋯ ⋮ 0 d 1 1 0 0 ⋯ 0 d 1 d 1 0 d 1 d 1 + 1 ⋯ 0 d 1 n 0 d 1 + 11 ⋯ ⋯ ⋯ 0 d 1 + 1 d 1 Π d 1 + 1 d 1 + 1 r ⋯ Π d 1 + 1 n r ⋮ ⋱ ⋱ ⋱ ⋮ ⋮ ⋱ ⋮ 0 n 1 ⋯ ⋯ ⋯ 0 n d 1 Π n d 1 + 1 r ⋯ Π n n r .

  2. If π 2 ⊂ T x ℬ 2 d 2 , then

    (3.2) δ ℬ 2 d 2 ( x ) ≤ n 2 2 ∣ ∣ ℋ ∣ ∣ 2 + d 2 ∣ ∣ ∇ ( ln f ) ∣ ∣ 2 − d 2 Δ ( ln f ) + d 2 2 ( d 2 + 2 d 1 − 1 ) − ( d 2 − 1 ) ‖ T ‖ 2 − 1 ε .

It follows that the above equation has equalities if and only if there exists an orthonormal basis { e 1 , … , e n } of T x ℬ n and an orthonormal basis { e n + 1 , … , e m } of T x ⊥ ℬ n such that

  1. π 2 = Span { e d 1 + 1 , e d 1 + 2 } and

  2. the Shape operators are represented as follows:

( i i i ) A e n + 1 = Π 11 n + 1 ⋯ ⋯ Π 1 d 1 n + 1 0 1 d 1 + 1 ⋯ ⋯ ⋯ 0 1 n ⋮ ⋱ ⋮ ⋮ ⋱ ⋱ ⋱ ⋮ ⋮ ⋱ ⋮ ⋮ ⋱ ⋱ ⋱ ⋮ Π d 1 1 n + 1 ⋯ ⋯ Π d 1 d 1 n + 1 0 d 1 d 1 + 1 ⋯ ⋯ ⋯ 0 d 1 n 0 d 1 + 11 ⋯ ⋯ 0 d 1 + 1 d 1 μ 1 Π d 1 + 1 d 1 + 2 n + 1 0 ⋯ 0 d 1 + 1 n ⋮ ⋱ ⋱ ⋮ Π d 1 + 2 d 1 + 1 n + 1 μ 2 0 ⋯ ⋮ ⋮ ⋱ ⋱ ⋮ 0 0 μ ⋯ ⋮ ⋮ ⋱ ⋱ ⋮ ⋮ ⋮ 0 ⋱ 0 0 n 1 ⋯ ⋯ 0 n d 1 0 n d 1 + 1 0 ⋯ 0 μ ,

where μ = μ 1 + μ 2 . If r ∈ { n + 2 , … , m } , thus we have

( i v ) A e r = Π 11 r ⋯ ⋯ Π 1 d 1 r 0 1 d 1 + 1 ⋯ ⋯ ⋯ 0 1 n ⋮ ⋱ ⋮ ⋮ ⋱ ⋱ ⋱ ⋮ ⋮ ⋱ ⋮ ⋮ ⋱ ⋱ ⋱ ⋮ Π d 1 1 r ⋯ ⋯ Π d 1 d 1 r 0 d 1 d 1 + 1 ⋯ ⋯ ⋯ 0 d 1 n 0 d 1 + 11 ⋯ ⋯ 0 d 1 + 1 d 1 Π d 1 + 1 d 1 + 1 r Π d 1 + 1 d 1 + 2 r 0 ⋯ 0 d 1 + 1 n ⋮ ⋱ ⋱ ⋮ Π d 1 + 2 d 1 + 1 r − Π d 1 + 1 d 1 + 1 r 0 ⋯ ⋮ ⋮ ⋱ ⋱ ⋮ 0 0 0 ⋯ ⋮ ⋮ ⋱ ⋱ ⋮ ⋮ ⋮ 0 ⋱ 0 0 n 1 ⋯ ⋯ 0 n d 1 0 n d 1 + 1 0 ⋯ 0 0 .

(v) In the case of equality in (1) or (2), the warped product ℬ 1 d 1 × f ℬ 2 d 1 is mixed totally geodesic in Q ε m × R . Moreover, ℬ 1 d 1 × f ℬ 2 d 2 is both ℬ 1 d 1 -minimal and ℬ 2 d 2 -minimal. Thus, ℬ 1 d 1 × f ℬ 2 d 2 is a minimal warped product submanifold in Q ε m × R .

Proof

Let π 1 ⊂ T x ℬ 1 d 1 be a 2-plane for x ∈ ℬ n , then we consider the orthonormal basis { e 1 , … , e d 1 , e d 1 + 1 , … , e n } of T x ℬ n such that { e 1 , … , e d 1 } is an orthonormal basis for T x ℬ 1 d 1 and { e d 1 + 1 , … , e n } is for T x ℬ 2 d 2 . Similarly, { e n + 1 , … , e m } is an orthonormal basis for T x ⊥ ℬ n . Given π = Span { e 1 , e 2 } , the normal vector e n + 1 lies in the direction of the mean curvature vector H , thus from (2.1) and (2.26), we obtain

(3.3) n 2 ∣ ∣ H ∣ ∣ 2 + n ( n − 1 ) 1 − 2 n ‖ T ‖ 2 ε = 2 τ ( x ) + ∣ ∣ Π ∣ ∣ 2 ,

which implies that

(3.4) ∑ i = 1 d 1 Π i i n + 1 2 = 2 τ ( x ) + ∣ ∣ Π ∣ ∣ 2 − n ( n − 1 ) 1 − 2 n ‖ T ‖ 2 ε − ∑ j = d 1 + 1 n Π j j n + 1 2 − 2 ∑ A = 1 d 1 ∑ B = d 1 + 1 n Π A A n + 1 Π B B n + 1 .

Taking these factors into account,

(3.5) Ω = 2 τ ( x ) − n ( n − 1 ) 1 − 2 n ‖ T ‖ 2 ε − ( d 1 − 2 ) ( d 1 − 1 ) ∑ i = 1 d 1 Π i i n + 1 2 − ∑ j = d 1 + 1 n Π j j n + 1 2 − 2 ∑ A = 1 d 1 ∑ B = d 1 + 1 n Π A A n + 1 Π B B n + 1 .

The following is the result of looking at (3.4) and (3.5):

(3.6) ∑ i = 1 d 1 Π i i n + 1 2 = ( d 1 − 1 ) ( Ω + ∣ ∣ Π ∣ ∣ 2 ) .

The above equation can be expressed as

(3.7) ∑ i = 1 d 1 Π i i n + 1 2 = ( d 1 − 1 ) Ω + ∑ i = 1 d 1 ( Π i i n + 1 ) 2 + ∑ j = d 1 + 1 n ( Π i j n + 1 ) 2 + ∑ i ≠ j = 1 n ( Π i j n + 1 ) 2 + ∑ r = n + 2 m ∑ i , j = 1 n ( Π i j r ) 2 .

Thus, if Lemma 2.1 is applied to the above equation, it becomes, i.e.,

t α = Π α α n + 1 , ∀ α ∈ { 1 , … , d 1 }

and

s = Ω + ∑ j = d 1 + 1 n ( Π j j n + 1 ) 2 + ∑ i ≠ j = 1 n ( Π i j n + 1 ) 2 + ∑ r = n + 2 m ∑ i , j = 1 n ( Π i j r ) 2 .

Thus, we obtain that

(3.8) Π 11 n + 1 Π 22 n + 1 ≥ 1 2 Ω + ∑ j = d 1 + 1 n ( Π j j n + 1 ) 2 + ∑ i ≠ j = 1 n ( Π i j n + 1 ) 2 + ∑ r = n + 2 m ∑ i , j = 1 n ( Π i j r ) 2 .

We derive the following values from (2.1) and (2.12):

(3.9) K ( π 1 ) = ( 1 − ‖ T ‖ 2 ) ε + ∑ r = n + 1 m ( Π 11 r Π 22 r − ( Π 12 r ) 2 ) .

Equations (3.8) and (3.9) are combined to give

(3.10) K ( π 1 ) ≥ ( 1 − ‖ T ‖ 2 ) ε + 1 2 Ω + 1 2 ∑ j = d 1 + 1 n ( Π j j n + 1 ) 2 + ∑ r = n + 1 m ( Π 11 r Π 22 r − ( Π 12 r ) 2 ) + 1 2 ∑ i ≠ j = 1 n ( Π i j n + 1 ) 2 + 1 2 ∑ r = n + 2 m ∑ i , j = 1 n ( Π i j r ) 2 .

Taking the last two terms from the equation above, we can obtain the following result:

(3.11) ∑ i , j = 1 i ≠ j n ( Π i j n + 1 ) 2 = ∑ i , j = 3 i ≠ j n ( Π i j n + 1 ) 2 + 2 ∑ j = 3 n ( Π 1 j n + 1 ) 2 + 2 ( Π 12 n + 1 ) 2 + 2 ∑ j = 3 n ( Π 2 j n + 1 ) 2 .

Moreover, for the last term, we obtain

(3.12) ∑ r = n + 2 m ∑ i , j = 1 n ( Π i j r ) 2 = ∑ r = n + 2 m ∑ i , j = 3 n ( Π i j r ) 2 + 2 ∑ r = n + 2 m ∑ j = 3 n ( Π 1 j r ) 2 + 2 ∑ r = n + 2 m ∑ j = 3 n ( Π 2 j r ) 2 + 2 ( Π 12 r ) 2 + ∑ r = n + 2 m ( ( Π 11 r ) 2 + ( Π 22 r ) 2 ) .

Furthermore, we have

(3.13) ∑ r = n + 2 m Π 11 r Π 22 r + 1 2 ∑ r = n + 2 m ( ( Π 11 r ) 2 + ( Π 22 r ) 2 ) = 1 2 ∑ r = n + 2 m ( Π 11 r + Π 22 r ) 2 ,

(3.14) ∑ j = 3 n ( ( Π 1 j n + 1 ) 2 + ( Π 2 j n + 1 ) 2 ) + ∑ r = n + 2 m ∑ j = 3 n ( Π 1 j r ) 2 + ∑ r = n + 2 m ∑ j = 3 n ( Π 2 j r ) 2 = ∑ r = n + 1 m ∑ j = 3 n { ( Π 1 j r ) 2 + ( Π 2 j r ) 2 } .

After adding (3.11) and (3.12), then using (3.13) and (3.14), and taking account that ( Π 12 n + 1 ) 2 + ∑ r = n + 2 m ( Π 12 n + 1 ) 2 = ∑ r = n + 1 m ( Π 12 n + 1 ) 2 , we obtain

(3.15) ∑ i , j = 1 i ≠ j n ( Π i j n + 1 ) 2 + ∑ r = n + 2 m ∑ i , j = 1 n ( Π i j r ) 2 = 2 ∑ r = n + 1 m ∑ j = 3 n { ( Π 1 j r ) 2 + ( Π 2 j r ) 2 } + ∑ i , j = 3 i ≠ j n ( Π i j n + 1 ) 2 + ∑ r = n + 2 m ∑ i , j = 3 n ( Π j j r ) 2 − 2 ∑ r = n + 2 m { Π 11 r Π 22 r − ( Π 12 r ) 2 } + ∑ r = n + 2 m ( Π 11 r + Π 22 r ) 2 .

It follows from (3.10) and (3.15), one derives

K ( π 1 ) ≥ ( 1 − ‖ T ‖ 2 ) ε + 1 2 Ω + 1 2 ∑ β = d 1 + 1 n ( Π β β n + 1 ) 2 + ∑ r = n + 1 m ∑ j = 3 n { ( Π 2 j r ) 2 + ( Π 2 j r ) 2 } + 1 2 ∑ i , j = 3 i ≠ j n ( Π i j n + 1 ) 2 + ∑ r = n + 2 m ∑ i , j = 3 n ( Π j j r ) 2 + 1 2 ∑ r = n + 2 m ( Π 11 r + Π 22 r ) 2 ,

which implies that

K ( π 1 ) ≥ ( 1 − ‖ T ‖ 2 ) ε + 1 2 Ω + ∑ i , j = 3 i ≠ j n ( Π i j n + 1 ) 2 + ∑ r = n + 2 m ∑ i , j = 3 n ( Π j j r ) 2 + ∑ β = d 1 + 1 n ( Π β β n + 1 ) 2 .

From (3.5), we arrive at

(3.16) K ( π 1 ) ≥ ( 1 − ‖ T ‖ 2 ) ε + τ ( x ) + 1 2 ( d 1 − 1 ) ∑ α = 1 n Π α α n + 1 2 − n 2 2 ∣ ∣ ℋ ∣ ∣ 2 − n ( n − 1 ) 2 1 − 2 n ‖ T ‖ 2 ε + 1 2 ∑ i , j = 3 i ≠ j n ( Π i j n + 1 ) 2 + ∑ r = n + 2 m ∑ i , j = 3 n ( Π j j r ) 2 + ∑ β = d 1 + 1 n ( Π β β n + 1 ) 2 .

We can obtain the following equation using (2.9) and (2.17) in (3.16):

K ( π 1 ) ≥ τ 1 ( x ) + τ 2 ( x ) + d 2 ∇ f f − n 2 2 ∣ ∣ ℋ ∣ ∣ 2 + 1 − n ( n − 1 ) 2 + ( n − 2 ) ‖ T ‖ 2 ε + 1 2 ∑ i , j = 3 i ≠ j n ( Π i j n + 1 ) 2 + ∑ r = n + 2 m ∑ i , j = 3 n ( Π i j r ) 2 + ∑ β = d 1 + 1 n ( Π β β n + 1 ) 2 ,

where τ i ( x ) , i = 1 , 2 , is the scalar curvature of ℬ i d i . This implies that

(3.17) τ 1 ( x ) − K ( π 1 ) ≤ n 2 2 ∣ ∣ ℋ ∣ ∣ 2 − τ 2 ( x ) − d 2 ∇ f f + n ( n − 1 ) 2 − ( n − 2 ) ‖ T ‖ 2 − 1 ε − 1 2 ∑ i , j = 3 i ≠ j n ( Π i j n + 1 ) 2 + ∑ r = n + 2 m ∑ i , j = 3 n ( Π i j r ) 2 + ∑ β = d 1 + 1 n ( Π β β n + 1 ) 2 .

The Gauss equation (2.4) for τ 2 ( x ) gives us

(3.18) τ 2 ( x ) = d 2 ( d 2 − 1 ) 2 1 − 2 d 2 ‖ T ‖ 2 ε − 1 2 ∑ r = n + 1 m ∑ A , B = d 1 + 1 n ( Π A B r ) 2 − 1 2 ∑ r = n + 1 m ( Π d 1 + 1 d 1 + 1 r + … + Π n n r ) .

According to equations (3.17) and (3.18), we can state the following.

(3.19) τ 1 ( x ) − K ( π 1 ) ≤ n 2 2 ∣ ∣ ℋ ∣ ∣ 2 − d 2 ( d 2 − 1 ) 2 1 − 2 d 2 ‖ T ‖ 2 ε − 1 2 ∑ i , j = 3 i ≠ j n ( Π i j n + 1 ) 2 + ∑ r = n + 2 m ∑ i , j = 3 n ( Π i j r ) 2 + ∑ β = d 1 + 1 n ( Π β β n + 1 ) 2 − ∑ r = n + 1 m ∑ A , B = d 1 + 1 n ( Π A B r ) 2 + n ( n − 1 ) 2 − ( n − 2 ) ‖ T ‖ 2 − 1 ε − d 2 ∇ f f .

Then, the last relation turns into

(3.20) τ 1 ( x ) − K ( π 1 ) ≤ n 2 2 ∣ ∣ ℋ ∣ ∣ 2 − d 2 ( d 2 − 1 ) 2 1 − 2 d 2 ‖ T ‖ 2 ε + n ( n − 1 ) 2 − ( n − 2 ) ‖ T ‖ 2 − 1 ε − 1 2 ∑ k , l = 3 k ≠ l d 1 ( Π k l n + 1 ) 2 + 2 ∑ k = 3 m ∑ l = d 1 + 1 n ( Π k l n + 1 ) 2 + ∑ A , B = 1 A ≠ B d 1 ( Π k l n + 1 ) 2 + ∑ r = n + 2 m ∑ k , l = 3 d 1 ( Π k l r ) 2 + 2 ∑ r = n + 2 m ∑ k = 3 d 1 ∑ A = d 1 + 1 n ( Π k l r ) 2 + ∑ r = n + 2 m ∑ A , B = d 1 + 1 n ( Π A B r ) 2 + ∑ β = d 1 + 1 n ( Π β β n + 1 ) 2 − ∑ r = n + 1 m ∑ A , B = d 1 + 1 n ( Π A B r ) 2 − d 2 ∇ f f .

In order to solve the previous equation, we can use the following two relations:

∑ A = d 1 + 1 n ( Π A A n + 1 ) 2 + ∑ A , B = d 1 + 1 A ≠ B n ( Π A B n + 1 ) 2 = ∑ A , B = d 1 + 1 n ( Π A B n + 1 ) 2

and

∑ A , B = d 1 + 1 n ( Π A B n + 1 ) 2 + ∑ r = n + 2 m ∑ A , B = d 1 + 1 n ( Π A B r ) 2 = ∑ r = n + 1 m ∑ A , B = d 1 + 1 n ( Π A B r ) 2 .

Assertion (3.20) follows as

(3.21) τ 1 ( x ) − K ( π 1 ) ≤ d 1 2 ( d 1 + 2 d 2 − 1 ) − ( d 1 − 1 ) ‖ T ‖ 2 − 1 ε − 1 2 ∑ k , l = 3 k ≠ l d 1 ( Π k l n + 1 ) 2 + ∑ r = n + 2 m ∑ k , l = 3 d 1 ( Π k l r ) 2 + 2 ∑ α = 3 d 1 ∑ β = d 1 + 1 n ( Π α β n + 1 ) 2 + 2 ∑ r = n + 1 m ∑ A = 3 d 1 ∑ B = d 1 + 1 n ( Π A B r ) 2 + n 2 2 ∣ ∣ ℋ ∣ ∣ 2 − d 2 ∇ f f .

The first inequality of Theorem 3.1 holds from the above equation and (2.13). For the second case if π 2 ⊂ T x ℬ 2 d 2 , we consider π 2 = Span { e d 1 + 1 , e d 1 + 2 } . Similar to the first case, the following results can be obtained:

∑ α = d 1 + 1 n Π α α n + 1 2 = 2 τ ( x ) + ∣ ∣ Π ∣ ∣ 2 − n ( n − 1 ) 1 − 2 n ‖ T ‖ 2 ε − ∑ β = 1 d 1 Π β β n + 1 2 − 2 ∑ α = 1 d 1 ∑ β = d 1 + 1 n Π α α n + 1 Π β β n + 1 .

Next, we will consider

Ψ = 2 τ ( x ) − n ( n − 1 ) 1 − 2 n ‖ T ‖ 2 ε − ( d 1 − 2 ) ( d 1 − 1 ) ∑ α = d 1 + 1 n Π α α n + 1 2 − ∑ β = d 1 + 1 n Π β β n + 1 2 − 2 ∑ α = 1 d 1 ∑ β = d 1 + 1 n Π α α n + 1 Π β β n + 1 .

The last two equation implies that

∑ α = d 1 + 1 n Π α α n + 1 2 = ( d 2 − 1 ) ( Ψ + ∣ ∣ Π ∣ ∣ 2 ) ,

which implies that

(3.22) ∑ α = d 1 + 1 n Π α α n + 1 2 = ( d 2 − 1 ) Ψ + ∑ α = 1 d 1 Π α α n + 1 2 + ∑ β = d 1 + 1 n Π β β n + 1 2 + ∑ α , β = 1 α ≠ β n ( Π α β n + 1 ) 2 + ∑ r = n + 2 m ∑ α = β = 1 n ( Π α β r ) 2 .

Similarly, we obtain the following equation when we apply Lemma 2.1:

(3.23) Π d 1 + 1 d 1 + 1 n + 1 Π d 1 + 2 d 1 + 2 n + 1 ≥ 1 2 Ψ + ∑ α = 1 d 1 Π α α n + 1 2 + ∑ α , β = 1 α ≠ β n ( Π α β n + 1 ) 2 + ∑ r = n + 2 m ∑ α = β = 1 n ( Π α β r ) 2 .

From (2.1) and (2.12), we find that

(3.24) K ( π 2 ) = ( 1 − ‖ T ‖ 2 ) ε + ∑ r = n + 1 m ( Π d 1 + 1 d 1 + 1 r Π d 1 + 2 d 1 + 2 r − ( Π d 1 + 1 d 1 + 2 r ) 2 ) .

Using equations (3.23) and, respectively, (3.24), we obtain

(3.25) K ( π 2 ) ≥ ( 1 − ‖ T ‖ 2 ) ε + ∑ r = n + 1 m ( Π d 1 + 1 d 1 + 1 r Π d 1 + 2 d 1 + 2 r − ( Π d 1 + 1 d 1 + 2 r ) 2 ) 1 2 Ψ + ∑ α = 1 d 1 Π α α n + 1 2 + ∑ α , β = 1 α ≠ β n ( Π α β n + 1 ) 2 + ∑ r = n + 2 m ∑ α = β = 1 n ( Π α β r ) 2 .

The second inequality of Theorem 3.1 can be found using the same method as (3.5) and (3.21). Considering the case, π 1 ⊂ T x ℬ 1 d 1 , then the equality holds if and only if in (3.8), (3.10), (3.17), (3.18), and (3.21), equalities are preserved. Based on this, we obtain the following result:

(3.26) Π 11 n + 1 + Π 22 n + 1 = Π 33 n + 1 = … = Π d 1 d 1 n + 1 ,

(3.27) ∑ r = n + 2 m ∑ j = 3 n ( ( Π 2 j r ) 2 + ( Π 2 j r ) 2 ) + ∑ r = n + 2 m ( Π 11 r + Π 22 r ) 2 = 0 ,

(3.28) ∑ r = n + 1 m ( Π d 1 + 1 d 1 + 1 r + … + Π n n r ) = ∑ α = 1 d 1 Π α α n + 1 2 = 0 ,

(3.29) ∑ k , l = 3 k ≠ l d 1 ( Π k l n + 1 ) 2 + ∑ r = n + 2 m ∑ k , l = 3 d 1 ( Π k l r ) 2 + ∑ α = 3 d 1 ∑ β = d 1 + 1 n ( Π α β n + 1 ) 2 + ∑ r = n + 2 m ∑ A = 3 d 1 ∑ B = d 1 + 1 n ( Π A B r ) 2 = 0 .

Equation (3.28) confirms that the warped product ℬ 1 d 1 × f ℬ 2 d 1 is both ℬ 1 d 1 -minimal and ℬ 2 d 2 -minimal warped product submanifold in Q ε m × R . Based on our results, we conclude that the warped product submanifold ℬ 1 d 1 × f ℬ 2 d 1 is minimal in Q ε m × R . The other case will be classified into two ways based on vector fields e r . Based on the assumption that r = n + 1 , here is what we define

Π 11 n + 1 + Π 22 n + 1 = Π 33 n + 1 = … = Π d 1 d 1 n + 1

and

∑ j = 3 n ( Π 1 j n + 1 ) 2 = ∑ j = 3 n ( Π 2 j n + 1 ) 2 = ∑ k , l = 3 k ≠ l d 1 ( Π k l n + 1 ) 2 = ∑ α = 3 d 1 ∑ β = d 1 + 1 n ( Π α β n + 1 ) 2 = 0 .

This condition can be represented by the following matrices:

( i ) A e n + 1 = μ 1 Π 12 n + 1 0 ⋯ 0 1 d 1 0 1 d 1 + 1 ⋯ 0 1 n Π 12 n + 1 μ 2 0 ⋯ ⋮ ⋮ ⋯ ⋮ 0 0 μ ⋯ ⋮ ⋮ ⋯ ⋮ ⋮ ⋮ ⋮ ⋱ ⋮ ⋮ ⋯ ⋮ 0 d 1 1 0 0 ⋯ μ 0 d 1 d 1 + 1 ⋯ 0 d 1 n 0 d 1 + 11 ⋯ ⋯ ⋯ 0 d 1 + 1 d 1 Π d 1 + 1 d 1 + 1 n + 1 ⋯ Π d 1 + 1 n n + 1 ⋮ ⋱ ⋱ ⋱ ⋮ ⋮ ⋱ ⋮ 0 n 1 ⋯ ⋯ ⋯ 0 n d 1 Π n d 1 + 1 n + 1 ⋯ Π n n n + 1

where μ = μ 1 + μ 2 gives (i) of Theorem 3.1. As well, if r ∈ { n + 2 , … , m } , then it must follow that

Π 11 r + Π 22 r = ∑ j = 3 n ( Π 1 j r ) 2 = ∑ j = 3 n ( Π 2 j r ) 2 = ∑ k , l = 3 k ≠ l d 1 ( Π k l r ) 2 = ∑ α = 3 d 1 ∑ β = d 1 + 1 n ( Π α β r ) 2 = 0 .

That is equivalent to the second metric:

( i i ) A e r = Π 11 r Π 12 r 0 ⋯ 0 1 d 1 0 1 d 1 + 1 ⋯ 0 1 n Π 21 r − Π 11 r 0 ⋯ ⋮ ⋮ ⋯ ⋮ 0 0 0 33 ⋯ ⋮ ⋮ ⋯ ⋮ ⋮ ⋮ ⋮ ⋱ ⋮ ⋮ ⋯ ⋮ 0 d 1 1 0 0 ⋯ 0 d 1 d 1 0 d 1 d 1 + 1 ⋯ 0 d 1 n 0 d 1 + 11 ⋯ ⋯ ⋯ 0 d 1 + 1 d 1 Π d 1 + 1 d 1 + 1 r ⋯ Π d 1 + 1 n r ⋮ ⋱ ⋱ ⋱ ⋮ ⋮ ⋱ ⋮ 0 n 1 ⋯ ⋯ ⋯ 0 n d 1 Π n d 1 + 1 r ⋯ Π n n r .

It is evident from the first two conditions that ℬ 1 d 1 × f ℬ 2 d 1 is mixed totally geodesic submanifold in Q ε m × R .

It is also necessary for the equality sign in (ii) to hold if the following two matrices are satisfied:

( i i i ) A e n + 1 = Π 11 n + 1 ⋯ ⋯ Π 1 d 1 n + 1 0 1 d 1 + 1 ⋯ ⋯ ⋯ 0 1 n ⋮ ⋱ ⋮ ⋮ ⋱ ⋱ ⋱ ⋮ ⋮ ⋱ ⋮ ⋮ ⋱ ⋱ ⋱ ⋮ Π d 1 11 n + 1 ⋯ ⋯ Π d 1 d 1 n + 1 0 d 1 d 1 + 1 ⋯ ⋯ ⋯ 0 d 1 n 0 d 1 + 11 ⋯ ⋯ 0 d 1 + 1 d 1 μ 1 Π d 1 + 1 d 1 + 2 n + 1 0 ⋯ 0 d 1 + 1 n ⋮ ⋱ ⋱ ⋮ Π d 1 + 2 d 1 + 1 n + 1 μ 2 0 ⋯ ⋮ ⋮ ⋱ ⋱ ⋮ 0 0 μ ⋯ ⋮ ⋮ ⋱ ⋱ ⋮ ⋮ ⋮ 0 ⋱ 0 0 n 1 ⋯ ⋯ 0 n d 1 0 n d 1 + 1 0 ⋯ 0 μ ,

where μ = μ 1 + μ 2 . If r ∈ { n + 2 , … , m } , thus we have

( i v ) A e r = Π 11 r ⋯ ⋯ Π 1 d 1 r 0 1 d 1 + 1 ⋯ ⋯ ⋯ 0 1 n ⋮ ⋱ ⋮ ⋮ ⋱ ⋱ ⋱ ⋮ ⋮ ⋱ ⋮ ⋮ ⋱ ⋱ ⋱ ⋮ Π d 1 11 r ⋯ ⋯ Π d 1 d 1 r 0 d 1 d 1 + 1 ⋯ ⋯ ⋯ 0 d 1 n 0 d 1 + 11 ⋯ ⋯ 0 d 1 + 1 d 1 Π d 1 + 1 d 1 + 1 r Π d 1 + 1 d 1 + 2 r 0 ⋯ 0 d 1 + 1 n ⋮ ⋱ ⋱ ⋮ Π d 1 + 2 d 1 + 1 r − Π d 1 + 1 d 1 + 1 r 0 ⋯ ⋮ ⋮ ⋱ ⋱ ⋮ 0 0 0 ⋯ ⋮ ⋮ ⋱ ⋱ ⋮ ⋮ ⋮ 0 ⋱ 0 0 n 1 ⋯ ⋯ 0 n d 1 0 n d 1 + 1 0 ⋯ 0 0 .

It is also determined by the above that ℬ 1 d 1 × f ℬ 2 d 1 is both ℬ 1 d 1 -minimal and ℬ 2 d 2 -minimal warped product submanifold in Q ε m × R , which suggest that the warped product submanifold ℬ 1 d 1 × f ℬ 2 d 1 is minimal in Q ε m × R .□

A wide range of distinct geometrical properties can be obtained from immersion in warped product manifolds. It is now possible to find the inequalities for the Riemannian manifolds with constant sectional curvature ∈ { 1 , 0 , − 1 } that can be presented as a product manifold Q ε m × R . The following result is obtained, the result is relevant to the application of the warped product submanifold in S n × R with ε = 1 .

Theorem 3.2

Assume that ℬ n = ℬ 1 d 1 × f ℬ 2 d 2 is an isometric immersion into a Euclidean sphere S n × R . Then, for each point x ∈ ℬ n and each plane section π i ⊂ T x ℬ i d i , for i = 1 , 2 , we obtain the following for

  1. π 1 ⊂ T x ℬ 1 d 1

    δ ℬ 1 d 1 ( x ) ≤ n 2 2 ∣ ∣ ℋ ∣ ∣ 2 + d 2 ∣ ∣ ∇ ( ln f ) ∣ ∣ 2 − d 2 Δ ( ln f ) + d 1 2 ( d 1 + 2 d 2 − 1 ) − ( d 1 − 1 ) ‖ T ‖ 2 − 1 .

  2. For π 2 ⊂ T x ℬ 2 d 2

    δ ℬ 2 d 2 ( x ) ≤ n 2 2 ∣ ∣ ℋ ∣ ∣ 2 + d 2 ∣ ∣ ∇ ( ln f ) ∣ ∣ 2 − d 2 Δ ( ln f ) + d 2 2 ( d 2 + 2 d 1 − 1 ) − ( d 2 − 1 ) ‖ T ‖ 2 − 1 .

According to Theorem  3.1, the above inequality implies equalities.

Proof

Now, we consider the constant sectional curvature ε = 1 and Q ε m = S m for the product manifold S m × R . Then, inserting the proceeding value in (3.1) and (3.2), we obtain the result.□

Here, we present the application of the warped product submanifold in H m × R with ε = − 1 .

Theorem 3.3

Assume that ℬ n = ℬ 1 d 1 × f ℬ 2 d 2 is an isometric immersion into a Hyperbolic spaces H m × R . Then, for each point x ∈ ℬ n and each plane section π i ⊂ T x ℬ i n i , for i = 1 , 2 , we obtain the following for

  1. π 1 ⊂ T x ℬ 1 d 1 or π 2 ⊂ T x ℬ 2 d 2

    δ ℬ 1 d 1 ( x ) ≤ n 2 2 ∣ ∣ H ∣ ∣ 2 + d 2 ∣ ∣ ∇ ( ln f ) ∣ ∣ 2 − d 2 Δ ( ln f ) − d 1 2 ( d 1 + 2 d 2 − 1 ) − ( d 1 − 1 ) ‖ T ‖ 2 − 1 .

  2. For π 2 ⊂ T x ℬ 2 d 2

    δ ℬ 2 d 2 ( x ) ≤ n 2 2 ∣ ∣ H ∣ ∣ 2 + d 2 ∣ ∣ ∇ ( ln f ) ∣ ∣ 2 − d 2 Δ ( ln f ) − d 2 2 ( d 2 + 2 d 1 − 1 ) − ( d 2 − 1 ) ‖ T ‖ 2 − 1 .

According to Theorem 3.1, the above inequality implies equalities.

Proof

Now we assume that Q ε m = H m and constant sectional curvature ε = − 1 for the product manifold H m × R . Then using these values in (3.1) and (3.2), we obtain required result.□

4 Generalized the results for space form Q ε m

Our final result is obtained by concluding that R = 0 , which implies that T = 0 , given Theorem 3.1.

Theorem 4.1

Assume that ℬ n = ℬ 1 d 1 × f ℬ 2 d 2 is an isometric immersion into a space form Q ε m . Then, for each point x ∈ ℬ n and each plane section π i ⊂ T x ℬ i d i , for i = 1 , 2 , we obtain

  1. Let π 1 ⊂ T x ℬ 1 d 1 , then

    (4.1) δ ℬ 1 d 1 ( x ) ≤ n 2 2 ∣ ∣ ℋ ∣ ∣ 2 + d 2 ∣ ∣ ∇ ( ln f ) ∣ ∣ 2 + 1 2 d 1 ( d 1 + 2 d 2 − 1 ) ε − ε − d 2 Δ ( ln f ) .

    It is only possible to compute the equality of the above inequality at x ∈ ℬ n if and only if there exists an orthonormal basis { e 1 , … , e n } of T x ℬ n and an orthonormal basis { e n + 1 , … , e m } of T x ⊥ ℬ n such that

    1. π 1 = Span { e 1 , e 2 } and

    2. Shape operators can be expressed as follows:

      ( i ) A e n + 1 = μ 1 Π 12 n + 1 0 ⋯ 0 1 d 1 0 1 d 1 + 1 ⋯ 0 1 n Π 12 n + 1 μ 2 0 ⋯ ⋮ ⋮ ⋯ ⋮ 0 0 μ ⋯ ⋮ ⋮ ⋯ ⋮ ⋮ ⋮ ⋮ ⋱ ⋮ ⋮ ⋯ ⋮ 0 d 1 1 0 0 ⋯ μ 0 d 1 d 1 + 1 ⋯ 0 d 1 n 0 d 1 + 11 ⋯ ⋯ ⋯ 0 d 1 + 1 d 1 Π d 1 + 1 d 1 + 1 n + 1 ⋯ Π d 1 + 1 n n + 1 ⋮ ⋱ ⋱ ⋱ ⋮ ⋮ ⋱ ⋮ 0 n 1 ⋯ ⋯ ⋯ 0 n d 1 Π n d 1 + 1 n + 1 ⋯ Π n n n + 1 ,

      where μ = μ 1 + μ 2 . If r ∈ { n + 2 , … , m } , then we have the matrix

      ( i i ) A e r = Π 11 r Π 12 r 0 ⋯ 0 1 d 1 0 1 d 1 + 1 ⋯ 0 1 n Π 21 r − Π 11 r 0 ⋯ ⋮ ⋮ ⋯ ⋮ 0 0 0 33 ⋯ ⋮ ⋮ ⋯ ⋮ ⋮ ⋮ ⋮ ⋱ ⋮ ⋮ ⋯ ⋮ 0 d 1 1 0 0 ⋯ 0 d 1 d 1 0 d 1 d 1 + 1 ⋯ 0 d 1 n 0 d 1 + 11 ⋯ ⋯ ⋯ 0 d 1 + 1 d 1 Π d 1 + 1 d 1 + 1 r ⋯ Π d 1 + 1 n r ⋮ ⋱ ⋱ ⋱ ⋮ ⋮ ⋱ ⋮ 0 n 1 ⋯ ⋯ ⋯ 0 n d 1 Π n d 1 + 1 r ⋯ Π n n r .

  2. If π 2 ⊂ T x ℬ 2 d 2 , then

    (4.2) δ ℬ 2 d 2 ( x ) ≤ n 2 2 ∣ ∣ ℋ ∣ ∣ 2 + d 2 ∣ ∣ ∇ ( ln f ) ∣ ∣ 2 + 1 2 d 2 ( d 2 + 2 d 1 − 1 ) ε − ε − d 2 Δ ( ln f ) .

It follows that the above equation has equalities if and only if there exists an orthonormal basis { e 1 , … , e n } of T x ℬ n and an orthonormal basis { e n + 1 , … , e m } of T x ⊥ ℬ n such that

  1. π 2 = Span { e d 1 + 1 , e d 1 + 2 } and

  2. The shape operators are represented as follows:

    ( i i i ) A e n + 1 = Π 11 n + 1 ⋯ ⋯ Π 1 d 1 n + 1 0 1 d 1 + 1 ⋯ ⋯ ⋯ 0 1 n ⋮ ⋱ ⋮ ⋮ ⋱ ⋱ ⋱ ⋮ ⋮ ⋱ ⋮ ⋮ ⋱ ⋱ ⋱ ⋮ Π d 1 11 n + 1 ⋯ ⋯ Π d 1 d 1 n + 1 0 d 1 d 1 + 1 ⋯ ⋯ ⋯ 0 d 1 n 0 d 1 + 11 ⋯ ⋯ 0 d 1 + 1 d 1 μ 1 Π d 1 + 1 d 1 + 2 n + 1 0 ⋯ 0 d 1 + 1 n ⋮ ⋱ ⋱ ⋮ Π d 1 + 2 d 1 + 1 n + 1 μ 2 0 ⋯ ⋮ ⋮ ⋱ ⋱ ⋮ 0 0 μ ⋯ ⋮ ⋮ ⋱ ⋱ ⋮ ⋮ ⋮ 0 ⋱ 0 0 n 1 ⋯ ⋯ 0 n d 1 0 n d 1 + 1 0 ⋯ 0 μ ,

    where μ = μ 1 + μ 2 . If r ∈ { n + 2 , … , m } , thus we have

    ( i v ) A e r = Π 11 r ⋯ ⋯ Π 1 d 1 r 0 1 d 1 + 1 ⋯ ⋯ ⋯ 0 1 n ⋮ ⋱ ⋮ ⋮ ⋱ ⋱ ⋱ ⋮ ⋮ ⋱ ⋮ ⋮ ⋱ ⋱ ⋱ ⋮ Π d 1 11 r ⋯ ⋯ Π d 1 d 1 r 0 d 1 d 1 + 1 ⋯ ⋯ ⋯ 0 d 1 n 0 d 1 + 11 ⋯ ⋯ 0 d 1 + 1 d 1 Π d 1 + 1 d 1 + 1 r Π d 1 + 1 d 1 + 2 r 0 ⋯ 0 d 1 + 1 n ⋮ ⋱ ⋱ ⋮ Π d 1 + 2 d 1 + 1 1 − Π d 1 + 1 d 1 + 1 r 0 ⋯ ⋮ ⋮ ⋱ ⋱ ⋮ 0 0 0 ⋯ ⋮ ⋮ ⋱ ⋱ ⋮ ⋮ ⋮ 0 ⋱ 0 0 n 1 ⋯ ⋯ 0 n d 1 0 n d 1 + 1 0 ⋯ 0 0 ,

    (v) If the equality holds in (1) or (2), then ℬ 1 d 1 × f ℬ 2 d 2 is mixed totally geodesic in space form Q ε m . Moreover, ℬ 1 d 1 × f ℬ 2 d 2 is both ℬ 1 d 1 -minimal and ℬ 2 d 2 -minimal. Thus, ℬ 1 d 1 × f ℬ 2 d 2 is a minimal warped product submanifold in space form Q ε m .

5 Several applications of the obtained Dirichlet eigenvalue inequalities

Determining the upper bound of the eigenvalue of the Laplacian on a specific manifold is an essential aspect of Riemannian geometry. This pursuit aims to examine eigenvalues, which arise as solutions to the Dirichlet boundary value problems for curvature functions. Understanding these eigenvalues is crucial, and one key objective is to establish their upper bound. This approach is particularly valuable when considering the diverse range of boundary conditions on a manifold and focusing on the Dirichlet boundary condition. An appropriate Laplacian limit on the given manifold can be defined by determining the upper bound of the eigenvalue. As an example, when we have the compact domain Σ in a complete noncompact Riemannian manifold, then we have the Dirichlet boundary condition whose first eigenvalue is υ 1 ( Σ ) > 0

(5.1) Δ σ + υ σ = 0 on Σ and σ = 0 on ∂ Σ ,

This equation corresponds to Δ where σ is a nonzero function defined on ℬ n . Then, υ 1 ( ℬ n ) is expressed as inf Σ υ 1 ( Σ ) .

If Dirichlet boundary conditions are considered, the Dirichlet eigenvalues are identical to the Laplace eigenvalues. The eigenvalues of differential equations are of significant interest to many branches of mathematics, including number theory, mathematical physics, and number theory. Among their most notable results is their ability to characterize a domain’s geometry. For instance, the first Dirichlet eigenvalue of a domain relates to its diameter, while the higher eigenvalues reflect the domain’s curvature and its embedding within Euclidean space. Additionally, the Dirichlet eigenvalues play a crucial role in solving the heat equation on a domain, where they determine both the eigenfunctions and the rate of decay of the solution. Based on the assumption that σ is the non-constant warping function on the compact warped product submanifold ℬ n . In this case, the minimum principle on υ 1 leads to (see, e.g., [19])

(5.2) ∫ ℬ n ∣ ∣ ∇ σ ∣ ∣ 2 d V ≥ υ 1 ∫ ℬ n ( σ ) 2 d V .

As a result, equality can only be achieved if and only if

(5.3) Δ σ = υ 1 σ .

We know that the boundary for compact support is covered by ℬ 1 × { d 2 } . Implementation of the integration along the base manifold ℬ d 1 in Eqs (3.1) and (3.2), we obtain the following result.

Theorem 5.1

Assume that ℬ n = ℬ 1 d 1 × f ℬ 2 d 2 is a compact warped product submanifold into a product manifold Q ε m × R . Then, we have

(5.4) ∫ ℬ 1 d 1 × { d 2 } δ ℬ 1 d 1 ( x ) d V ≤ n 2 2 ∫ ℬ 1 d 1 × { d 2 } ∣ ∣ H ∣ ∣ 2 d V + υ 1 d 2 ∫ ℬ 1 d 1 × { d 2 } ( ln f ) 2 d V + ∫ ℬ 1 d 1 × { d 2 } d 1 2 ( d 1 + 2 d 2 − 1 ) − ( d 1 − 1 ) ‖ T ‖ 2 − 1 ε d V ,

for π 1 ⊂ T ℬ 1 d 1 . Moreover, we have

(5.5) ∫ { d 1 } × ℬ 2 d 2 δ ℬ 2 d 2 ( x ) d V ≤ n 2 2 ∫ { d 1 } × ℬ 2 d 2 ∣ ∣ H ∣ ∣ 2 d V + υ 1 d 2 ∫ { d 1 } × ℬ 2 d 2 ( ln f ) 2 d V + ∫ { d 1 } × ℬ 2 d 2 d 2 2 ( d 2 + 2 d 1 − 1 ) − ( d 2 − 1 ) ‖ T ‖ 2 − 1 ε d V ,

for π 2 ⊂ T ℬ 2 d 2 .

Proof

We obtain the result easily by replacing σ = ln f in (3.1) and (3.2) with the Stokes theorem condition ∫ Δ σ d V = 0 for compact support.□

In the following, we present the applications of Brochler formulas:

Theorem 5.2

Assuming that ℬ n = ℬ 1 d 1 × f ℬ 2 d 2 is a compact warped product submanifold into a product manifold Q ε m × R . Then, we have

(5.6) ∫ ℬ 1 d 1 × { d 2 } Ric ( ∇ ln f , ∇ ln f ) d V ≥ υ 1 d 2 ∫ ℬ 1 d 1 × { d 2 } δ ℬ 1 d 1 ( x ) d V − n 2 υ 1 2 d 2 ∫ ℬ 1 d 1 × { d 2 } ∣ ∣ H ∣ ∣ 2 d V + υ 1 d 2 ∫ ℬ 1 d 1 × { d 2 } 1 + ( d 1 − 1 ) ‖ T ‖ 2 − d 1 2 ( d 1 + 2 d 2 − 1 ) ε d V − ∫ ℬ 1 d 1 × { d 2 } ‖ ∇ 2 ln f ‖ 2 d V ,

for π 1 ⊂ T ℬ 1 d 1 . Moreover, we have

(5.7) ∫ { d 1 } × ℬ 2 d 2 Ric ( ∇ ln f , ∇ ln f ) d V ≥ υ 1 d 2 ∫ { d 1 } × ℬ 2 d 2 δ ℬ 2 d 2 ( x ) d V ( x ) d V − n 2 υ 1 2 d 2 ∫ { d 1 } × ℬ 2 d 2 ∣ ∣ H ∣ ∣ 2 d V + υ 1 d 2 ∫ { d 1 } × ℬ 2 d 2 1 + ( d 2 − 1 ) ‖ T ‖ 2 − d 2 2 ( d 2 + 2 d 1 − 1 ) ε d V − ∫ { d 1 } × ℬ 2 d 2 ‖ ∇ 2 ln f ‖ 2 d V ,

for π 2 ⊂ T ℬ 2 d 2 .

Proof

A Laplacian whose first eigenfunction is σ is equal to div ( ∇ σ ) for ℬ n connected to the first non zero eigenvalue υ 1 , so that, Δ σ = − υ 1 σ . As a result, recalling Bochner’s formula (see [1]), the differentiable function σ at the Riemannian manifold has the following relation:

1 2 Δ ‖ ∇ σ ‖ 2 = ‖ ∇ 2 σ ‖ 2 + Ric ( ∇ σ , ∇ σ ) + g ( ∇ σ , ∇ ( Δ σ ) ) .

The Stokes theorem tells us that we have to integrate the previous equation to obtain the following result:

(5.8) ∫ ℬ 1 × { d 2 } ‖ ∇ 2 σ ‖ 2 d V + ∫ ℬ 1 × { d 2 } Ric ( ∇ σ , ∇ σ ) d V + ∫ ℬ 1 × { d 2 } g ( ∇ σ , ∇ ( Δ σ ) ) d V = 0 .

Now, using Δ σ = − υ 1 σ and making some rearrangement in Eq. (5.8), we derive

(5.9) ∫ ℬ 1 × { d 2 } ‖ ∇ σ ‖ 2 d V = 1 υ 1 ∫ ℬ 1 × { d 2 } ‖ ∇ 2 σ ‖ 2 d V + ∫ ℬ 1 × { d 2 } Ric ( ∇ σ , ∇ σ ) d V .

Integrating in (3.1) and (3.2) and using the above equation, we obtain the desired results.□

6 Chern’s problem: Finding the conditions under which warped products must be minimal

Our solution to the Chern problem [20] in this section is to provide a partial solution to why a warped submanifold must be minimal in a product manifold Q ε m × R .

Corollary 6.1

Let ℬ n = ℬ 1 d 1 × f ℬ 2 d 2 be an isometric immersion of a warped product submanifold into a product manifold Q ε m × R . Then, for each point x ∈ ℬ n and each π 1 ⊂ T x ℬ 1 d 1 , we have

δ ℬ 1 d 1 ( x ) + d 2 Δ ( ln f ) ≤ d 2 ∣ ∣ ∇ ( ln f ) ∣ ∣ 2 + d 1 2 ( d 1 + 2 d 2 − 1 ) − ( d 1 − 1 ) ‖ T ‖ 2 − 1 ε ,

and if the equality satisfies, then ℬ n is minimal.

The second result is:

Corollary 6.2

Let ℬ n = ℬ 1 d 1 × f ℬ 2 d 2 be an isometric immersion of a warped product submanifold into a product manifold Q ε m × R . Then, for each point x ∈ ℬ n and each π 2 ⊂ T x ℬ 2 d 2 , we have

δ ℬ 2 d 2 ( x ) + d 1 Δ ( ln f ) ≤ d 2 ∣ ∣ ∇ ( ln f ) ∣ ∣ 2 + d 2 2 ( d 2 + 2 d 1 − 1 ) − ( d 2 − 1 ) ‖ T ‖ 2 − 1 ε ,

and if the equality satisfies, then ℬ n is minimal.

Remark 6.1

Finally, we noticed that Theorem 3.1 is the solution of Problem 1.1. Furthermore, Corollaries 6.1 and 6.2 are the solution of Problem 6.2.

7 Conclusion remarks

Riemannian submanifolds have intrinsic and extrinsic invariants. Creating connections between these invariants is one of the fundamental problems of submanifold theory. This pursuit is motivated by Nash’s renowned theory of isometric immersion, which suggests that viewing each Riemannian manifold as a submanifold in a Euclidean space is significant and influential [21]. In this context, the squared mean curvature is the primary extrinsic invariant, while the Ricci curvature and the scalar curvature act as the primary intrinsic invariants [5]. Furthermore, the Chen delta invariant, one of the numerical invariants in algebraic topology, assumes significance in measuring the degree to which a loop in space does not represent the boundary of a surface. Specifically, if a loop functions as a boundary, the Chen delta invariant attains zero value. Otherwise, it quantifies how much the loop deviates from being a boundary. The applications of the delta invariant span a broad spectrum within mathematics, including differential geometry, differential topology, and algebraic geometry and algebraic topology. For instance, researchers have employed it to investigate the topology and geometry of moduli spaces of algebraic curves, examine the geometry of the Kähler-Einstein metric on a complex manifold, and explore the topology and geometry of configuration spaces of particles in a Euclidean space. Moreover, the delta invariant finds practical applications in physics, particularly in topological field theories. Several applications have also been made to physics, including studies of topological field theories [2,22,23].

Acknowledgement

The authors extend their appreciation to the Deanship of Research and Graduate Studies at King Khalid University for funding this work through a Large Research Project under grant number R.G.P.2/12/45.

  1. Funding information: The authors extend their appreciation to the Deanship of Scientific Research at King Khalid University for funding this work through the research groups program under grant number R.G.P.2/12/45.

  2. Author contributions: All authors have accepted responsibility for the entire content of this manuscript and consented to its submission to the journal, reviewed all the results and approved the final version of the manuscript.

  3. Conflict of interest: The authors state no conflicts of interest.

References

[1] R. L. Bishop and B. O’Neill, Manifolds of negative curvature, Trans. Amer. Math. Soc. 145 (1969), 1–49. Search in Google Scholar

[2] B. Chen, Pseudo-Riemannian Geometry, δ-Invariants, and Applications, World Scientific, Hackensack, New Jersey, 2011. Search in Google Scholar

[3] B. Chen, A. Blaga, and G. Vilcu, Differential geometry of submanifolds in complex space forms involving δ-invariants, Mathematics 10 (2022), 591. Search in Google Scholar

[4] B. Chen, On isometric minimal immersions from warped products into real space forms, Proc. Edinb. Math. Soc. 45 (2002), no. 3, 579–587. Search in Google Scholar

[5] M. Tripathi, Improved Chen-Ricci inequality for curvature-like tensors and its applications, Differential Geom. Appl. 29 (2011), no. 5, 685–698. Search in Google Scholar

[6] G. Vilcu, On Chen invariants and inequalities in quaternionic geometry, J. Inequal. Appl. 2013 (2013), 66. Search in Google Scholar

[7] B. Chen, An optimal inequality for CR-warped products in complex space forms involving CR δ-invariant, Int. J. Math. 23 (2012), no. 3, 1250045. Search in Google Scholar

[8] A. Mustafa, C. Ozel, A. Pigazzini, R. Kaur, and G. Shanker, First Chen inequality for general warped product submanifolds of a Riemannian space form and applications, 2021, arXiv: http://arXiv.org/abs/arXiv:2109.08911. Search in Google Scholar

[9] F. Alghamdi, L. Alqahtani, A. Alkhaldi, and A. Ali, An invariant of Riemannian type for Legendrian Warped product submanifolds of Sasakian space forms, Mathematics 11 (2023), 4718. Search in Google Scholar

[10] F. Alghamdi, L. Alqahtani, and A. Ali, Chen inequalities on warped product Legendrian submanifolds in Kenmotsu space forms and applications, J. Inequal. Appl. 2024 (2024), 63. Search in Google Scholar

[11] M. Fatemah and A. Ali, Chen inequality for general warped product submanifold of Riemannian warped products, Phys. Scr. 99 (2024), no. 4, 045229. Search in Google Scholar

[12] H. Chen, G. Chen, and H. Li, Some pinching theorems for minimal submanifolds in Sm(1)×R, Sci. China Math. 56 (2013), no. 8, 1679–1688. Search in Google Scholar

[13] H. Lin and X. Wang, Gap theorems for submanifolds in Hn×R, J. Geom. Phys. 160 (2021), 103998. Search in Google Scholar

[14] C. Qun and C. Qing, Normal scalar curvature and a pinching theorem in Sm×R and Sm×R, Sci. China Math. 54 (2011), 1977–1984. Search in Google Scholar

[15] Z. Hou, X. Zhan, and W. Qiu, Pinching problems of minimal submanifolds in a product space, Vietnam J. Math. 47 (2019), no. 2, 227–253. Search in Google Scholar

[16] X. Zhan, A DDVV type inequality and a pinching theorem for compact minimal submanifolds in a generalized cylinder Sn1(c)×Rn2, Results Math. 74 (2019), no. 3, 24. Search in Google Scholar

[17] U. Abresch and H. Rosenberg, The Hopf differential for constant mean curvature surfaces in S2×R and Hn×R, Acta Math. 193 (2004), 141–174. Search in Google Scholar

[18] R. Pedrosa and M. Ritoré, Isoperimetric domains in the Riemannian product of a circle with a simply connected space form and applications to free boundary problems, Indiana Univ. Math. J. 48 (1999), no. 4, 1357–1394. Search in Google Scholar

[19] R. Bryant, Second order families of special Lagrangian 3-folds, in perspectives in Riemannian geometry, CRM Proc. Lecture Notes, Amer. Math. Soc., Providence, RI, vol. 40, 2006, pp. 63–98. Search in Google Scholar

[20] S. S. Chern, Minimal submanifold in a Riemannian manifold, Lecture notes, University of Kansas: Lawrence, KS, USA, 1968. Search in Google Scholar

[21] J. Nash, The embedding problem for Riemannian manifolds, Ann. of Math. 63 (1956), no. 1, 20–63. Search in Google Scholar

[22] B. Daniel, Isometric immersions into Sn×R and Hn×R and applications to minimal surfaces, Trans. Amer. Math. Soc. 361 (2009), 6255–6282. Search in Google Scholar

[23] Y. Li, N. Turki, S. Deshmukh, and O. Belova, Euclidean hypersurfaces isometric to spheres, AIMS Math. 9 (2024), 28306–28319. Search in Google Scholar
Received: 2023-12-19
Revised: 2024-08-29
Accepted: 2024-08-30
Published Online: 2024-10-30

© 2024 the author(s), published by De Gruyter

This work is licensed under the Creative Commons Attribution 4.0 International License.

Articles in the same Issue

  1. Special Issue on Contemporary Developments in Graph Topological Indices
  2. On the maximum atom-bond sum-connectivity index of graphs
  3. Upper bounds for the global cyclicity index
  4. Zagreb connection indices on polyomino chains and random polyomino chains
  5. On the multiplicative sum Zagreb index of molecular graphs
  6. The minimum matching energy of unicyclic graphs with fixed number of vertices of degree two
  7. Special Issue on Convex Analysis and Applications - Part I
  8. Weighted Hermite-Hadamard-type inequalities without any symmetry condition on the weight function
  9. Scattering threshold for the focusing energy-critical generalized Hartree equation
  10. (p, q)-Compactness in spaces of holomorphic mappings
  11. Characterizations of minimal elements of upper support with applications in minimizing DC functions
  12. Some new Hermite-Hadamard-type inequalities for strongly h-convex functions on co-ordinates
  13. Global existence and extinction for a fast diffusion p-Laplace equation with logarithmic nonlinearity and special medium void
  14. Extension of Fejér's inequality to the class of sub-biharmonic functions
  15. On sup- and inf-attaining functionals
  16. Regularization method and a posteriori error estimates for the two membranes problem
  17. Rapid Communication
  18. Note on quasivarieties generated by finite pointed abelian groups
  19. Review Articles
  20. Amitsur's theorem, semicentral idempotents, and additively idempotent semirings
  21. A comprehensive review of the recent numerical methods for solving FPDEs
  22. On an Oberbeck-Boussinesq model relating to the motion of a viscous fluid subject to heating
  23. Pullback and uniform exponential attractors for non-autonomous Oregonator systems
  24. Regular Articles
  25. On certain functional equation related to derivations
  26. The product of a quartic and a sextic number cannot be octic
  27. Combined system of additive functional equations in Banach algebras
  28. Enhanced Young-type inequalities utilizing Kantorovich approach for semidefinite matrices
  29. Local and global solvability for the Boussinesq system in Besov spaces
  30. Construction of 4 x 4 symmetric stochastic matrices with given spectra
  31. A conjecture of Mallows and Sloane with the universal denominator of Hilbert series
  32. The uniqueness of expression for generalized quadratic matrices
  33. On the generalized exponential sums and their fourth power mean
  34. Infinitely many solutions for Schrödinger equations with Hardy potential and Berestycki-Lions conditions
  35. Computing the determinant of a signed graph
  36. Two results on the value distribution of meromorphic functions
  37. Zariski topology on the secondary-like spectrum of a module
  38. On deferred f-statistical convergence for double sequences
  39. About j-Noetherian rings
  40. Strong convergence for weighted sums of (α, β)-mixing random variables and application to simple linear EV regression model
  41. On the distribution of powered numbers
  42. Almost periodic dynamics for a delayed differential neoclassical growth model with discontinuous control strategy
  43. A new distributionally robust reward-risk model for portfolio optimization
  44. Asymptotic behavior of solutions of a viscoelastic Shear beam model with no rotary inertia: General and optimal decay results
  45. Silting modules over a class of Morita rings
  46. Non-oscillation of linear differential equations with coefficients containing powers of natural logarithm
  47. Mutually unbiased bases via complex projective trigonometry
  48. Hyers-Ulam stability of a nonlinear partial integro-differential equation of order three
  49. On second-order linear Stieltjes differential equations with non-constant coefficients
  50. Complex dynamics of a nonlinear discrete predator-prey system with Allee effect
  51. The fibering method approach for a Schrödinger-Poisson system with p-Laplacian in bounded domains
  52. On discrete inequalities for some classes of sequences
  53. Boundary value problems for integro-differential and singular higher-order differential equations
  54. Existence and properties of soliton solution for the quasilinear Schrödinger system
  55. Hermite-Hadamard-type inequalities for generalized trigonometrically and hyperbolic ρ-convex functions in two dimension
  56. Endpoint boundedness of toroidal pseudo-differential operators
  57. Matrix stretching
  58. A singular perturbation result for a class of periodic-parabolic BVPs
  59. On Laguerre-Sobolev matrix orthogonal polynomials
  60. Pullback attractors for fractional lattice systems with delays in weighted space
  61. Singularities of spherical surface in R4
  62. Variational approach to Kirchhoff-type second-order impulsive differential systems
  63. Convergence rate of the truncated Euler-Maruyama method for highly nonlinear neutral stochastic differential equations with time-dependent delay
  64. On the energy decay of a coupled nonlinear suspension bridge problem with nonlinear feedback
  65. The limit theorems on extreme order statistics and partial sums of i.i.d. random variables
  66. Hardy-type inequalities for a class of iterated operators and their application to Morrey-type spaces
  67. Solving multi-point problem for Volterra-Fredholm integro-differential equations using Dzhumabaev parameterization method
  68. Finite groups with gcd(χ(1), χc (1)) a prime
  69. Small values and functional laws of the iterated logarithm for operator fractional Brownian motion
  70. The hull-kernel topology on prime ideals in ordered semigroups
  71. ℐ-sn-metrizable spaces and the images of semi-metric spaces
  72. Strong laws for weighted sums of widely orthant dependent random variables and applications
  73. An extension of Schweitzer's inequality to Riemann-Liouville fractional integral
  74. Construction of a class of half-discrete Hilbert-type inequalities in the whole plane
  75. Analysis of two-grid method for second-order hyperbolic equation by expanded mixed finite element methods
  76. Note on stability estimation of stochastic difference equations
  77. Trigonometric integrals evaluated in terms of Riemann zeta and Dirichlet beta functions
  78. Purity and hybridness of two tensors on a real hypersurface in complex projective space
  79. Classification of positive solutions for a weighted integral system on the half-space
  80. A quasi-reversibility method for solving nonhomogeneous sideways heat equation
  81. Higher-order nonlocal multipoint q-integral boundary value problems for fractional q-difference equations with dual hybrid terms
  82. Noetherian rings of composite generalized power series
  83. On generalized shifts of the Mellin transform of the Riemann zeta-function
  84. Further results on enumeration of perfect matchings of Cartesian product graphs
  85. A new extended Mulholland's inequality involving one partial sum
  86. Power vector inequalities for operator pairs in Hilbert spaces and their applications
  87. On the common zeros of quasi-modular forms for Γ+0(N) of level N = 1, 2, 3
  88. One special kind of Kloosterman sum and its fourth-power mean
  89. The stability of high ring homomorphisms and derivations on fuzzy Banach algebras
  90. Integral mean estimates of Turán-type inequalities for the polar derivative of a polynomial with restricted zeros
  91. Commutators of multilinear fractional maximal operators with Lipschitz functions on Morrey spaces
  92. Vector optimization problems with weakened convex and weakened affine constraints in linear topological spaces
  93. The curvature entropy inequalities of convex bodies
  94. Brouwer's conjecture for the sum of the k largest Laplacian eigenvalues of some graphs
  95. High-order finite-difference ghost-point methods for elliptic problems in domains with curved boundaries
  96. Riemannian invariants for warped product submanifolds in Q ε m × R and their applications
  97. Generalized quadratic Gauss sums and their 2mth power mean
  98. Euler-α equations in a three-dimensional bounded domain with Dirichlet boundary conditions
  99. Enochs conjecture for cotorsion pairs over recollements of exact categories
  100. Zeros distribution and interlacing property for certain polynomial sequences
  101. Random attractors of Kirchhoff-type reaction–diffusion equations without uniqueness driven by nonlinear colored noise
  102. Study on solutions of the systems of complex product-type PDEs with more general forms in ℂ2
  103. Dynamics in a predator-prey model with predation-driven Allee effect and memory effect
  104. A note on orthogonal decomposition of 𝔰𝔩n over commutative rings
  105. On the δ-chromatic numbers of the Cartesian products of graphs
  106. Binomial convolution sum of divisor functions associated with Dirichlet character modulo 8
  107. Commutator of fractional integral with Lipschitz functions related to Schrödinger operator on local generalized mixed Morrey spaces
  108. System of degenerate parabolic p-Laplacian
  109. Stochastic stability and instability of rumor model
  110. Certain properties and characterizations of a novel family of bivariate 2D-q Hermite polynomials
  111. Stability of an additive-quadratic functional equation in modular spaces
  112. Monotonicity, convexity, and Maclaurin series expansion of Qi's normalized remainder of Maclaurin series expansion with relation to cosine
  113. On k-prime graphs
  114. On the existence of tripartite graphs and n-partite graphs
  115. Classifying pentavalent symmetric graphs of order 12pq
  116. Almost periodic functions on time scales and their properties
  117. Some results on uniqueness and higher order difference equations
  118. Coding of hypersurfaces in Euclidean spaces by a constant vector
  119. Cycle integrals and rational period functions for Γ0+(2) and Γ0+(3)
  120. Degrees of (L, M)-fuzzy bornologies
  121. A matrix approach to determine optimal predictors in a constrained linear mixed model
  122. On ideals of affine semigroups and affine semigroups with maximal embedding dimension
  123. Solutions of linear control systems on Lie groups
  124. A uniqueness result for the fractional Schrödinger-Poisson system with strong singularity
  125. On prime spaces of neutrosophic extended triplet groups
  126. On a generalized Krasnoselskii fixed point theorem
  127. On the relation between one-sided duoness and commutators
  128. Non-homogeneous BVPs for second-order symmetric Hamiltonian systems
  129. Erratum
  130. Erratum to “Infinitesimals via Cauchy sequences: Refining the classical equivalence”
  131. Corrigendum
  132. Corrigendum to “Matrix stretching”
  133. Corrigendum to “A comprehensive review of the recent numerical methods for solving FPDEs”
https://doi.org/10.1515/math-2024-0063
Keywords for this article
warped products; Riemannian product manifolds; delta-invariant; gradient Ricci curvature; Dirichlet eigenvalues
Creative Commons
BY 4.0

Articles in the same Issue

  1. Special Issue on Contemporary Developments in Graph Topological Indices
  2. On the maximum atom-bond sum-connectivity index of graphs
  3. Upper bounds for the global cyclicity index
  4. Zagreb connection indices on polyomino chains and random polyomino chains
  5. On the multiplicative sum Zagreb index of molecular graphs
  6. The minimum matching energy of unicyclic graphs with fixed number of vertices of degree two
  7. Special Issue on Convex Analysis and Applications - Part I
  8. Weighted Hermite-Hadamard-type inequalities without any symmetry condition on the weight function
  9. Scattering threshold for the focusing energy-critical generalized Hartree equation
  10. (p, q)-Compactness in spaces of holomorphic mappings
  11. Characterizations of minimal elements of upper support with applications in minimizing DC functions
  12. Some new Hermite-Hadamard-type inequalities for strongly h-convex functions on co-ordinates
  13. Global existence and extinction for a fast diffusion p-Laplace equation with logarithmic nonlinearity and special medium void
  14. Extension of Fejér's inequality to the class of sub-biharmonic functions
  15. On sup- and inf-attaining functionals
  16. Regularization method and a posteriori error estimates for the two membranes problem
  17. Rapid Communication
  18. Note on quasivarieties generated by finite pointed abelian groups
  19. Review Articles
  20. Amitsur's theorem, semicentral idempotents, and additively idempotent semirings
  21. A comprehensive review of the recent numerical methods for solving FPDEs
  22. On an Oberbeck-Boussinesq model relating to the motion of a viscous fluid subject to heating
  23. Pullback and uniform exponential attractors for non-autonomous Oregonator systems
  24. Regular Articles
  25. On certain functional equation related to derivations
  26. The product of a quartic and a sextic number cannot be octic
  27. Combined system of additive functional equations in Banach algebras
  28. Enhanced Young-type inequalities utilizing Kantorovich approach for semidefinite matrices
  29. Local and global solvability for the Boussinesq system in Besov spaces
  30. Construction of 4 x 4 symmetric stochastic matrices with given spectra
  31. A conjecture of Mallows and Sloane with the universal denominator of Hilbert series
  32. The uniqueness of expression for generalized quadratic matrices
  33. On the generalized exponential sums and their fourth power mean
  34. Infinitely many solutions for Schrödinger equations with Hardy potential and Berestycki-Lions conditions
  35. Computing the determinant of a signed graph
  36. Two results on the value distribution of meromorphic functions
  37. Zariski topology on the secondary-like spectrum of a module
  38. On deferred f-statistical convergence for double sequences
  39. About j-Noetherian rings
  40. Strong convergence for weighted sums of (α, β)-mixing random variables and application to simple linear EV regression model
  41. On the distribution of powered numbers
  42. Almost periodic dynamics for a delayed differential neoclassical growth model with discontinuous control strategy
  43. A new distributionally robust reward-risk model for portfolio optimization
  44. Asymptotic behavior of solutions of a viscoelastic Shear beam model with no rotary inertia: General and optimal decay results
  45. Silting modules over a class of Morita rings
  46. Non-oscillation of linear differential equations with coefficients containing powers of natural logarithm
  47. Mutually unbiased bases via complex projective trigonometry
  48. Hyers-Ulam stability of a nonlinear partial integro-differential equation of order three
  49. On second-order linear Stieltjes differential equations with non-constant coefficients
  50. Complex dynamics of a nonlinear discrete predator-prey system with Allee effect
  51. The fibering method approach for a Schrödinger-Poisson system with p-Laplacian in bounded domains
  52. On discrete inequalities for some classes of sequences
  53. Boundary value problems for integro-differential and singular higher-order differential equations
  54. Existence and properties of soliton solution for the quasilinear Schrödinger system
  55. Hermite-Hadamard-type inequalities for generalized trigonometrically and hyperbolic ρ-convex functions in two dimension
  56. Endpoint boundedness of toroidal pseudo-differential operators
  57. Matrix stretching
  58. A singular perturbation result for a class of periodic-parabolic BVPs
  59. On Laguerre-Sobolev matrix orthogonal polynomials
  60. Pullback attractors for fractional lattice systems with delays in weighted space
  61. Singularities of spherical surface in R4
  62. Variational approach to Kirchhoff-type second-order impulsive differential systems
  63. Convergence rate of the truncated Euler-Maruyama method for highly nonlinear neutral stochastic differential equations with time-dependent delay
  64. On the energy decay of a coupled nonlinear suspension bridge problem with nonlinear feedback
  65. The limit theorems on extreme order statistics and partial sums of i.i.d. random variables
  66. Hardy-type inequalities for a class of iterated operators and their application to Morrey-type spaces
  67. Solving multi-point problem for Volterra-Fredholm integro-differential equations using Dzhumabaev parameterization method
  68. Finite groups with gcd(χ(1), χc (1)) a prime
  69. Small values and functional laws of the iterated logarithm for operator fractional Brownian motion
  70. The hull-kernel topology on prime ideals in ordered semigroups
  71. ℐ-sn-metrizable spaces and the images of semi-metric spaces
  72. Strong laws for weighted sums of widely orthant dependent random variables and applications
  73. An extension of Schweitzer's inequality to Riemann-Liouville fractional integral
  74. Construction of a class of half-discrete Hilbert-type inequalities in the whole plane
  75. Analysis of two-grid method for second-order hyperbolic equation by expanded mixed finite element methods
  76. Note on stability estimation of stochastic difference equations
  77. Trigonometric integrals evaluated in terms of Riemann zeta and Dirichlet beta functions
  78. Purity and hybridness of two tensors on a real hypersurface in complex projective space
  79. Classification of positive solutions for a weighted integral system on the half-space
  80. A quasi-reversibility method for solving nonhomogeneous sideways heat equation
  81. Higher-order nonlocal multipoint q-integral boundary value problems for fractional q-difference equations with dual hybrid terms
  82. Noetherian rings of composite generalized power series
  83. On generalized shifts of the Mellin transform of the Riemann zeta-function
  84. Further results on enumeration of perfect matchings of Cartesian product graphs
  85. A new extended Mulholland's inequality involving one partial sum
  86. Power vector inequalities for operator pairs in Hilbert spaces and their applications
  87. On the common zeros of quasi-modular forms for Γ+0(N) of level N = 1, 2, 3
  88. One special kind of Kloosterman sum and its fourth-power mean
  89. The stability of high ring homomorphisms and derivations on fuzzy Banach algebras
  90. Integral mean estimates of Turán-type inequalities for the polar derivative of a polynomial with restricted zeros
  91. Commutators of multilinear fractional maximal operators with Lipschitz functions on Morrey spaces
  92. Vector optimization problems with weakened convex and weakened affine constraints in linear topological spaces
  93. The curvature entropy inequalities of convex bodies
  94. Brouwer's conjecture for the sum of the k largest Laplacian eigenvalues of some graphs
  95. High-order finite-difference ghost-point methods for elliptic problems in domains with curved boundaries
  96. Riemannian invariants for warped product submanifolds in Q ε m × R and their applications
  97. Generalized quadratic Gauss sums and their 2mth power mean
  98. Euler-α equations in a three-dimensional bounded domain with Dirichlet boundary conditions
  99. Enochs conjecture for cotorsion pairs over recollements of exact categories
  100. Zeros distribution and interlacing property for certain polynomial sequences
  101. Random attractors of Kirchhoff-type reaction–diffusion equations without uniqueness driven by nonlinear colored noise
  102. Study on solutions of the systems of complex product-type PDEs with more general forms in ℂ2
  103. Dynamics in a predator-prey model with predation-driven Allee effect and memory effect
  104. A note on orthogonal decomposition of 𝔰𝔩n over commutative rings
  105. On the δ-chromatic numbers of the Cartesian products of graphs
  106. Binomial convolution sum of divisor functions associated with Dirichlet character modulo 8
  107. Commutator of fractional integral with Lipschitz functions related to Schrödinger operator on local generalized mixed Morrey spaces
  108. System of degenerate parabolic p-Laplacian
  109. Stochastic stability and instability of rumor model
  110. Certain properties and characterizations of a novel family of bivariate 2D-q Hermite polynomials
  111. Stability of an additive-quadratic functional equation in modular spaces
  112. Monotonicity, convexity, and Maclaurin series expansion of Qi's normalized remainder of Maclaurin series expansion with relation to cosine
  113. On k-prime graphs
  114. On the existence of tripartite graphs and n-partite graphs
  115. Classifying pentavalent symmetric graphs of order 12pq
  116. Almost periodic functions on time scales and their properties
  117. Some results on uniqueness and higher order difference equations
  118. Coding of hypersurfaces in Euclidean spaces by a constant vector
  119. Cycle integrals and rational period functions for Γ0+(2) and Γ0+(3)
  120. Degrees of (L, M)-fuzzy bornologies
  121. A matrix approach to determine optimal predictors in a constrained linear mixed model
  122. On ideals of affine semigroups and affine semigroups with maximal embedding dimension
  123. Solutions of linear control systems on Lie groups
  124. A uniqueness result for the fractional Schrödinger-Poisson system with strong singularity
  125. On prime spaces of neutrosophic extended triplet groups
  126. On a generalized Krasnoselskii fixed point theorem
  127. On the relation between one-sided duoness and commutators
  128. Non-homogeneous BVPs for second-order symmetric Hamiltonian systems
  129. Erratum
  130. Erratum to “Infinitesimals via Cauchy sequences: Refining the classical equivalence”
  131. Corrigendum
  132. Corrigendum to “Matrix stretching”
  133. Corrigendum to “A comprehensive review of the recent numerical methods for solving FPDEs”
Sign up now to receive a 20% welcome discount
Institutional Access
How does access work?
Have an idea on how to improve our website?
Please write us.
© 2025 De Gruyter Brill
Downloaded on 10.9.2025 from https://www.degruyterbrill.com/document/doi/10.1515/math-2024-0063/html
Scroll to top button