Integral inequalities for closed linear Weingarten submanifolds in the product spaces

Abstract An integral inequality for closed linear Weingarten 𝑚-submanifolds with parallel normalized mean curvature vector field (pnmc lw-submanifolds) in the product spaces 𝑀𝑛(𝑐) × ℝ, 𝑛 > 𝑚 ≥ 4, where 𝑀𝑛(𝑐) is a space form of constant sectional curvature 𝑐 ∈ {−1, 1}, is proved. As an application is shown that the sharpness in this inequality is attained in the totally umbilical hypersurfaces, and in a certain family of standard product of the form 𝕊1(√1 − 𝑟2) × 𝕊𝑚−1(𝑟) with 0 < 𝑟 < 1 when 𝑐 = 1. In the case where 𝑐 = −1, is obtained an integral inequality whose sharpness is attained only in the totally umbilical hypersurfaces. When 𝑚 = 2 and 𝑚 = 3, an integral inequality is also obtained with equality happening in the totally umbilical hypersurfaces.


INTRODUCTION
Within the theory of isometric immersions, the characterization of closed submanifolds (compact with empty boundaries) with one of their constant curvatures using integral inequalities constitutes a classical research topic.Notable among these is Simons' integral inequality (see Simons 1968), which establishes a relationship between the squared norm of the second fundamental form and the dimension and codimension of the minimal submanifold in the unit sphere.It is worth highlighting that Simons' tool has proven effective not only in the study of minimal closed submanifolds in the sphere but also in the investigation of submanifolds with other constant curvatures, as well as in more general ambient spaces (see, for example, Chern et al. 1970, Lawson 1969, Ôtsuki 1970, dos Santos & da Silva 2021, and references therein).
In the context of hypersurfaces, Cheng & Yau (1977) investigated the rigidity of hypersurfaces with constant scalar curvature in a space form.They introduced a new second-order differential operator known as the square operator.Building upon Cheng-Yau's technique, Li (1996) studied the pinching problem concerning the square norm of the second fundamental form for complete hypersurfaces with constant scalar curvature.Later, Wei (2008) derived a Simons' type integral inequality for closed -minimal rotational hypersurfaces immersed in  +1 , characterizing the equality through the standard product  1 (√1 −  2 ) ×  −1 ().In higher codimension, Guo & Li (2013) extended the results of Li (1996) and showed that the only closed submanifolds with parallel normalized mean curvature (pnmc) in the unit sphere  + with constant scalar curvature, and whose second fundamental form On the other hand, a natural extension of the submanifolds with constant second mean curvature is the linear Weingarten submanifolds.A submanifold is said to be linear Weingarten (here we will denote by lw-submanifolds) when the first and the second mean curvatures satisfy a certain linear relation.Here, we deal with -dimensional closed pnmc lw-submanifolds immersed in a Riemannian product space   () × ℝ, where   () is a space form of constant sectional curvature  = −1, 1 with  >  ≥ 4. In this setting, we extend the technique developed by the first two authors in dos Santos & da Silva (2021, 2022) in order to prove a sharp integral inequality for pnmc lw-submanifolds obtaining natural generalizations of the main results of Alías & Meléndez (2020) and dos Santos & da Silva (2021).Furthermore, we also obtain integral inequalities when  = −1, which is not contemplated in dos Santos & da Silva (2021).This manuscript is organized as follows: In Section 1, we provide a brief review of fundamental concepts related to submanifolds immersed in a Riemannian product space   () × ℝ.Subsequently, we establish a Simons' type formula for pnmc lw-submanifolds in   () × ℝ (see Proposition 1.2).In Section 2, we present auxiliary lemmas concerning pnmc lw-submanifolds in   () × ℝ. Moving on to Section 3, we provide a lower estimate for a Cheng-Yau modified operator acting on the square norm of the traceless second fundamental form of such submanifolds (see Proposition 3.1).We then apply this result to establish our characterization theorems for closed pnmc lw-submanifolds in   () × ℝ with a constant angle between the normalized mean curvature and the unit vector field tangent to ℝ (see Theorems 3.3 and 3.4).Finally, in the last section, we examine the cases of two and three dimensions (see Theorems 4.1 and 4.2).

-A SIMONS TYPE FORMULA FOR SUBMANIFOLDS IN 𝑀 𝑁 (𝐶) × ℝ
Along this manuscript, we will always deal with an -dimensional connected submanifold Σ  immersed in a Riemannian manifold  +1 with  ≥ .We choose a local field of orthonormal frames Now, restricting all the tensors to Σ  ,   = 0 on Σ  .Hence, ∑    ∧   =   = 0 and as it is well known we get This gives with  denoting the second fundamental form of Σ  in  +1 .The square length of the shape operator is (3) Furthermore, we define the mean curvature vector ℎ and the mean curvature function  of Σ  in  +1 , respectively by where tr(  ) = ∑  ℎ   .As it is well known, the basic equations of the submanifolds are the Gauss equation where   and   are the components of the curvature tensor of  +1 and Σ  , respectively, the Ricci equation where  ⟂  are the components of the normal curvature tensor of Σ  , and the Codazzi equation where ℎ   denote the first covariant derivatives of ℎ   .Additionally, where ∇ denotes the covariant derivative of the second fundamental form . In particular, we say that Σ  is a parallel submanifold of  +1 when ∇ = 0 (see van der Veken & Vrancken 2008).
In this setting, the following Simons-type formula is well-known (see dos Santos & da Silva 2021, 2022): Then, we have where () = tr(  ) for all matrix  = (  ).
From now on, let us consider the case where the ambient space is a product space.Let  +1 =   () × ℝ be a product space, where   () be a connected Riemannian manifold endowed with metric tensor ⟨ , ⟩  and of constant sectional curvature  = −1, 1 and ℝ is the real line.Thus, the product space   () × ℝ is the differential manifold   () × ℝ endowed with the Riemannian metric with (, ) ∈   () × ℝ and ,  ∈  (,) (  () × ℝ), where  ℝ and   denote the projections onto the corresponding factor.Associated with the product space, we know that, the vector field , (, ) ∈   () × ℝ (11) is parallel and unitary, that is, where ∇ is the Levi-Civita connection of the Riemannian metric of   () × ℝ.Using the notations established in Fetcu & Rosenberg (2013), we write the decomposition where  ∶=   and  ∶=  ⟂  denotes, respectively, the tangent and normal parts of the vector field   on the tangent and normal bundle of the submanifold Σ  in   () × ℝ.Moreover, from ( 12) and ( 13), we get the relation In what follows, we will denote by ∇ and ∇ ⟂ , respectively, the tangent and normal Levi-Civita connections along the tangent and normal bundle of Σ  , a direct computation by (13) give us ∇   =   () and ∇ ⟂   = −(, ), for all  ∈ (), where   = ∑  ⟨,   ⟩  denotes the Weingarten operator in the  direction.By this digression, our aim now is to get a Simons-type formula for a pnmc lw-submanifold Σ  in   () × ℝ.Firstly, since   () × ℝ locally symmetric, we have   =   = 0. On the other hand, a direct computation from (15), gives   = 0, for all , , , .Moreover, Next, we will also consider the traceless second fundamental form It is easy to check that each   =   − ⟨ℎ,   ⟩ is traceless and that Observe that || 2 = 0 if and only if Σ  is a totally umbilical submanifold of   () × ℝ.Within this context, a standard computation give us and Now, let Σ  be a  submanifolds immersed in product space   () × ℝ.This means that  > 0 and the normalized mean curvature vector field  = ℎ/ is parallel as a section of the normal bundle.In this setting, we will consider { +1 , … ,  +1 } be a local orthonormal frame field in the normal bundle such that  +1 = .By this, tr(  ) =  and tr(  ) = ⟨ℎ,   ⟩ = 0, for all  ≥  + 2, and by ( 19) Since  parallel, the Ricci equation ( 6) guarantees that     =     for all  ≥  + 2. Using this, ( 20) and ( 24), Therefore, inserting ( 17), ( 18), ( 21), ( 22) and ( 25) in Proposition 1.1 we get According to Grosjean (2002) and Cao & Li (2007), we define the r-th mean curvature function   of an -dimensional submanifold immersed in a Riemannian space, as follows: for any even integer  ∈ {0, 1, … ,  − 1}, the -th are given by where 1 …  is the generalized Kronecker symbol and   = ∑ ,, ℎ     with { +1 , … ,  +1 } an orthonormal frame on the normal bundle.By convention,  0 =  0 = 1.For our study on submanifolds Σ  in the product space   () × ℝ, we will consider the second mean curvature function  2 , which is given by On the other hand, a natural extension of submanifolds having constant second mean curvature is the so-called linear Weingarten, in short, lw-submanifolds.A submanifold is said to be linear Weingarten red if its first and second mean curvatures are linearly related, that is, for constants ,  ∈ ℝ. Observe that when  = 0, (29) reduces to  2 constant.
For the study of the lw-submanifolds, we will consider the following Cheng-Yau's modified differential operator given by where   stands for a component of the Hessian of  ∈  2 ().From the tensorial point of view, (30) can be written as where  is the identity in the algebra of smooth vector fields on Σ  and ℎ +1 = (ℎ +1  ) denotes the second fundamental form of Σ  in the direction  +1 .By (31), it is not difficult to see that for every ,  ∈  2 () and for every smooth function  ∶ ℝ → ℝ.
Hence, taking  =  in (30), by ( 28) and ( 29), we obtain From all these results we have the following Simons-type formula for Cheng-Yau's modified operator acting on the mean curvature function of Σ  in   () × ℝ which generalizes Proposition 2 of dos Santos & da Silva (2022): ) .

-KEY LEMMAS
In this section, we will present some necessary results for the proof of our results.
By taking the derivative in (37), and consequently It is not difficult to check that Thus by using (35), Now, from Kato's inequality we obtain Therefore, we have either or If the inequality ( 35) is strict, from (41) we get Now, let us assume in addition that the equality holds in (36) on Σ  .In this case, we wish to show that  is constant on Σ  .Suppose, by contradiction, that it does not occur.Consequently, there exists a point  ∈ Σ  such that |∇()| > 0. So, one deduces from (39) that for each principal curvature  +1  of Σ  ,  = 1, … , .On the other hand, with a straightforward computation, we verify that Now, we claim that  − −1 2  ≥ 0. For this, let us consider two cases.When  ≤ 0, our assertion is immediate.Otherwise, if  > 0, from (37) we see that since Σ  is a pnmc submanifold.Thus,  − ( − 1) > 0 and consequently,  − −1 2  ≥ 0 as claimed.So, from (49) we obtain Given a unit normal vector field  ∈ (Σ) ⟂ , we say that a submanifold Σ  of   () × ℝ has constant -angle if the angle between  and   is constant, that is, the function ⟨,   ⟩ is constant along of Σ  .We should notice that constant -angle submanifolds, where  = ℎ/, corresponds to a natural extension of hypersurfaces with constant angle in a product space, which was widely studied by Dillen and many other authors (see, for instance, Dillen et al. 2007, Dillen & Munteanu 2009, Navarro et al. 2016, Nistor 2017).By using this context, the next result is a suitable adaptation of Lemma 2.1 of dos Santos & da Silva (2021) which assures that the integral of the  operator acting on any nonnegative function is equal to zero.
The following two results are fundamental to our study and can be found in Li & Li (1992) and Santos (1994), respectively.) .
We will conclude this section by quoting the following codimension reduction result for submanifolds in the product space   () × ℝ, see Lemma 1.6 of Mendonça & Tojeiro (2013).

-MAIN RESULTS
In our first result, we obtain a suitable lower estimate for the operator  applied on the squared norm of the traceless operator of a lw-submanifold, which will be also essential to the proofs of our main results.
where  ,, is defined in (68).Now, Lemma 2.2 guarantees that the operator  is positive definite since  ≥ 0. So, by ( 33) and ( 75), we can write Hence, by inserting ( 84) in ( 85), we get (66).Finally, if equality holds in (66), considering that  > 0 and  is positive definite, we can deduce from (85) that  is constant.Moreover, (83) must also be satisfied as an equality.Since we already established that  is constant, this implies ∇ = 0, indicating that the second fundamental form is parallel.Additionally, in order to achieve equality in Lemma 2.5, (70) must also be an equality.Consequently, we conclude that Σ  is a parallel submanifold of   () × ℝ with exactly two distinct principal curvatures, one of which is simple.
Remark 3.2.Since the mean curvature vector field is normalized, it follows that  > 0. By using (75), If  =  = 0 and there exists a point  ∈ Σ  such that ||() = 0, then  must vanish, which contradicts the fact that  > 0. Therefore, we conclude that ||, , and  cannot vanish simultaneously.Now, we are ready to give proof of our first result.
From now on, for simplicity, we will denote  = || 2 .So, (85) can be rewritten as follows Taking into account that  ≥ 0 and  ≥ 0, from Remark 3.2 and (93) we get for every real number .By closedness of Σ  , we can integrate both sides of (94) in order to obtain Now, we will define the function where () is given by Since  > 2,  ≥ 0 and  is a smooth function, we have that  is well defined (see Remark 3.2) and  ≥ 0. Hence, taking into the integral, from (34) and Lemma 2.3, we have that is, Taking the first and second derivatives of (96), we have and Lemma 2.2 assures that the operator  is positive semidefinite, using (99), ( 100) and ( 101) in (95), we can estimate This proves inequality (87).We assume that the equality holds in (103) and  > 0. By (102), we get where with equality holding if and only if  > 2 and  = 0. Since  > 0, from Lemma 2.2,  is positive definite, consequently ⟨(∇), ∇⟩ ≥ 0 (106) with equality if and only if ∇ = 0. Therefore, it follows from (104) that: which implies that the function  = || 2 must be constant, either  ≡ 0 or  ≡  0 > 0.
Remark 3.5.Let us recall that a submanifold Σ  of   () × ℝ is said to be a vertical cylinder over  −1 if Σ  =  −1  ( −1 ) where  −1 is a submanifold of   ().It is not difficult to check that Σ  is a non-minimal parallel vertical cylinder in   () × ℝ if, and only if,  −1 is a non-minimal parallel submanifold in   ().Moreover, its mean curvature vector field ℎ is given by ℎ = −1  ℎ 0 , where ℎ 0 denotes the mean curvature vector field of  −1 .Hence, Σ  is a pnmc lw-submanifold of   () × ℝ having constant -angle and that is not lies in a slice provided vertical cylinders are characterized by the fact that   is always tangent to Σ  (see Fetcu & Rosenberg 2013).Therefore, we conclude that the hypothesis of the submanifold to be closed in Theorems 3.3 and 3.4 is, indeed, necessary.

-FURTHER RESULTS FOR 𝑀 = 2 AND 𝑀 = 3
We should notice that when  = 2 and  = 3, the integral inequalities obtained in Theorems 3.3 and 3.4 holds.To see this, it is sufficient to do a rereading on the first inequality of (72).In fact, from ( 72 Proof.The proof follows the same steps as the proof of Theorem 3.3 until we reach inequality (103), changing the function  ,, by  ,, along of the computations.If the equality in (125) holds, then also occurs equality in (119) and hence, Σ  is a totally umbilical.Besides this, the equality also occurs in ( 91), from where we conclude that  = 0. Therefore, Σ  is a totally umbilical hypersurface in  +1 × { 0 } ↪   × ℝ for some  0 ∈ ℝ.
are the eigenvalues of , follows that  is positive semidefinite.Similarly if  > 0.

FÁBIO
R. DOS SANTOS, SYLVIA F. DA SILVA & ANTONIO F. DE SOUSA PNMC LW-SUBMANIFOLDS IN THE PRODUCT SPACES DOS SANTOS, SYLVIA F. DA SILVA & ANTONIO F. DE SOUSA PNMC LW-SUBMANIFOLDS IN THE PRODUCT SPACES Moreover, the following inequality is well known (see Equation 3.5 of Guo & Li 2013)

)
It is clear that, if  = 0 then,   is normal to Σ  and, hence Σ  lies in   ().
The first ones are extensions of the Lemmas 1 and 2 of dos Santos & da Silva (2022) (see also Lemma 2.3 of dos Santos & da Silva (2021) and Lemmas 4.1 and 4.3 of dos Santos (2021)) to lw-submanifolds.Let Σ  be an lw-submanifold in the product space   () × ℝ, such that  2 =  +  with Moreover, if the inequality (35) is strict and the equality occurs in (36), then Σ  is an open piece of a parallel submanifold of   () × ℝ. FÁBIO R. DOS SANTOS, SYLVIA F. DA SILVA & ANTONIO F. DE SOUSA PNMC LW-SUBMANIFOLDS IN THE PRODUCT SPACES Proof.Inserting  2 =  +  in (28) we have we arrive at a contradiction.Hence, in this case, we conclude that  must be constant on Σ  .Let us consider { 1 , … ,   } an orthonormal frame on Σ  such that ℎ +1 Lemma 2.2.Let Σ  be a pnmc lw-submanifold in the product space   () × ℝ, such that  2 =  +  with  ≥ 0. Then the operator  defined in (32) is positive semidefinite.In the case where  > 0, we have that  is positive definite.FÁBIO R. DOS SANTOS, SYLVIA F. DA SILVA & ANTONIO F. DE SOUSA PNMC LW-SUBMANIFOLDS IN THE PRODUCT SPACES Proof.