Bounds of generalized normalized $$\delta $$δ-Casorati curvatures for real hypersurfaces in the complex quadric

In the present paper, we first characterize real hypersurfaces in the complex quadric $$Q^{m}$$Qm by giving an inequality in terms of the scalar curvature and the Mean curvature vector field. We also obtain the condition under which this inequality becomes an equality. Further, we develop two extremal inequalities involving the normalized $$\delta $$δ-Casorati curvatures and the extrinsic generalized normalized $$\delta $$δ-Casorati curvatures for real hypersurfaces in $$Q^{m}$$Qm. Finally, we derive the necessary and sufficient condition for the equality in both cases.


Introduction
In 1993, Chen [7] introduced the notion of Chen invariants (or δ-invariants) and obtained some optimal inequalities consisting of intrinsic and some extrinsic invariants for any Riemannian submanifold [8].
Moreover, Casorati introduced the theory of the extrinsic invariant given by the normalized square of the second fundamental form, which is known as Casorati curvature of a submanifold in a Riemannian manifold [6]. This theory of Casorati curvature is an extended version of the notion of principal curvatures of a hypersurface in a Riemannian manifold. It is a very fast growing area of research to obtain geometric relations or inequalities concerning the esteemed curvature. Thus, it is both important and very interesting to obtain some extremal inequalities for the Casorati curvatures of submanifolds in any ambient Riemannian manifold.
In the present paper, we have characterized the real hypersurfaces in Q m by giving an inequality concerning the scalar curvature and the Mean curvature vector field and proved that the equality holds for a U-principal normal vector field if and only if the real hypersurface M is totally geodesic in Q m . Moreover in a continuation we have obtained the optimal inequalities for real hypersurfaces of the complex quadric Q m involving the normalized δ-Casorati curvatures and the extrinsic generalized normalized δ-Casorati curvatures. The equality cases are also considered.

The complex quadric Q m
The complex quadric Q m is the complex hypersurface in C P m+1 defined by the relation z 2 1 + · · · + z 2 m+1 = 0, where z 1 , . . . , z m+1 are homogeneous coordinates on C P m+1 . Then, the Kähler structure on C P m+1 naturally induces a standard Kähler structure (J, g) on Q m [18]. The 1-dimensional Q 1 and 2-dimensional Q 2 are congruent to the round 2-sphere S 2 and the Riemannian product S 2 × S 2 , respectively. Thus, we will assume that m is greater than or equal to 3 throughout the paper.
Apart from the complex structure J , there is one more geometric structure on Q m , namely a parallel rank-two vector bundle U which contains an S 1 -bundle of real structures, that is, complex conjugations A on the tangent spaces of Q m . Here the notion of parallel vector bundle U means that (∇ X A)Y = q(X )AY for any vector fields X and Y on Q m , where ∇ and q denote a connection and a certain 1-form defined on T p Q m , p ∈ Q m , respectively. Now, a non-zero tangent vector W ∈ T p Q m is known as singular tangent vector if it is tangent to more than one maximal flat in Q m . There are two types of singular tangent vectors for the complex quadric Q m [20]: (1) If there exists a complex conjugation A ∈ U such that W ∈ V( A), then W is singular. Such a singular tangent vector is called U-principal. (2) If there exists a conjugation A ∈ U and orthonormal vectors U, V ∈ V( A) such that W/||W || = (U + J V )/ √ 2, then W is singular. Such a singular tangent vector is called U-isotropic,

Some general fundamental equations
Here, we recall some notions for a real hypersurface M in Q m . Let M be a real hypersurface of Q m with the connection ∇ induced from the Levi-Civita connection ∇ in Q m . Then where N is the unit normal vector field on M and φ X denotes the tangential component of J X for X ∈ (T M).
Here, M is endowed with an almost contact metric structure (φ, ξ, η, g) satisfying [5]: On the other hand, the fundamental Gauss and Weingarten formulas for M are respectively, for X, Y ∈ (T M) and N ∈ (T ⊥ M), where h is the second fundamental form and S is the shape operator of M, that are related by Moreover, the structure (φ, ξ, η, g) satisfies and 0 ≤ θ ≤ π 4 (see Proposition 3 [16]) is a function on M. Since ξ = −J N, we have the following relations: from which it follows that g(ξ, AN ) = 0. Now, from the Gauss equation for Q m ⊂ C P m+1 , the Riemannian curvature tensor R of the connection ∇ has the following form [19]: where K (π) stands for the sectional curvature of M associated with a plane section π ⊂ (T M) and is spanned by tangent vectors {e i , e j } and K (e i ∧ e j ) = g(R(e i , e j )e j , e i ) for 1 ≤ i < j ≤ 2m − 1.
The normalized scalar curvature ρ of M is given by .
We denote the mean curvature vector field by H , which is given as

Conveniently, we take h
. . , 2m − 1} and α = 2m. Then, one defines the squared mean curvature ||H || 2 of M in Q m and the squared norm ||h|| 2 of the second fundamental form h as: . It is well known that 1 2m−1 times the squared norm of h is called the Casorati curvature C of M in Q m [6]. Thus, we have Since ||h|| 2 = tr(S 2 ), the above expression can be reexpressed by The real hypersurface M of Q m is said to be invariantly quasi-umbilical if there exists a local orthonormal normal frame {e 2m } of M in Q m such that the shape operators S e 2m have an eigenvalue of multiplicity 2m − 2 for α = 2m and the distinguished eigendirection of S e 2m is the same for α = 2m [21]. Now, let us suppose that L is a k-dimensional subspace of (T M), k ≥ 2, such that {e 1 , . . . , e k } is an orthonormal basis of L. Then, the scalar curvature τ (L) and the Casorati curvature C(L) of the k-plane L are given by where s = 2m − 2 are given as [12]: From the above two relations, one can note that the generalized normalized δ-Casorati curvatures δ c (r ; 2m −2) andδ c (r ; 2m − 2) are the generalized versions of the normalized δ-Casorati curvatures δ c (2m − 2) and δ c (2m − 2), respectively, by substituting r by (2m−1)(2m−2) 2 as: for p ∈ M.

Main results
In this section, we will obtain some extremal inequalities involving the scalar curvature, the normalized scalar curvature and the generalized normalized δ-Casorati curvature for real hypersurfaces M of the complex quadric Q m .
To begin with, contracting with respect to X and W in (3.1), the Ricci curvature is given by [19] Ric(Y,

U-principal normal vector field
In this subsection, we will assume M to be a Hopf hypersurface (i.e., for a smooth function α = g(Sξ, ξ ) on M, one has Sξ = αξ ) admitting a U-principal unit vector field N satisfying AN = N (or Aξ = −ξ ). Then, after putting Z = ξ in (4.1) and since Sξ = αξ , we obtain Also from (4.1), we derive Again, using our assumption and applying the geometric condition of isometric Reeb flow (φ S = Sφ), we have where we have used Finally, we deduce that Moreover, we have

U-isotropic normal vector field
Here, again we will assume M to be a Hopf hypersurface (i.e., for any smooth function α = g(Sξ, ξ ) on M, one has Sξ = αξ ) and M admits a U-isotropic unit vector field N satisfying g(AN, N ) = 0. Then, after putting Z = ξ in (4.1) and since Sξ = αξ , we obtain Again, using our assumption together with the condition of isometric Reeb flow (φ S = Sφ), we have Finally, we get Summarizing, we have the following successive results:

Theorem 4.2 The Ricci curvature of a Hopf hypersurface M in Q m with isometric Reeb flow satisfies the following:
(a) If the unit normal vector field N is U-principal, then

Lemma 4.3 Given a real hypersurface M in Q m with φQ = Qφ, where Q is the Ricci operator, one has
Proof We know that the Ricci operator Q is defined by Ric(Y, Z ) = g(QY, Z ). So, we have Now, assume that one has φQ = Qφ on M, then which completes the proof.
By setting Y = Z = e i in (4.1), the scalar curvature of a real hypersurface M in the complex quadric Q m has the form where we have used ||h|| 2 = g(h(e i , e j ), h(e i , e j )) = g(g(Se i , e j )N , g(Se i , e j )N ) = tr(S 2 ).

Case 1:
If the unit normal vector field N is U-principal, then the scalar curvature given by (4.3) has the reduced form From this, we have the inequality and equality holds if and only if the real hypersurface M is totally geodesic in Q m .

respectively. Moreover, both relations (i) and (ii) become equalities if and only if M is an invariantly quasi-
where M is the diagonal matrix of order 2m − 2 with entries a.
Proof In terms of the Casorati curvature, we have the relation Consider the quadratic polynomial in the components of the second fundamental form with n = 2m − 1 Let us assume that L is spanned by {e 1 , . . . , e n−1 } and take e α = N = e n+1 for α = n + 1. Then, we have With a simple calculation, we have Then, with some computations, we derive Now, the critical points h c = (h α 11 , h α 22 , . . . , h α nn ) of P are the solutions of the system of linear homogeneous equations: So, from (4.8) it follows that any solution satisfies h α i j = 0 for i = j ∈ {1, 2, . . . , n}. Moreover, the Hessian matrix of the system (4.8) has the following form: whose diagonal block submatrices are given by Since the Hessian H( p) is positive semidefinite for all points, the function or the polynomial P is convex. Due to the convexity of P, the critical point h c is a minimum and in fact a global minimum. Thus, the polynomial P is of parabolic type and has a minimum at any solution h c of (4.8).
After this, applying (4.8) on (4.7) follows that h c is a solution of P, i.e. P(h c ) = 0. So, P ≥ 0 and thus from (4.6), we obtain Finally, we have Similarly, one can easily get the inequality (ii).

Furthermore, we can easily check that the equality arises in the theorem if and only if
Thus, we get the equality case if and only if the real hypersurface M is invariantly quasi-umbilical having flat normal connection ∇ ⊥ in Q m such that the shape operator takes the form (4.4).

Moreover, both relations (i) and (ii) become equalities if and only if M is an invariantly quasi-umbilical real
hypersurface having flat normal connection ∇ ⊥ in Q m such that with respect to some orthonormal basis {e 1 , . . . , e 2m−1 } and {e 2m = N } of (T M) and (T ⊥ M) respectively, the shape operator S N takes the following form: where M is the diagonal matrix of order 2m − 2 with entries a.

Lemma 4.8 For a real hypersurface M of Q m , we have
.

Moreover, relation (i) becomes an equality if and only if M is an invariantly quasi-umbilical real hypersurface
having flat normal connection ∇ ⊥ in Q m such that with respect to some orthonormal basis {e 1 , . . . , e 2m−1 } and {e 2m = N } of (T M) and (T ⊥ M) respectively, the shape operator S N takes the following form: .

Moreover, relation (ii) becomes an equality if and only if M is an invariantly quasi-umbilical real hypersurface
having flat normal connection ∇ ⊥ in Q m such that with respect to some orthonormal basis {e 1 , . . . , e 2m−1 } and {e 2m = N } of (T M) and (T ⊥ M), respectively, the shape operator S N takes the form: where M is the diagonal matrix of order 2m − 2 with entries a.

Corollary 4.9 Let M be a real hypersurface of Q m .
(i) The normalized δ-Casorati curvature δ c (2m − 2) for a U-principal normal (resp. U-isotropic) vector field satisfies Moreover, relation (i) becomes an equality if and only if M is an invariantly quasi-umbilical real hypersurface having flat normal connection ∇ ⊥ in Q m such that with respect to some orthonormal basis {e 1 , . . . , e 2m−1 } and {e 2m = N } of (T M) and (T ⊥ M) respectively, the shape operator S N takes the following form where M is the diagonal matrix of order 2m − 2 with entries 2a.

Moreover, relation (ii) becomes an equality if and only if M is an invariantly quasi-umbilical real hypersurface
having flat normal connection ∇ ⊥ in Q m such that with respect to some orthonormal basis {e 1 , . . . , e 2m−1 } and {e 2m = N } of (T M) and (T ⊥ M), respectively, the shape operator S N takes the following form: where M is the diagonal matrix of order 2m − 2 with entries a.

Further studies
Here, we present some open problems. Similar problems can be formulated in some different situations, where the expressions of the curvature tensor have some well-known different form.

Problem
To study different submanifolds of the complex quadric Q m and obtain Casorati's inequalities for the same ambient manifold.

Problem
To study slant submanifolds of the complex quadric Q m and obtain Casorati's inequalities for the same ambient manifold, if possible.

Problem
To obtain Casorati's inequalities for submanifolds of locally conformal Kähler space forms.

Problem
To study some different connections like semi-symmetric non-metric connections and obtain Casorati's inequalities for submanifolds of the complex quadric Q m equipped with semi-symmetric non-metric connection.
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