Bounds for Generalized Normalized δ-Casorati Curvatures for Submanifolds in Generalized (κ, µ)-space Forms

pISSN 1225-6951 eISSN 0454-8124

⃝ Kyungpook Mathematical Journalc

Bounds for Generalized Normalized δ-Casorati Curvatures for Submanifolds in Generalized (κ, µ)-space Forms

Mohd Aquib^∗ and Mohammad Hasan Shahid

Department of Mathematics, Faculty of Natural Sciences, Jamia Millia Islamia, New Delhi-110025, India

e-mail : aquib80@gmail.com and mshahid@jmi.ac.in

Abstract. In this paper, we prove the optimal inequalities for the generalized normalized δ-Casorati curvature and the normalized scalar curvature for different submanifolds in generalized (κ, µ)-space forms. The proof is based on an optimization procedure involving a quadratic polynomial in the components of the second fundamental form. We also characterize the submanifolds on which equalities hold.

1. Introduction

The theory of Chen invariants, which establish the simple relationships between the main intrinsic invariants and the main extrinsic invariants of the submanifolds is one of the most interesting research areas of differential geometry started by Chen [5] in 1993. In the initial papers, Chen established inequalities between the scalar curvature, the sectional curvature (intrinsic invariants) and the squared norm of the mean curvature (the main extrinsic invariant) of a submanifold in a real space form. The same author obtained the inequalities for submanifolds between the k- Ricci curvature, the squared mean curvature, and the shape operator in the real space form with arbitrary codimension [4]. Since then, different geometers proved the similar inequalities for different submanifolds and ambient spaces [9, 12].

The Casorati curvature (extrinsic invariant) of a submanifold of a Riemannian manifold, introduced by Casorati is defined as the normalized square length of the second fundamental form [3]. The concept of Casorati curvature extends the concept of the principal direction of a hypersurface of a Riemannian manifold [8].

The geometrical meaning and the importance of the Casorati curvature has been

* Corresponding Author.

Received July 7, 2017; accepted February 8, 2018.

2010 Mathematics Subject Classification: 53B05, 53B20, 53C40.

Key words and phrases: Casorati curvature, generalized (κ, µ)-space forms, normalized scalar curvature.

167

discussed by distinguished geometers [6, 10, 13]. Therefore, it attracts the attention of geometers to obtain the optimal inequalities for the Casorati curvatures of the submanifolds of different ambient spaces [1, 7, 11].

In this article, we obtain the optimal inequalities which relates the generalized normalized δ-Casorati curvature and the normalized scalar curvature for different submanifolds in generalized (κ, µ)-space form and consider the equality case of the inequality.

2. Preliminaries

In this section we will introduce some basic facts and notions for the later use.

A differentiable manifold M of dimension (2m + 1) is called an almost contact metric manifold if there is an almost contact metric structure (ϕ, ξ, η, g) consisting of a (1, 1) tensor field ϕ, a vector field ξ, a 1-form η and a compatible Riemannian metric g satisfying

ϕ²=−I + η ⊗ ξ, η(ξ) = 1,

ϕξ = 0, η◦ ϕ = 0,

g(ϕX, ϕY ) = g(X, Y )− η(X)η(Y ),

g(X, ϕY ) =−g(ϕX, Y ), g(X, ξ) = η(X),

for all X, Y ∈ M. Cariazzo et. al. [2], introduced generalized (κ, µ)-space forms as an almost contact metric manifold (M , ϕ, ξ, η, g) whose curvature tensor R can be written as

R(X, Y )Z = f1[g(Y, Z)X− g(X, Z)Y ]

+ f2[g(X, ϕZ)ϕY − g(Y, ϕZ)ϕX + 2g(X, ϕY )ϕZ]

+ f3[η(X)η(Z)Y − η(Y )η(Z)X + g(X, Z)η(Y )ξ

−g(Y, Z)η(X)ξ]

+ f4[g(Y, Z)hX− g(X, Z)hY + g(hY, Z)X − g(hX, Z)Y ] (2.1)

+ f₅[g(hY, Z)hX− g(hX, Z)hY + g(ϕhX, Z)ϕhY

−g(ϕhY, Z)ϕhX]

+ f6[η(X)η(Z)hY − η(Y )η(Z)hX + g(hX, Z)η(Y )ξ

−g(hY, Z)η(X)ξ],

where fi, (i = 1, 2, 3, 4, 5, 6) are smooth functions on M , h = ¹₂Lξ is a symmetric (1, 1)-tensor field on M andLξ is Lie differentiation in the direction of ξ.

The Equation of Gauss for submanifold M of M is given by R(X, Y, Z, W ) = R(X, Y, Z, W ) + g(σ(X, Z), σ(Y, W ))

−g(σ(X, W ), σ(Y, Z)) (2.2)

for any vectors X, Y ,Z and W tangent to M . Where we denote R, the curvature tensor of M and σ as the second fundamental tensor.

From (2.1) and Gauss equation (2.2), we have

R(X, Y, Z, W ) = f1[g(Y, Z)g(X, W )− g(X, Z)g(Y, W )]

+ f₂[g(X, ϕZ)g(ϕY, W )− g(Y, ϕZ)g(ϕX, W ) +2g(X, ϕY )g(ϕZ, W )]

+ f₃[η(X)η(Z)g(Y, W )− η(Y )η(Z)g(X, W ) +g(X, Z)η(Y )g(ξ, W )− g(Y, Z)η(X)g(ξ, W )]

+ f4[g(Y, Z)g(hX, W )− g(X, Z)g(hY, W ) (2.3)

+g(hY, Z)g(X, W )− g(hX, Z)g(Y, W )]

+ f5[g(hY, Z)g(hX, w)− g(hX, Z)g(hY, W ) +g(ϕhX, Z)g(ϕhY, W )− g(ϕhY, Z)g(ϕhX, W )]

+ f6[η(X)η(Z)g(hY, W )− η(Y )η(Z)g(hX, W ) +g(hX, Z)η(Y )g(ξ, W )− g(hY, Z)η(X)g(ξ, W )]

−g(σ(X, Z), σ(Y, W )) + g(σ(X, W ), σ(Y, Z)).

Let M be an n-dimensional submanifold of a generalized (κ, µ)-space form M of dimension 2m + 1. Let ∇ and ∇ be the Levi-Civita connection on M and M respectively. The Gauss and Weingarten equations are respectively defined as

∇XY =∇XY + σ(X, Y ),

∇Xν =−SνX +∇^⊥XY,

for vector fields X, Y ∈ T M and ν ∈ T^⊥M . Where S and∇^⊥is the shape operator and the normal connection respectively. The second fundamental form and the shape operator are related by the following equation

g(σ(X, Y ), ν) = g(SνX, Y ), for vector fields X, Y ∈ T M and ν ∈ T^⊥M .

Let M be an n-dimensional submanifold of a generalized (κ, µ)−space form M of dimension 2m + 1. For any tangent vector field X ∈ T M, we can write ϕX = P X + F X, where P X and F X are the tangential and normal components of ϕX respectively. If P = 0, the submanifold is said to be an anti-invariant

submanifold and if F = 0, the submanifold is said to be an invariant submanifold.

The squared norm of P at p∈ M is defined as

∥P ∥²=

∑n i,j=1

g²(ϕe_i, e_j), (2.4)

where{e1, . . . , e_n} is any orthonormal basis of the tangent space TpM.

A submanifold M of an almost Hermitian manifold M is said to be a slant submanifold if for any p∈ M and a non zero vector X ∈ TpM , the angle between J X and TpM is constant, i.e., the angle does not depend on the choice of p ∈ M and X ∈ TpM . The angle θ∈ [0,^π₂] is called the slant angle of M in M .

A submanifold M of an almost Hermitian manifold M is said to be a bi-slant submanifold, if there exist two orthogonal distributions D1and D2, such that

(i) T M admits the orthogonal direct decomposition, i.e., T M = D1+ D2, (ii) for i=1,2, Di is the slant distribution with slant angle θi.

In fact, semi-slant submanifolds, hemi-slant submanifolds, CR-submanifolds, slant submanifolds can be obtained from bi-slant submanifolds in particular. We can see the case in the following table:

Table 1: Defination

S.N.

M M D1 D2 θ1 θ2

(1) M bi-slant slant slant slant angle slant angle

(2) M semi-

slant

invariant slant 0 slant angle

(3) M hemi-

slant

slant anti- invariant

slant angle ^π₂

(4) M CR invariant anti-

invariant

0 ^π₂

(5) M slant either D1 = 0 or D2= 0 either θ1= θ2= θ or θ1= θ2 ̸= θ Invariant and anti-invariant submanifolds are the slant submanifolds with slant angle θ = 0 and θ = ^π₂ respectively and when 0 < θ < ^π₂, then slant submanifold is called proper slant submanifold.

If M is a bi-slant submanifold in generalized (κ, µ)-space form M , then one can easily see that

∥P ∥²=

∑n i,j

g²(P e_i, e_j) = 2(d₁cos²θ₁+ d₂cos²θ₂).

(2.5)

Let M be a Riemannian manifold and K(π) denotes the sectional curvature of M of the plane section π ⊂ TpM at a point p ∈ M. If {e1, . . . , e_n} and {en+1, . . . , e_2m+1} be the orthonormal basis of TpM and T_p^⊥M at any p ∈ M, then the scalar curvature τ at that point is given by

τ (p) = ∑

1≤i<j≤n

K(e_i∧ ej)

and the normalized scalar curvature ρ is defined as ρ = 2τ

n(n− 1).

The mean curvature vector denoted by H is defined as

H = 1 n

∑n i,j=1

σ(e_i, e_i).

We also put

σ^γ_ij= g(σ(ei, ej), eγ), i, j ∈ 1, 2, .., n, γ ∈ {n + 1, n + 2, ..., 2m + 1}.

The norm of the squared mean curvature of the submanifold is defined by

∥H∥²= 1 n²

2m+1∑

γ=n+1

(∑n

i=1

σ_ii^γ )2

and the squared norm of second fundamental form h is denoted byC defined as

C = 1 n

2m+1∑

γ=n+1

∑n i,j=1

(σ_ij^γ)2

known as Casorati curvature of the submanifold.

If we suppose that L is an r-dimensional subspace of T M , r ≥ 2, and {e1, e₂, . . . , e_r} is an orthonormal basis of L. then the scalar curvature of the r- plane section L is given as

τ (L) = ∑

1≤γ<β≤r

K(e_γ∧ eβ)

and the Casorati curvatureC of the subspace L is as follows

C(L) = 1 r

2m+1∑

γ=n+1

∑n i,j=1

(σ_ij^γ)2

.

A point p∈ M is said to be an invariantly quasi-umbilical point if there exist 2m− n + 1 mutually orthogonal unit normal vectors ξn+1, . . . , ξ_2m+1 such that the shape operators with respect to all directions ξ_γ have an eigenvalue of multiplicity n− 1 and that for each ξγ the distinguished eigne direction is the same. The submanifold is said to be an invariantly quasi-umbilical submanifold if each of its points is an invariantly quasi-umbilical point.

The normalized δ-Casorati curvature δc(n− 1) and bδc(n− 1) are defined as [δ_c(n− 1)]p=1

2Cp+n + 1

2n inf{C(L)|L : a hyperplane of TpM} (2.6)

and

[bδc(n− 1)]p= 2Cp+2n− 1

2n sup{C(L)|L : a hyperplane of TpM}.

(2.7)

For a positive real number t̸= n(n − 1), put a(t) = 1

nt(n− 1)(n + t)(n²− n − t), (2.8)

then the generalized normalized δ-Casorati curvatures δ_c(t; n− 1) and bδc(t; n− 1) are given as

[δ_c(t; n− 1)]p= tCp+ a(t)inf{C(L)|L : a hyperplane of TpM}, if 0 < t < n²− n, and

[bδc(t; n− 1)]p= tCp+ a(t)sup{C(L)|L : a hyperplane of TpM}, if t > n²− n.

3. Main Theorems

In this section, we obtain optimal inequalities for generalized normalized δ- Casorati curvatures for different n-dimensional submanifolds M in (2m + 1)- dimensional generalized (κ, µ)-space forms M taking ξ as tangent vector of M . Theorem 3.1. Let M be a submanifold in generalized (κ, µ)-space form M . Then

(i) the generalized normalized δ-Casorati curvature δc(t; n− 1) satisfies ρ ≤ δc(t; n− 1)

n(n− 1) + f₁+ 3

n(n− 1)f₂∥P ∥²−2 nf₃+ 2

nf₄tr(h^T)

+ 1

n(n− 1)f5

[(tr(h^T))²− ∥h^T∥²− ∥(ϕh)^T∥²− (tr(ϕh)^T)²] (3.1)

− 2

n(n− 1)f6tr(h^T),

for any real number t such that 0 < t < n(n− 1), and

(ii) the generalized normalized δ-Casorati curvature bδc(t; n− 1) satisfies

ρ ≤ δb_c(t; n− 1)

n(n− 1) + f1+ 3

n(n− 1)f2∥P ∥²−2 nf3+2

nf4tr(h^T)

+ 1

n(n− 1)f₅[

(tr(h^T))²− ∥h^T∥²− ∥(ϕh)^T∥²− (tr(ϕh)^T)²] (3.2)

− 2

n(n− 1)f6tr(h^T), for any real number t > n(n− 1).

Moreover, the equality holds in (3.1) and (3.2) if and only if M is an invariantly quasi-umbilical submanifold with trivial normal connection in M , such that with respect to suitable tangent orthonormal frame{e1, . . . , en} and normal orthonormal frame {en+1, . . . , e2m+1}, the shape operator Sr ≡ Se_r, r ∈ {n + 1, . . . , 2m + 1}, takes the following form

Sn+1=











a 0 0 . . . 0 0 0 a 0 . . . 0 0 0 0 a . . . 0 0 ... ... ... . .. ... ... 0 0 0 . . . a 0 0 0 0 . . . 0 ⁿ⁽ⁿ_t⁻¹⁾a











, Sn+2=· · · = S2m+1= 0.

(3.3)

Proof. Let {e1, . . . , e_n} and {en+1, . . . , e_2m+1} be the orthonormal basis of TpM and T_p^⊥M respectively at any point p∈ M. Then from (2.3) and (2.5), we have

2τ = n²∥H∥²− (n)C + n(n − 1)f1+ 3f2∥P ∥²

−2(n − 1)f3+ 2f₄(n− 1)tr(h^T) + f₅[

(tr(h^T))² (3.4)

−∥h^T∥²− ∥(ϕh)^T∥²− (tr(ϕh)^T)²]

− 2f6tr(h^T).

Define the following function, denoted by Q, a quadratic polynomial in the compo- nents of the second fundamental form

Q = tC + a(t)C(L) − 2τ + n(n − 1)f1+ 3f2∥P ∥²

−2(n − 1)f3+ 2f4(n− 1)tr(h^T) + f5

[(tr(h^T))² (3.5)

−∥h^T∥²− ∥(ϕh)^T∥²− (tr(ϕh)^T)²]

− 2f6tr(h^T),

where L is the hyperplane of TpM . Without loss of generality, we suppose that L is spanned by e1, . . . , en+s−1, it follows from (3.5) that

Q = n + t n

2m+1∑

γ=n+1

∑n i,j=1

(σ^γ_ij)²+ a(t) n− 1

2m+1∑

γ=n+1 n−1

∑

i,j=1

(σ^γ_ij)²−

2m+1∑

γ=n+1

(∑n

i=1

σ_ii^γ )2

,

which can be easily written as Q =

2m+1∑

γ=n+1 n∑−1

i=1

[(n + t

n + a(t) n− 1

)

(σ_ii^γ)²+2(n + t) n (σ^γ_in)²

]

+

2m+1∑

n+1

[ 2

(n + t

n + a(t) n− 1

) ∑n

(i<j)=1

(σ^γ_ij)²− 2

∑n (i<j)=1

σ^γ_iiσ_jj^γ + t n(σ_nn^γ )²

] . (3.6)

From(3.6), we can see that the critical points

σ^c= (σⁿ⁺¹₁₁ , σⁿ⁺¹₁₂ , . . . , σ_nnⁿ⁺¹, . . . , σ₁₁^2m+1, . . . , σ^2m+1_nn ) ofQ are the solutions of the following system of homogenous equations:



















∂Q

∂σ^γ_ii = 2 (

n+t n +_n^b(t)₋₁

)

(σ_ii^γ)− 2∑n

k=1σ^γ_kk= 0

∂Q

∂σ^γ_nn = ^2t_nσ^γ_nn− 2∑n−1

k=1σ^γ_kk= 0

∂Q

∂σ^γ_ij = 4 (

n+t n +_n^b(t)₋₁

)

(σ^γ_ij) = 0

∂Q

∂σ^γ_in = 4(^n+t_n )(σ^γ_in) = 0, (3.7)

where i, j ={1, 2, . . . , n − 1}, i ̸= j, and γ ∈ {n + 1, . . . , 2m + 1}.

Hence, every solution σ^c has σ_ij^γ = 0 for i̸= j and the corresponding determi- nant to the first two equations of the above system is zero. Moreover, the Hessian matrix ofQ is of the following form

H(Q) =



 H1 O O

O H2 O

O O H₃



 ,

where

H1=









2 (

n+t n +^a(t)_n−1

)

− 2 −2 . . . −2 −2

−2 2

( n+t

n +_n−1^a(t) )

− 2 . . . −2 −2

.. .

.. .

.. .

.. .

.. .

−2 −2 . . . 2

( n+t

n +_n−1^a(t) )

− 2 −2

−2 −2 . . . −2 ^2t_n







 ,

H2 and H3 are the diagonal matrices and O is the null matrix of the respective dimensions. H₂and H₃ are respectively given as

H₂= diag (

4 (n + t

n + a(t) n− 1

) , 4

(n + t

n + a(t) n− 1

) , . . . , 4

(n + t

n + a(t) n− 1

)) , and

H3= diag

(4(n + t)

n ,4(n + t)

n , . . . ,4(n + t) n

) .

Hence, we find thatH(Q) has the following eigenvalues

λ11= 0, λ22= 2 (2t

n + a(t) n− 1

)

, λ33=· · · = λnn= 2 (n + t

n + a(t) n− 1

) ,

λ_ij = 4 (n + t

n + a(t) n− 1

)

, λ_in=4(n + t)

n , ∀ i, j ∈ {1, 2, . . . , n − 1}, i ̸= j.

Thus, Q is parabolic and reaches at minimum Q(σ^c) = 0 for the solution σ^c of the system (3.7). HenceQ ≥ 0 and hence

2τ ≤ tC + a(t)C(L) + n(n − 1)f1+ 3f₂∥P ∥²

−2(n − 1)f3+ 2f4(n− 1)tr(h^T) + f5

[(tr(h^T))²

−∥h^T∥²− ∥(ϕh)^T∥²− (tr(ϕh)^T)²]

− 2f6tr(h^T), whereby, we obtain

ρ ≤ t

(n)(n− 1)C + a(t)

(n)(n− 1)C(L) + f1+ 3

n(n− 1)f₂∥P ∥²

−2 nf3+2

nf4tr(h^T) + 1 n(n− 1)f5

[(tr(h^T))²

−∥h^T∥²− ∥(ϕh)^T∥²− (tr(ϕh)^T)²]

− 2

n(n− 1)f6tr(h^T),

for every tangent hyperplane L of M . If we take the infimum over all tangent hyperplanes L, the result trivially follows. Moreover the equality sign holds if and only if

σ^γ_ij= 0, ∀ i, j ∈ {1, . . . , n}, i ̸= j and γ ∈ {n + 1, . . . , 2m + 1}

(3.8) and

σ_nn^γ = (n)(n− 1)

t σ^γ₁₁=· · · = (n)(n− 1)

t σ^γ_n−1n−1,∀γ ∈ {n + 1, . . . , 2m + 1}.

(3.9)

From (3.8) and (3.9), we obtain that the equality holds if and only if the subman- ifold is invariantly quasi-umbilical with normal connections in M , such that the shape operator takes the form (3.3) with respect to the orthonormal tangent and orthonormal normal frames.

In the same way, we can prove (ii). 2

Corollary 3.2. Let M be a submanifold in (κ, µ)-space form M . Then

(i) The normalized δ-Casorati curvature δc(n− 1) satisfies ρ ≤ δc(n− 1) + a(t)

(n)(n− 1)C(L) + f1+ 3

n(n− 1)f₂∥P ∥²

−2 nf₃+2

nf₄tr(h^T) + 1 n(n− 1)f₅[

(tr(h^T))²

−∥h^T∥²− ∥(ϕh)^T∥²− (tr(ϕh)^T)²]

− 2

n(n− 1)f6tr(h^T).

Moreover, the equality sign holds if and only if M is an invariantly quasi- umbilical submanifold with trivial normal connection in M , such that with respect to suitable tangent orthonormal frame {e1, . . . , en} and normal or- thonormal frame {en+1, . . . , e2m+1}, the shape operator Sr ≡ Se_r, r ∈ {n + 1, . . . , 2m + 1}, take the following form

S_n+1=













a 0 0 . . . 0 0

0 a 0 . . . 0 0

0 0 a . . . 0 0

.. . .. . .. . . .. ... ...

0 0 0 . . . a 0

0 0 0 . . . 0 2a













, S_n+2=· · · = S2m+1= 0.

(ii) The normalized δ-Casorati curvature bδc(n− 1) satisfies ρ ≤ bδc(n + s− 1) + a(t)

(n)(n− 1)C(L) + f1+ 3

n(n− 1)f2∥P ∥²

−2 nf3+2

nf4tr(h^T) + 1 n(n− 1)f5

[(tr(h^T))²

−∥h^T∥²− ∥(ϕh)^T∥²− (tr(ϕh)^T)²]

− 2

n(n− 1)f₆tr(h^T).

Moreover, the equality sign holds if and only if M is an invariantly quasi- umbilical submanifold with trivial normal connection in M , such that with respect to suitable tangent orthonormal frame {e1, . . . , e_n} a and normal orthonormal frame {en+1, . . . , e_2m+1}, the shape operator Sr ≡ Ser, r ∈ {n + 1, . . . , 2m + 1}, take the following form

Sn+1=













2a 0 0 . . . 0 0

0 2a 0 . . . 0 0

0 0 2a . . . 0 0 .. . .. . .. . . .. ... ...

0 0 0 . . . 2a 0

0 0 0 . . . 0 a













, Sn+2=· · · = S2m+1= 0.

Next, we prove the following.

Theorem 3.3. Let M be a bi-slant submanifold in generalized (κ, µ)-space forms M . Then

(i) The generalized normalized δ-Casorati curvature δc(t; n− 1) satisfies ρ ≤ δbc(t; n− 1)

n(n− 1) + f₁+ 3

n(n− 1)f₂(d₁cos²θ₁+ d₂cos²θ₂)

−2 nf3+ 2

nf4tr(h^T) + 1 n(n− 1)f5

[(tr(h^T))² (3.10)

−∥h^T∥²− ∥(ϕh)^T∥²− (tr(ϕh)^T)²]

− 2

n(n− 1)f6tr(h^T), for any real number t such that 0 < t < n(n− 1).

(ii) The generalized normalized δ-Casorati curvature bδ_c(t; n− 1) satisfies ρ ≤ δbc(t; n− 1)

n(n− 1) + f1+ 3

n(n− 1)f2(d1cos²θ1+ d2cos²θ2)

−2 nf3+ 2

nf4tr(h^T) + 1 n(n− 1)f5

[(tr(h^T))² (3.11)

−∥h^T∥²− ∥(ϕh)^T∥²− (tr(ϕh)^T)²]

− 2

n(n− 1)f6tr(h^T), for any real number t > n(n− 1). Moreover, the equality holds in (3.10) and (3.11) if and only if M is an invariantly quasi-umbilical submani- fold with trivial normal connection in M , such that with respect to suit- able tangent orthonormal frame {e1, . . . , en} and normal orthonormal frame {en+1, . . . , e2m+1}, the shape operator Sr≡ Ser, r∈ {n+1, . . . , 2m+1}, take the following form

Sn+1=











b 0 0 . . . 0 0 0 b 0 . . . 0 0 0 0 b . . . 0 0 ... ... ... . .. ... ... 0 0 0 . . . b 0 0 0 0 . . . 0 ⁿ⁽ⁿ_t⁻¹⁾b











, Sn+2=· · · = S2m+1= 0.

(3.12)

Proof. Proof of the above theorem is similar to the proof of Theorem 3.1 and is

obtained just using (2.5) in Theorem 3.1. 2

Further, we state and prove the following Theorems.

Theorem 3.4. Let M be a semi-slant submanifold in generalized (κ, µ)-space forms M . Then

(i) The generalized normalized δ-Casorati curvature δc(t; n− 1) satisfies ρ ≤ δb_c(t; n− 1)

n(n− 1) + f₁+ 3

n(n− 1)f₂(d₁+ d₂cos²θ₂)

−2 nf3+2

nf4tr(h^T) + 1 n(n− 1)f5

[(tr(h^T))² (3.13)

−∥h^T∥²− ∥(ϕh)^T∥²− (tr(ϕh)^T)²]

− 2

n(n− 1)f6tr(h^T), for any real number t such that 0 < t < n(n− 1).

(ii) The generalized normalized δ-Casorati curvature bδ_c(t; n− 1) satisfies ρ ≤ δb_c(t; n− 1)

n(n− 1) + f₁+ 3

n(n− 1)f₂(d₁+ d₂cos²θ₂)

−2 nf3+2

nf4tr(h^T) + 1 n(n− 1)f5

[(tr(h^T))² (3.14)

−∥h^T∥²− ∥(ϕh)^T∥²− (tr(ϕh)^T)²]

− 2

n(n− 1)f6tr(h^T), for any real number t > n(n− 1). Moreover , the equality holds in (3.13) and (3.14) if and only if M is an invariantly quasi-umbilical submani- fold with trivial normal connection in M , such that with respect to suit- able tangent orthonormal frame {e1, . . . , en} and normal orthonormal frame {en+1, . . . , e2m+1}, the shape operator Sr≡ Se_r, r∈ {n+1, . . . , 2m+1}, take the following form

Sn+1=











b 0 0 . . . 0 0 0 b 0 . . . 0 0 0 0 b . . . 0 0 ... ... ... . .. ... ... 0 0 0 . . . b 0 0 0 0 . . . 0 ⁿ⁽ⁿ_t⁻¹⁾b











, Sn+2=· · · = S2m+1= 0.

(3.15)

Theorem 3.5. Let M be a hemi-slant submanifold in generalized (κ, µ)-space forms M . Then

(i) The generalized normalized δ-Casorati curvature δ_c(t; n− 1) satisfies ρ ≤ δb_c(t; n− 1)

n(n− 1) + f₁+ 3

n(n− 1)f₂(d₁cos²θ₁)

−2 nf3+2

nf4tr(h^T) + 1 n(n− 1)f5

[(tr(h^T))² (3.16)

−∥h^T∥²− ∥(ϕh)^T∥²− (tr(ϕh)^T)²]

− 2

n(n− 1)f6tr(h^T),

for any real number t such that 0 < t < n(n− 1).

(ii) The generalized normalized δ-Casorati curvature bδc(t; n− 1) satisfies

ρ ≤ δbc(t; n− 1)

n(n− 1) + f1+ 3

n(n− 1)f2(d1cos²θ1)

−2 nf3+ 2

nf4tr(h^T) + 1 n(n− 1)f5

[(tr(h^T))² (3.17)

−∥h^T∥²− ∥(ϕh)^T∥²− (tr(ϕh)^T)²]

− 2

n(n− 1)f₆tr(h^T), for any real number t > n(n− 1). Moreover , the equality holds in (3.16) and (3.17) if and only if M is an invariantly quasi-umbilical submani- fold with trivial normal connection in M , such that with respect to suit- able tangent orthonormal frame {e1, . . . , en} and normal orthonormal frame {en+1, . . . , e2m+1}, the shape operator Sr≡ Se_r, r∈ {n+1, . . . , 2m+1}, take the following form

Sn+1=











b 0 0 . . . 0 0 0 b 0 . . . 0 0 0 0 b . . . 0 0 ... ... ... . .. ... ... 0 0 0 . . . b 0 0 0 0 . . . 0 ⁿ⁽ⁿ_t⁻¹⁾b











, Sn+2=· · · = S2m+1= 0.

(3.18)

Theorem 3.6. Let M be a CR-submanifold in generalized (κ, µ)-space forms M . Then

(i) The generalized normalized δ-Casorati curvature δ_c(t; n− 1) satisfies

ρ ≤ δb_c(t; n− 1)

n(n− 1) + f₁+ 3

n(n− 1)f₂d₁

−2 nf3+ 2

nf4tr(h^T) + 1 n(n− 1)f5

[(tr(h^T))² (3.19)

−∥h^T∥²− ∥(ϕh)^T∥²− (tr(ϕh)^T)²]

− 2

n(n− 1)f6tr(h^T), for any real number t such that 0 < t < n(n− 1).

(ii) The generalized normalized δ-Casorati curvature bδc(t; n− 1) satisfies ρ ≤ δb_c(t; n− 1)

n(n− 1) + f₁+ 3

n(n− 1)f₂d₁

−2 nf3+2

nf4tr(h^T) + 1 n(n− 1)f5

[(tr(h^T))² (3.20)

−∥h^T∥²− ∥(ϕh)^T∥²− (tr(ϕh)^T)²]

− 2

n(n− 1)f6tr(h^T), for any real number t > n(n− 1). Moreover , the equality holds in (3.19) and (3.20) if and only if M is an invariantly quasi-umbilical submani- fold with trivial normal connection in M , such that with respect to suit- able tangent orthonormal frame {e1, . . . , e_n} and normal orthonormal frame {en+1, . . . , e2m+1}, the shape operator Sr≡ Se_r, r∈ {n+1, . . . , 2m+1}, take the following form

Sn+1=











b 0 0 . . . 0 0 0 b 0 . . . 0 0 0 0 b . . . 0 0 ... ... ... . .. ... ... 0 0 0 . . . b 0 0 0 0 . . . 0 ⁿ⁽ⁿ_t⁻¹⁾b











, Sn+2=· · · = S2m+1= 0.

(3.21)

Theorem 3.7. Let M be a slant submanifold in generalized (κ, µ)-space forms M . Then

(i) The generalized normalized δ-Casorati curvature δc(t; n− 1) satisfies ρ ≤ δb_c(t; n− 1)

n(n− 1) + f1+ 3

(n− 1)f2cos²θ

−2 nf₃+2

nf₄tr(h^T) + 1 n(n− 1)f₅[

(tr(h^T))² (3.22)

−∥h^T∥²− ∥(ϕh)^T∥²− (tr(ϕh)^T)²]

− 2

n(n− 1)f6tr(h^T), for any real number t such that 0 < t < n(n− 1).

(ii) The generalized normalized δ-Casorati curvature bδ_c(t; n− 1) satisfies ρ ≤ δb_c(t; n− 1)

n(n− 1) + f₁+ 3

(n− 1)f₂cos²θ

−2 nf3+2

nf4tr(h^T) + 1 n(n− 1)f5

[(tr(h^T))² (3.23)

−∥h^T∥²− ∥(ϕh)^T∥²− (tr(ϕh)^T)²]

− 2

n(n− 1)f6tr(h^T),

for any real number t > n(n− 1). Moreover , the equality holds in (3.22) and (3.23) if and only if M is an invariantly quasi-umbilical submani- fold with trivial normal connection in M , such that with respect to suit- able tangent orthonormal frame {e1, . . . , e_n} and normal orthonormal frame {en+1, . . . , e_2m+1}, the shape operator Sr≡ Ser, r∈ {n+1, . . . , 2m+1}, take the following form

Sn+1=











b 0 0 . . . 0 0 0 b 0 . . . 0 0 0 0 b . . . 0 0 ... ... ... . .. ... ... 0 0 0 . . . b 0 0 0 0 . . . 0 ⁿ⁽ⁿ_t⁻¹⁾b











, Sn+2=· · · = S2m+1= 0.

(3.24)

Remark. We obtain proof of Theorem 3.4 - Theorem 3.7, just using Table 1 and results of Theorem 3.3.

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