General Properties of Slant Submanifolds in Contact Metric Manifolds

Optimal Inequalities for Submanifolds in Trans-Sasakian Manifolds Endowed with a Semi-Symmetric Metric Connection

“…In [23], A. Lotta has introduced the notion of contact slant submanifolds into almost contact metric manifolds. A submanifold M tangent to ξ in an almost contact metric manifold M is called a contact slant submanifold if for any p ∈ M and any X ∈ T p M linearly independent of ξ, the angle between φX and T p M is a constant θ ∈ [0, π/2], called the slant angle of M.…”

“…R(e 0 , e j , e j , e 0 ) = n ∑ j=1 R(e 0 , e j , e j , e 0 ) + n ∑ j=1 g(h(e 0 , e 0 ), h(e j , e j )) − n ∑ j=1 g(h(e 0 , e j ), h(e 0 , e j )), (23) or equivalently,…”

Warped Product Pointwise Semi-slant Submanifolds of Almost Contact Manifolds

“…Afterwards, the geometry of slant submanifolds became an active topic of research in differential geometry. Later, A. Lotta [19] has extended this study for almost contact metric manifolds. J. L. Cabrerizo et al investigated slant submanifolds of a Sasakian manifold [6].…”

Pointwise Quasi Hemi-Slant Submanifolds of Cosymplectic Manifolds

“…In 1990, Chen has put forward the notion of slant submanifold, which totally real submanifolds and generalizes holomorphic [5]. Then, the theory of submanifolds is investigated by many geometers like [6][7][8][9][10][11][12][13][14]. As a generalization of slant submanifolds; semi-slant submanifolds, hemi-slant submanifolds, bislant submanifolds, quasi bi-slant submanifolds, quasi hemi-slant submanifolds, pointwise quasi bi-slant submanifolds, PQHS submanifolds [15][16][17][18][19][20][21][22][23][24][25][26][27][28][29] and many others.…”

“…), we obtain < [𝑈, 𝑉], 𝜉 >=< ∇ ̃𝑈𝑉, 𝜉 > −< 𝑉, ∇ ̃𝑈𝜉, >= 0.Next, for every 𝑈, 𝑉 ∈ Γ(𝔇) and 𝑍 = 𝑄𝑍 + 𝑅𝑍 ∈ Γ(𝔇 ⊕ 𝔇 𝜃 ). Using (5),(8) and 𝐹𝑉 = 0 for all 𝑉 ∈ Γ(𝔇), we get< [𝑈, 𝑉], 𝑍 >=< ∇ ̃𝑈𝜑𝑉, 𝜑𝑍 > −< ∇ ̃𝑉𝜑𝑈, 𝜑𝑍 > =< ∇ ̃𝑈𝑇𝑉, 𝑇𝑄𝑍 + 𝐹𝑄𝑍 > +< ∇ ̃𝑈𝑇𝑉, 𝐹𝑅𝑍 > −< ∇ ̃𝑉𝑇𝑈, 𝜑𝑄𝑍 + 𝜑𝑅𝑍 >.By using (9) in the above equation, we have < [𝑈, 𝑉], 𝑍 >=< ∇ 𝑈 𝑇𝑉, 𝑇𝑄𝑍 > +< ℎ(𝑈, 𝑇𝑉), 𝐹𝑄𝑍 > +< ℎ(𝑈, 𝑇𝑉), 𝐹𝑅𝑍 > +< 𝜑(∇ ̃𝑉𝑇𝑈), 𝑅𝑍 > −< ∇ ̃𝑉𝑇𝑈, 𝑇𝑄𝑍 + 𝐹𝑄𝑍 > =< ∇ 𝑈 𝑇𝑉 − ∇ 𝑉 𝑇𝑈, 𝑇𝑄𝑍 > +< ℎ(𝑈, 𝑇𝑉), 𝐹𝑍 > +< 𝑇∇ 𝑉 𝑇𝑈 + 𝐵ℎ(𝑉, 𝑇𝑈), 𝑅𝑍 > −< ℎ(𝑉, 𝑇𝑈), 𝐹𝑄𝑍 >…”