Submanifolds of generalized $$(k,\mu )$$ ( k ,
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Submanifolds of generalized (k, )-space-forms Shyamal Kumar Hui1 · Siraj Uddin2 · Pradip Mandal1
© Akadémiai Kiadó, Budapest, Hungary 2018
Abstract In this paper, we have studied submanifolds especially, totally umbilical submanifolds of generalized (k, μ)-space-forms. We have found a necessary and sufficient condition for such submanifolds to be either invariant or anti-invariant. It is also shown that every totally umbilical submanifold of a generalized (k, μ)-space-form is a pseudo quasi-Einstein manifold. Keywords Generalized (k, μ)-space-forms · Invariant · Anti-invariant · Totally umbilical submanifolds · Quasi-Einstein manifold Mathematics Subject Classification 53C15 · 53C25 · 53C40
1 Introduction In 1995 Blair et al. [5] introduced the notion of contact metric manifold with characteristic vector field ξ belonging to the (k, μ)-nullity distribution and such type of manifolds are called (k, μ)-contact metric manifolds. A contact metric manifold M˜ is said to be a generalized (k, μ)-space [6] if its curvature tensor R˜ satisfies the condition ˜ , Y )ξ = k{η(Y )X − η(X )Y } + μ{η(Y )h X − η(X )hY } R(X
(1.1)
for some smooth functions k and μ on M˜ independent of the choice of vector fields X and Y . If k and μ are constants then the manifold M˜ is called a (k, μ)-space. A (k, μ)-space M˜ of dimension greater than 3 with constant ϕ-sectional curvature c is called (k, μ)-space-form [12] and the curvature tensor R˜ of such a manifold is given by [12]
B
Shyamal Kumar Hui [email protected] Siraj Uddin [email protected] Pradip Mandal [email protected]
1
Department of Mathematics, The University of Burdwan, Burdwan, West Bengal 713104, India
2
Department of Mathematics, King Abdulaziz University, Jeddah 21589, Saudi Arabia
123
S. K. Hui et al.
˜ , Y )Z = c + 3 R1 (X , Y )Z + c − 1 R2 (X , Y )Z R(X 4 4 c + 3 1 + − k R3 (X , Y )Z + R4 (X , Y )Z + R5 (X , Y )Z 4 2 + (1 − μ)R6 (X , Y )Z ,
(1.2)
where R1 , R2 , R3 , R4 , R5 , R6 are defined as R1 (X , Y )Z = g(Y , Z )X − g(X , Z )Y , R2 (X , Y )Z = g(X , ϕ Z )ϕY − g(Y , ϕ Z )ϕ X + 2g(X , ϕY )ϕ Z , R3 (X , Y )Z = η(X )η(Z )Y − η(Y )η(Z )X + g(X , Z )η(Y )ξ − g(Y , Z )η(X )ξ, R4 (X , Y )Z = g(Y , Z )h X − g(X , Z )hY + g(hY , Z )X − g(h X , Z )Y , R5 (X , Y )Z = g(hY , Z )h X − g(h X , Z )hY + g(ϕh X , Z )ϕhY − g(ϕhY , Z )ϕh X , R6 (X , Y )Z = η(X )η(Z )hY − η(Y )η(Z )h X + g(h X , Z )η(Y )ξ − g(hY , Z )η(X )ξ ˜ where h = 1 £ξ ϕ and £ is the usual Lie derivative. As a for all vector fields X , Y , Z on M, 2 generalization of (k, μ)-space-form, in [7] Carriazo et al. introduced and studied the notion of generalized (k, μ)-space-form with the existence of such notion by several interesting ˜ examples. An almost contact metric manifold M(ϕ, ξ, η, g) is called generalized (k, μ)˜ the ring of smooth functions on space-form if there exist f 1 , f 2 , f 3 , f 4 , f 5 , f 6 ∈ C ∞ ( M), ˜ such that M, ˜ , Y )Z = f 1 R1 (X , Y )Z + f 2 R2 (X , Y )Z + f 3 R3 (X , Y )Z R(X + f 4 R4 (X , Y )Z + f 5 R5 (X , Y )Z + f 6 R6 (X , Y )Z ,
(1.3)
where R1
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