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Some Results About Concircular and Concurrent Vector Fields On Pseudo-Kaehler Manifolds
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International Conference on Quantum Science and Applications (ICQSA-2016) IOP Publishing Journal of Physics: Conference Series 766 (2016) 012034 doi:10.1088/1742-6596/766/1/012034
Some Results About Concircular and Concurrent Vector Fields On Pseudo-Kaehler Manifolds Sibel Sevin¸ c, G¨ ul¸sah Aydın S ¸ ekerci and A. Ceylan C ¸o ¨ken Cumhuriyet University, Department of Mathematics, Sivas, TURKEY S¨ uleyman Demirel University, Department of Mathematics, Isparta, TURKEY Akdeniz University, Department of Mathematics, Antalya, TURKEY E-mail:
[email protected] E-mail:
[email protected] E-mail:
[email protected] Abstract. Kaehler manifolds which are used in physics have a lot of application fields. In this study we only state concircular and concurrent vector field that are defined on these manifolds. A vector field on a pseudo-Riemannian manifold N is called concircular, if it satisfies ∇X υ = µX for any vector X tangent to N, where ∇ is the Levi-Civita connection of N . Furthermore, a concircular vector field υ is called a concurrent vector field if the function µ is non-constant. So, we provide some results on submanifolds of pseudo-Kaehler manifolds with respect to a concircular vector field or a concurrent vector field. Morever, we investigate this problem for another manifolds and proof some theorems.
1. Introduction The notion of concircular vector field on a Riemannian manifold was introduced by A. Fialkow [7] by ∇X υ = µX (1) where X is tangent to N , ∇ denotes the Levi-Civita connection of N and µ is a non-trivial function on N . A concircular vector field satisfying (1) is called non-trivial if the function µ is non-constant. Furthermore, a concircular vector field υ is called a concurrent vector field if the function µ is equal to one [2]. For simplicity, we call as a concurrent vector field v if the function µ in (1) is a non-zero constant. Concircular vector fields have an important role in the theory of projective and conformal transformations. Additionally, such vector fields have interesting applications in general relativity. Particularly, a Lorentzian manifold is a generalized Robertson-Walker spacetime if and only if it admits a timelike concircular vector field. For some further results related to concircular vector fields, you can see [7, 3, 8]. A pseudo-Riemannian metric g on a complex manifold (M, J) is called pseudo-Hermitian if the metric g and the complex structure J on M are given by g (JX, JY ) = g (X, Y ) , X, Y ∈ Tp M, p ∈ M
(2)
A pseudo-Hermitian manifold is called a pseudo-Kaehler manifold if its complex structure J is parallel with respect to its Levi-Civita connection ∇. So it is clear that ∇J = 0. Content from this work may be used under the terms of the Creative Commons Attribution 3.0 licence. Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI. Published under licence by IOP Publishing Ltd 1
International Conference on Quantum Science and Applications (ICQSA-2016) IOP Publishing Journal of Physics: Conference Series 766 (2016) 012034 doi:10.1088/1742-6596/766/1/012034
Let N be an n-dimensional manifold with a 3-dimensional vector bundle Q consisting of three tensors J1 , J2 and J3 of type (1, 1) over N . Suppose, in any coordinate neighborhood U of N , there is a local basis {J1 , J2 , J3 }of Q such that J12 = J22 = J32 = −I and J2 J3 = −J3 J2 = J1 , J3 J1 = −J1 J3 = J2 , J1 J2 = −J2 J1 = J3 Such a basis{J1 , J2 , J3 }is called a canonical local basis of the bundle Q in U . We say that the bundle Q has an almost quaternion structure in N and (N, Q) is called an almost quaternion manifold whose dimension is n = 4m, (m ≥ 1). In such a case, the given structure Q is called an integrable quaternionic structure. Suppose g is an indefinite metric on (N, Q) such that g (φX, φY ) = g(X, Y ), ∀X, Y ∈ T pN , p ∈ N and φ = J1 , J2 , J3 , with{J1 , J2 , J3 }being a basis of Q at p. Then,(N, g, Q) is called an indefinite almost quaternion manifold.If the Levi-Civita ˜ , g, Q satisfies connection ∇ of M ∇X J1 = r (X) J2 − q (X) J3 ∇X J2 = −r (X) J1 − p (X) J3 ∇X J3 = q (X) J1 − p (X) J2
(3)
˜ , then M ˜ is called an indefinite quaternion Kaehler manifold, where for any vector field X on M p, q, r are certain local 1−forms and {J1 , J2 , J3 } is a local canonical basis of Q. Let M be a lightlike submanifold of an indefinite quaternion Kaehler manifold N . We say that M is a QR-lightlike submanifold if the following conditions are fulfilled: Ja RadT M ∩ RadT M = {0}, e ⊥ D0 , S(T M ) = {Ja RadT M ⊕ D} e ⊂ S(T M ), Ja (D0 ) = D0 , Ja (L1 ⊥ L2 ) = D where L1 = ltr(T M ) and L2 is a vector sub-bundle of S(T M ⊥ ). Now, we consider D = {RadT M ⊕orth Ja RadT M } ⊕orth D0 , where D0 is non-degenerate distribution. It is easy to check that D0 is invariant with respect to each Ja .Thus we have e TM = D ⊕ D [6]. 2. Some results for concurent vector fields Assume that N is a indefinite quaternionic Kaehler manifold and M is a pseudo-Kaehler submanifold of N which admits a concurrent vector field υ. We have the following: Proposition 2.1. Let N be a indefinite quaternionic Kaehler manifold and M be a submanifold with the indefinite metric of N . Then we have a) For any concurrent vector field v on M , Ja υ (a = 1, 2, 3) is never a concurrent vector field on M. b) The complex distribution D = Sp {v, J1 v, J2 v, J3 v} of N is always integrable.
2
International Conference on Quantum Science and Applications (ICQSA-2016) IOP Publishing Journal of Physics: Conference Series 766 (2016) 012034 doi:10.1088/1742-6596/766/1/012034
¯ Levi-Civita connection of N Proof. a) Let υ be a concurrent vector field on M . For ∇ ¯ X Ja υ = Ja ∇ ¯ X υ = cJa X + h (X, υ) Ja Wi ∇ ¯ (since ∇ ¯ X Jυ 6= cX). which implies that Ja υ is not a concurrent vector field for ∇ ⊥ b) Due to the Gauss equation, for Wi ∈ T M ¯ Ja υ υ = ∇Ja υ υ + h (Ja υ, υ) Wi = cJa υ + h (Ja υ, υ) Wi ∇ ¯ υ Ja υ = ∇υ Ja υ + h (υ, Ja υ) Wi ∇ ¯ X Ja υ = Ja ∇X υ = Ja cX = cJa X ∇ then we find ¯ Ja υ υ − ∇ ¯ υ Ja υ ∇
= cJa υ + h (Ja υ, υ) Wi − cJa υ − h (υ, Ja υ) Wi = 0 ⇒ [Ja υ, υ] = 0
(4)
¯ Levi-Civita connection of N , the distribution D is integrable. which implies that for ∇ Corollary. Let υ is a concurrent vector field on an indefinite quaternionic Kaehler manifold N . Then υ is a concurrent vector field on a submanifold of N if and only if h (X, υ) = 0. Proof. Let N be an indefinite quaternionic Kaehler manifold and M be a submanifold of N . ¯ Levi-Civita connection of N we can write ∇ ¯ X υ = cX, then For ∇ ¯ X υ = ∇X υ + h (X, υ) Wi = cX ∇ and we find h (X, υ) = 0 Obviously, the opposite of the theorem is true. Now we define the Ricci tensor of N . Let e1 , ..., e2n be an orthonormal frame on M , then the Ricci tensor Ric of M is defined by Ric (X, Y ) =
2n X
εi g (R (X, ei ) Y, ei ) +
i=1
2m X
εj g (R (X, ej ) Y, ej )
(5)
j=2n
where g (ei , ej ) = εi δij . Let R denote the Riemannian curvature tensor of M which is given by R (X, Y ) Z = ∇X ∇Y Z − ∇Y ∇X Z − ∇[X,Y ] Z
(6)
for vector fields X, Y, Z tangent to M . Furthermore, it is well-known that the Riemannian curvature tensor R satisfies i)R (X, Y ) = −R (Y, X) ii)R (X, Y ) JZ = J (R (X, Y ) Z) iii)R (JX, JY ) = R (X, Y ) Z iv)g (R (X, Y ) Z, W ) = g (R (Z, W ) X, Y )
(7)
Since υ is a concurrent vector field, ¯ (X, υ) υ = ∇ ¯ X∇ ¯ υυ − ∇ ¯ υ∇ ¯ Xυ − ∇ ¯ [X,υ] υ R ¯ X (cυ + h (υ, υ) Wi ) − ∇ ¯ υ (cX + h (X, υ) Wi ) = ∇ −c [X, υ] − h ([X, υ] , υ) Wi = X (h (υ, υ)) Wi + h (υ, υ) ∇⊥ X Wi − h (υ, υ) AWi X − υ (h (X, υ)) Wi ⊥ −h (X, υ) ∇υ Wi + h (X, υ) AWi υ − h ([X, υ] , υ) Wi 3
(8)
International Conference on Quantum Science and Applications (ICQSA-2016) IOP Publishing Journal of Physics: Conference Series 766 (2016) 012034 doi:10.1088/1742-6596/766/1/012034
So, we can write Ric (υ, υ) =
2m X
εj g (R (υ, ej ) υ, ej ) = −
j=2n
2m X
εj g (R (ej , υ) υ, ej )
(9)
j=2n
Multiplying with ej we find g (R (ej , υ) υ, ej ) = g (−h (υ, υ) AWi X + h (X, υ) AWi υ, ej )
(10)
From this, Ric (υ, υ) = −
2m X
εj g (−h (υ, υ) AWi X + h (X, υ) AWi υ, ej )
(11)
j=2n
Consequently, we have the following: Proposition 2.2. Let N be a indefinite quaternionic Kaehler manifold and M be a submanifold with the indefinite metric of N which admits a concurrent vector field (e.i. M is Ricci-flat). If −h (υ, υ) AWi X + h (X, υ) AWi υ = 0
(12)
then N is Ricci-flat (Ric = 0). Theorem 2.3. Let N be an indefinite quatenionic Kaehler manifold and M be a QR-lightlike submanifold of manifold N . Then, there are the followings. a) For any concurent vector field v on M , Ja υ is never a concurent vector field on M . b) The complex distribution D of N is always integrable. ¯ Levi-Civita connection of N Proof. a) Let υ be a concurent vector field on M . For ∇ ¯ X Ja υ = Ja ∇ ¯ X υ = Ja ∇X υ + hl (X, υ) Nj + hs (X, υ) Wi ∇ = Ja cX + hl (X, υ) Nj + hs (X, υ) Wi = cJa X + hl (X, υ) Ja Nj + hs (X, υ) Ja Wi ¯ X Ja υ 6= cX, Ja υ is not a concurent vector field where Nj ∈ ltr (T M ) , Wi ∈ S (T M )⊥ . Since ∇ ¯ for ∇. b) ¯ Ja υ υ = ∇Ja υ υ + hl (X, υ) Nj + hs (X, υ) Wi ∇ ¯ υ Ja υ = ∇υ Ja υ + hl (X, υ) Nj + hs (X, υ) Wi ∇ then we find ¯ Ja υ υ − ∇ ¯ υ Ja υ = cJa υ − ∇υ Ja υ = cJa υ − Ja ∇υ υ = cJa υ − Ja cv = 0 ⇒ [Ja υ, υ] = 0 ∇ ¯ Levi-Civita connection of N , the distribution D is integrable. which implies that for ∇ Corollary. Let υ is a concurrent vector field on a indefinite quaternionic Kaehler manifold N . Then υ is a concurrent vector field on a QR-lightlike submanifold of N if and only if hl (X, υ) = 0 and hs (X, υ) = 0 Proof. Let N be a indefinite quaternionic Kaehler manifold and M be a submanifold of N . For ¯ Levi-Civita connection of N we can write ∇ ¯ X υ = cX, then ∇ ¯ X υ = ∇X υ + hl (X, υ) Nj + hs (X, υ) Wi = cX ∇ and we find hl (X, υ) = 0 and hs (X, υ) = 0 Obviously, the opposite of the theorem is true. 4
International Conference on Quantum Science and Applications (ICQSA-2016) IOP Publishing Journal of Physics: Conference Series 766 (2016) 012034 doi:10.1088/1742-6596/766/1/012034
3. QR-lightlike hypersurfaces with concircular vector fields Theorem 3.1. Let M be a QR-lightlike hypersurface of an indefinite Kaehler manifold N . Then we have, h (X, v) = 0, X ∈ Γ (D0 ) such that v is a concircular vector field. Proof. For v∈ Γ (T M ) and ξ ∈ Γ (RadT M ) , ˜ X v, Ja ξ ge ∇ = ge (∇X v + h (X, v) N, Ja ξ)
(13)
= ge (∇X v, Ja ξ) + ge (h (X, v) N, Ja ξ) Due to N ∈ Γ (ltrT M ) and Ja ξ ∈ Γ (S (T M )) , ge (N, Ja ξ) = 0. So, ˜ X v, Ja ξ = g (µX, Ja ξ) ge ∇
(14)
By using this equation, we find ˜ X v, Ja ξ = 0 ge ∇
(15)
From here, we get ˜ X v, Ja Jb ξ ge ∇ = ge (∇X v + h (X, v) N, Ja Jb ξ)
(16)
= ge (µX, Ja Jb ξ) + h (X, v) ge (N, Ja Jb ξ) This equation must equal to zero. Then, let us find the needed conditions for this. For a = b, ˜ X v, Ja Jb ξ = ge (µX, −ξ) + h (X, v) ge (N, −ξ) = h (X, v) ge ∇ (17) So, h (X, v) = 0 is. For a 6= b, ˜ X v, Ja Jb ξ = ge (µX, Jc ξ) + h (X, v) ge (N, Jc ξ) = 0 ge ∇
(18)
where Jc = Ja Jb . We say that M is a mixed geodesic QR-lightlike surface if h (X, Z) = 0 for any X ∈ Γ (D) , Z ∈ e . Γ D Theorem 3.2. Let M be a QR-lightlike hypersurface of an indefinite Kaehler manifold N . a) M is a mixed geodesic if X (Ja ) = 0 while Z is a concircular vector field. b) M is a mixed geodesic if ge ([Ja N, X] , ξ) + ge (AN X, Ja ξ) = X (Ja ) while X is a concircular vector field. Proof. a) Let Z be a concircular vector field and Z = Ja N. h (X, Ja N ) N
˜ X Ja N − ∇X Ja N = ∇ ˜ X N − µX = X (Ja N ) + Ja ∇ = X (Ja N ) + Ja ∇⊥ X N + Ja (AN X) − µX
So, we find that ge (h (X, Ja N ) N, ξ) = X (Ja )
5
International Conference on Quantum Science and Applications (ICQSA-2016) IOP Publishing Journal of Physics: Conference Series 766 (2016) 012034 doi:10.1088/1742-6596/766/1/012034
b) Let X be a concircular vector field. h (X, Ja N ) N
˜ Ja N X − ∇ Ja N X = ∇ ˜ X J a N − ∇ Ja N X = [Ja N, X] − ∇
So, we find that ˜ X (Ja N ) , ξ − ge (∇Ja N X, ξ) ge (h (X, Ja N ) N, ξ) = ge ([Ja N, X] , ξ) − ge ∇ ˜ X N , ξ − µe = ge ([Ja N, X] , ξ) − ge (X (Ja ) N, ξ) − ge Ja ∇ g (Ja N, ξ) ˜ X N, Ja ξ − µe g (Ja N, ξ) = ge ([Ja N, X] , ξ) − X (Ja ) − ge ∇ = ge ([Ja N, X] , ξ) − X (Ja ) − ge (AN X, Ja ξ)
4. References [1] Barros M and Romero A 1982 Math. Ann. 261 1 [2] Chen B Y 1973 Geometry of Submanifolds (New York: M. Dekker) p 298 [3] Chen B Y 2011 Pseudo-Riemannian Geometry, δ−Invariants and Applications (Singapore: World Scientific Publishing Co. Pte. Ltd.) p 477 [4] Chen B Y 2015 Bull. Korean Math. Soc. 52 5 [5] Chen B Y 2016 Kragujevac J. Math. 40 1 [6] Duggal K L and S ¸ ahin B 2010 Differential Geometry of Lightlike Submanifolds(Basel-Boston-Berlin: Birkhauser Verlag AG) p 300 [7] Fialkow A 1939 Trans. Amer. Math. Soc. 45 3 [8] Kobayashi S and Nomizu K 1969 Foundations of Differantial Geometry Vol. 1 (New York: Interscience Publishers) p 329 [9] K¨ upeli N D 1996 Singular Semi-Riemannian Geometry (The Netherlands: Kluwer Academic Publishers) p 177 [10] O’Neill B 1983 Semi-Riemannian Geometry with Applications to Relativity (New York: Academic Press Inc.) p 482 [11] Romero A and Suh Y 2004 Extracta Mathematicae 19 3 [12] Yano K 1940 Proc. Imp. Acad. 16 6
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