ON THE QUASI-CONFORMAL CURVATURE TENSOR OF A (k, µ)-CONTACT METRIC MANIFOLD

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ON THE QUASI-CONFORMAL CURVATURE TENSOR OF A (k, µ)-CONTACT METRIC MANIFOLD

U.C. DE and AVIJIT SARKAR

The object of the present paper is to study a quasi-conformally flat (k, µ)-contact metric manifold and such a manifold with vanishing extended quasi-conformal curvature tensor. Quasi-conformally semisymmetric and quasi-conformally recurrent (k, µ)-contact metric manifolds are also considered.

AMS 2010 Subject Classification: 53C15, 53C40.

Key words: (k, µ)-contact metric manifolds, quasi-conformal curvature tensor, extended quasi-conformal curvature tensor, quasi-conformally semisymmetric, quasi-conformally recurrent.

1. INTRODUCTION

In [4], the authors introduced a class of contact metric manifolds for which the characteristic vector fieldξ belongs to the (k, µ)-nullity distribution for some real numbers k and µ. Such manifolds are known as (k, µ)-contact metric manifolds. The class of (k, µ)-contact metric manifolds encloses both Sasakian and non-Sasakian manifolds. Before Boeckx [5], two classes of non- Sasakian (k, µ)-contact metric manifolds were known. The first class consists of the unit tangent sphere bundles of spaces of constant curvature, equipped with their natural contact metric structure, and the second class contains all the three-dimensional unimodular Lie groups, except the commutative one, admitting the structure of a left invariant (k, µ)-contact metric manifold [4], [5], [18]. A full classification of (k, µ)-contact metric manifolds was given by E. Boeckx [5]. (k, µ)-contact metric manifolds are invariant under D-homothetic transformations. Recently, in [15], the authors proved that a non-Sasakian contact metric manifold with η-parallel torsion tensor and sectional curvatures, of plane sections containing the Reeb vector field, different from 1 at some point, is a (k, µ)-contact manifold. In another recent paper [21], R. Sharma showed that if a (k, µ)-contact metric manifold admits a nonzero holomorphically planer conformal vector field, then it is either Sasakian, or, locally isometric to E³ orEⁿ⁺¹×Sⁿ(4).In [8], J.T. Cho studied a conformally flat contact Riemannian (k, µ)-space and such a space with vanishing

MATH. REPORTS14(64),2 (2012), 115–129

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116 U.C. De and Avijit Sarkar 2

C-Bochner curvature tensor. In [14], the authors investigated on the conformal curvature tensor of a (k, µ)-contact metric manifold. Conformal curvature tensor of a (k, µ)-contact metric manifold was also studied in the paper [9]. In [19], D. Perrone worked on a contact Riemannian manifold satisfying R(X, ξ)·R = 0. Again, in [20], B.J. Papantoniou dealt with a contact Rie- mannian manifold satisfying R(X, ξ)·R= 0 andξ belonging to (k, µ)-nullity distribution. In [1], the authors studied extended pseudo projective curvature tensor on a contact metric manifold. Recently, quasi-conformal curvature tensor on a Sasakian manifold has been studied by U.C. De, J.B. Jun and A.K. Gazi [10]. In this connection, we mention some works of H. Endo [12], [13]

regarding some curvature tensors of contact and K-contact manifolds. After the Riemannian curvature tensor, Weyl conformal curvature tensor plays an important role in differential geometry as well as in theory of relativity. Yano and Sawaki [23] introduced the notion of quasi-conformal curvature tensor which is generalization of conformal curvature tensor in a Riemannian manifold. In a recent paper De and Matsuyama [11] studied quasi-conformally flat manifold satisfying certain condition on the Ricci tensor. They proved that a quasi-conformally flat manifold satisfying

(1.1) S(X, Y) =rT(X)T(Y),

where S is the Ricci tensor, r is the scalar curvature and T is a nonzero 1- form defined by T(X) =g(X, ρ), ρ is a unit vector field, can be expressed as a locally warped product I×_e^q M^∗,whereM^∗ is an Einstein manifold. From this result, it easily follows that a quasi-conformally flat space-time satisfying (1.1) is a Robertson-Walker space time [17]. The aim of our present paper is to study a quasi-conformally flat (k, µ)-contact metric manifold. We also like to study a (k, µ)-contact metric manifold with vanishing extended quasi- conformal curvature tensor. We also consider quasi-conformally semisymmetric and quasi-conformally recurrent (k, µ)-contact metric manifolds. The present paper is organized as follows:

In Section 2, we discuss about a (k, µ)-contact metric manifold and give some preliminary results regarding such a manifold. Section 3 deals with a quasi-conformally flat (k, µ)-contact metric manifold. Section 4 is devoted to study a (k, µ)-contact metric manifold with vanishing extended quasi-conformal curvature tensor. Section 5 contains the study of a quasi-conformally semisymmetric (k, µ)-contact metric manifold. The last section is devoted to study a quasi-conformally recurrent (k, µ)-contact metric manifold.

2. PRELIMINARIES

LetM be a (2n+ 1)-dimensionalC^∞-differentiable manifold. The manifold is said to admit an almost contact metric structure (φ, ξ, η, g) if it satisfies

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3 On the quasi-conformal curvature tensor of a (k, µ)-contact metric manifold 117

the following relations:

φ²X =−X+η(X)ξ, η(ξ) = 1, g(X, ξ) =η(X), (2.1)

φξ= 0, ηφ= 0, g(X, φY) =−g(φX, Y), g(X, φX) = 0, (2.2)

g(φX, φY) =g(X, Y)−η(X)η(Y), (2.3)

where φ is a tensor field of type (1,1), ξ is a vector field, η is an 1-form and g is a Riemannian metric on M.A manifold equipped with an almost contact metric structure is called an almost contact metric manifold. An almost contact metric manifold is called a contact metric manifold if it satisfies

g(X, φY) =dη(X, Y).

Given a contact metric manifold M(φ, ξ, η, g), we consider a (1,1) tensor field h defined by h = ¹₂L_ξφ, where L denotes Lie differentiation. h is a symmetric operator and h satisfieshφ=−φh. Ifλ is an eigenvalue ofh with eigenvectorX,then−λis also an eigenvalue ofhwith eigenvectorφX.Again, we have trh= trφh= 0,and hξ= 0.Moreover, if ∇denotes the Riemannian connection of g,then the following relation holds [4]:

(2.4) ∇_Xξ =−φX−φhX.

The vector fieldξ is a Killing vector field with respect tog if and only if h = 0.A contact metric manifold M(φ, ξ, η, g) for whichξ is a Killing vector is said to be a K-contact manifold. AK-contact structure on M gives rise to an almost complex structure on the product M ×R. If this almost complex structure is integrable, the contact metric manifold is said to be Sasakian.

Equivalently, a contact metric manifold is said to be Sasakian if and only if R(X, Y)ξ =η(Y)X−η(X)Y

holds for all X, Y, where R denotes the Riemannian curvature tensor of the manifold M. The (k, µ)-nullity distribution of a contact metric manifold M(φ, ξ, η, g) is a distribution [4]

N(k, µ) :p→Np(k, µ) ={Z ∈Tp(M) :R(X, Y)Z= (2.5)

=k(g(Y, Z)X−g(X, Z)Y) +µ(g(Y, Z)hX−g(X, Z)hY)},

for any X, Y ∈T_pM. Hence, if the characteristic vector field ξ belongs to the (k, µ)-nullity distribution, we have

(2.6) R(X, Y)ξ=k[η(Y)X−η(X)Y] +µ[η(Y)hX−η(X)hY].

A contact metric manifold with ξ belonging to (k, µ)-nullity distribution is called a (k, µ)-contact metric manifold. If k = 1, µ = 0, then the manifold becomes Sasakian [4]. In particular, if µ= 0,then the notion of (k, µ)-nullity distribution reduces to k-nullity distribution, introduced by S. Tanno [22]. A

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contact metric manifold with ξ belonging tok-nullity distribution is known as N(k)-contact metric manifold.

In a (k, µ)-contact metric manifold we have the following [4]:

(2.7) h² = (k−1)φ², k≤1,

(2.8) (∇_Xφ)(Y) =g(X+hX, Y)ξ−η(Y)(X+hX), η(R(X, Y)Z) =

(2.9)

=k[g(Y, Z)η(X)−g(X, Z)η(Y)] +µ[g(hY, Z)η(X)−g(hX, Z)η(Y)], (2.10) R(ξ, X)Y =k[g(X, Y)ξ−η(Y)X] +µ[g(hX, Y)ξ−η(Y)hX], (2.11) R(ξ, X)ξ =k[η(X)ξ−X]−µhX,

Qφ−φQ= 2(2(n−1) +µ)hφ.

Also, for a contact metric manifold M²ⁿ⁺¹(φ, ξ, η, g) with ξ ∈ N(k, µ), the Ricci operator Qis given by [4]

QX = [2(n−1)−nµ]X+ [2(n−1) +µ]hX+ (2.12)

+ [2(1−n) +n(2k+µ)]η(X)ξ, n≥1,

S(X, Y) = [2(n−1)−nµ]g(X, Y) + [2(n−1) +µ]g(hX, Y)+

(2.13)

+ [2(1−n) +n(2k+µ)]η(X)η(Y), n≥1,

(2.14) S(X, ξ) = 2nkη(X),

where S is the Ricci tensor of the manifold. A (k, µ)-contact metric manifold is called an η-Einstein manifold if it satisfies

S(X, W) =a1g(X, W) +b1η(X)η(Y), where a1 and b1 are two scalars.

For a (2n+ 1)-dimensional Riemannian manifold, the quasi-conformal curvature tensor ˜C is given by

C(X, Y˜ )Z =aR(X, Y)Z+b h

S(Y, Z)X−S(X, Z)Y+ (2.15)

+g(Y, Z)QX −g(X, Z)QY i

− r 2n+ 1

h a 2n+ 2b

i

[g(Y, Z)X−g(X, Z)Y], where a and b are two scalars, and r is the scalar curvature of the manifold.

The notion of quasi-conformal curvature tensor was introduced by K. Yano and S. Sawaki [23]. If a= 1 and b=−_2n−1¹ , then quasi-conformal curvature tensor reduces to conformal curvature tensor.

ForX, Y orthogonal to ξ, we get from (2.9) and (2.15) (2.16) η(R(X, Y)Z) = 0, η( ˜C(X, Y)Z) = 0.

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Let us recall the following results:

Lemma 2.1.A contact metric manifoldM²ⁿ⁺¹(φ, ξ, η, g)withR(X, Y)ξ= 0,for all vector fields X, Y on the manifold and n >1,is locally isometric to the Riemannian product Eⁿ⁺¹×Sⁿ(4) [3].

Lemma 2.2.LetM²ⁿ⁺¹(φ, ξ, η, g)be a contact metric manifold withξ belonging to the (k, µ)-nullity distribution. Ifk <1, then for any X orthogonal to ξ, the ξ-sectional curvatureK(X, ξ) is given by

K(X, ξ) =k+µg(hX, X) =k+λµ if X ∈D(λ),

=k−λµ if X ∈D(−λ),

where D(λ) and D(−λ) are two mutually orthogonal distributions defined by the eigenspace of h, and λ=√

1−k [4].

A (k, µ)-contact metric manifold will be called a manifold of quasi- constant curvature if the Riemannian curvature tensor ˜Rof type (0,4) satisfies the condition

R(X, Y, Z, W˜ ) =p[g(Y, Z)g(X, W)−g(X, Z)g(Y, W)]+

(2.17)

+q[g(X, W)T(Y)T(Z)−g(X, Z)T(Y)T(W)+

+g(Y, Z)T(X)T(W)−g(Y, W)T(X)T(Z)],

where ˜R(X, Y, Z, W) = g(R(X, Y)Z, W), p, q are scalars, and there exists a unit vector field ρ satisfying g(X, ρ) = T(X). The notion of quasi-constant curvature for Riemannian manifolds was introduced by Chen and Yano [6].

Ann-dimensional Riemannian manifold whose curvature tensor ˜Rof type (0,4) satisfies the condition

R(X, Y, Z, W˜ ) =F(Y, Z)F(X, W)−F(X, Z)F(Y, W),

where F is a symmetric tensor of type (0,2) is called a special manifold with the associated symmetric tensorF,and is denoted by ψ(F)_n.

Such type of manifolds have been studied by S.S. Chern [7] in 1956.

Remark 2.1. An n-dimensional manifold of quasi-constant curvature is a ψ(F)_n.

Proof. Let us define√

pg(X, Y) +^√^q_pT(X)T(Y) =F(X, Y).Then, from (2.17), it follows that

R(X, Y, Z, W˜ ) =F(Y, Z)F(X, W)−F(X, Z)F(Y, W).

Hence, a manifold of quasi-constant curvature is aψ(F)_n.This completes the proof of the remark.

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120 U.C. De and Avijit Sarkar 6 3. QUASI-CONFORMALLY FLAT(k, µ)-CONTACT METRIC MANIFOLDS

Theorem 3.1. A (2n+ 1)-dimensional (n > 1) quasi-conformally flat (k, µ)-contact metric manifold cannot be an η-Einstein manifold.

Proof. For a (2n+ 1)-dimensional (n >1) quasi-conformally flat (k, µ)- contact metric manifold we have

R(X, Y, Z, W˜ ) = b

a[S(X, Z)g(Y, W)−S(Y, Z)g(X, W)+

(3.1)

+S(Y, W)g(X, Z)−S(X, W)g(Y, Z)] +

+ r

a(2n+ 1) h a

2n+ 2b i

[g(Y, Z)g(X, W)−g(X, Z)g(Y, W)]. By using (2.1), (2.6) and (2.14), and putting Z =ξ in (3.1), we get

k(g(X, W)η(Y)−g(Y, W)η(X))+µ(η(Y)g(hX, W)−η(X)g(hY, W)) = (3.2)

= b

a[2nkg(Y, W)η(X)−2nkg(X, W)η(Y) +S(Y, W)η(X)−S(X, W)η(Y)] +

+ r

a(2n+ 1) h a

2n + 2b i

[g(X, W)η(Y)−g(Y, W)η(X)]. Putting Y =ξ in (3.2), and using (2.1) and (2.14), we obtain

S(X, W) = a b

r a(2n+ 1)

a 2n+ 2b

−2nkb a −k

g(X, W)+

(3.3) +a

b

k+4nkb

a − r

a(2n+ 1) a

2n + 2b

η(X)η(W)− aµ

b g(hX, W).

From (3.3), it follows that if µ = 0, the manifold is η-Einstein. Con- versely, if the manifold is η-Einstein, then we can write

S(X, W) =a1g(X, W) +b1η(X)η(Y),

wherea1 andb1are two scalars. From the above equation and (3.3),we obtain a₁g(X, W) +b₁η(X)η(W) = a

b

r a(2n+ 1)

a

2n + 2b

−2nkb a −k

(3.4) ·

·g(X, W) +a b

k+4nkb

a − r

a(2n+ 1) a

2n + 2b

η(X)η(W)− aµ

b g(hX, W).

Putting W =φX and using (2.2),we get from (3.4) aµ

b g(hX, φX) = 0, for all X. Consequently, µ= 0.

Hence, we see that a (2n+ 1)-dimensional (n >1) quasi-conformally flat (k, µ)-contact metric manifold is an η-Einstein manifold, if and only if µ= 0.

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But from (2.13),it follows that a (k, µ)-contact metric manifold is η-Einstein if and only if 2(n−1) +µ= 0.If we consider a (2n+ 1)-dimensional (n >1) quasi-conformally flat η-Einstein (k, µ)-contact metric manifold, then n= 1, which contradicts the fact thatn >1.Hence, the theorem is established.

Theorem 3.2. A (2n+ 1)-dimensional (n > 1) quasi-conformally flat (k, µ)-contact metric manifold, which is not conformally flat, is of quasi- constant curvature.

Proof. In (3.3),let us denote

(3.5) A= r

b(2n+ 1) a

2n + 2b

−2nk−ak b and

B = ak

b + 4nk− r b(2n+ 1)

a

2n+ 2b . Then, we see that

(3.6) A+B= 2nk.

In (3.3), putting X = W = e_i, where {e_i} is an orthonormal basis of the tangent space at each point of the manifold, and taking summation over i, i= 1,2,3, . . . ,(2n+ 1),we get

(3.7) r=A(2n+ 1) +B.

From (3.6) and (3.7), we get

(3.8) A= r

2n −k.

From (3.5) and (3.8), it follows that r

b(2n+ 1) a

2n+ 2b

−2nk−ak b = r

2n−k.

The above relation gives

(a+ 2nb−b)(r−2nk(2n+ 1)) = 0.

Hence, either a+ 2nb−b= 0,or,r = 2nk(2n+ 1).

Let us suppose that a+ 2nb−b = 0. Then, we see that b = _2n−1^−a . Hence, from (2.15),it follows that ˜C(X, Y)Z =aC(X, Y)Z, whereC(X, Y)Z is the Weyl conformal curvature tensor. This means that in this case quasi- conformally flat manifold is equivalent to conformally flat manifold. In [4], the authors showed that a contact metric manifold M²ⁿ⁺¹ with ξ belonging to (k, µ)-nullity distribution is a contact strongly pseudo convex integrable CR- manifold. In [14], the authors proved that a conformally flat contact pseudo- convex integrable CR-manifold of dimension greater than three is a space of constant curvature 1, from which it follows that a conformally flat contact

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metric manifold M²ⁿ⁺¹ (n >1) withξ belonging to (k, µ)-nullity distribution is a space of constant curvature 1.Hence, in this case, the quasi-conformally flat manifold under consideration is also of constant curvature 1.

For the second possibility, that is, for r = 2nk(2n + 1), we obtain from (3.3)

(3.9) S(X, W) = 2nkg(X, W)−aµ

b g(hX, W).

Using (3.9) in (3.1), we have

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

(3.10)

−µ[g(hX, Z)g(Y, W)−g(hY, Z)g(X, W)+

+g(hY, W)g(X, Z)−g(hX, W)g(Y, Z)].

From (2.13) and (3.9), it follows that

(3.11) g(hX, W) =l(g(X, W)−η(X)η(W)), where l= b(2nk+nµ−2n+2)

2nb+µb+µa−2b .

Using (3.11) in (3.10), we obtain

R(X, Y, Z, W˜ ) =p[g(Y, Z)g(X, W)−g(X, Z)g(Y, W)]+

(3.12)

+q[g(X, W)η(Y)η(Z)−g(X, Z)η(Y)η(W)+

+g(Y, Z)η(X)η(W)−g(Y, W)η(X)η(Z)], where p=k+ 2lµ andq =−lµ.This completes the proof.

For anN(k)-contact metric manifold,µ= 0.Consequently,q= 0.Hence, we have the following:

Corollary 3.1. A (2n+ 1)-dimensional (n >1)quasi-conformally flat N(k)-contact metric manifold is of constant curvature.

Fork= 1 andµ= 0,a (k, µ)-contact metric manifold becomes Sasakian.

Therefore, we are in a position to state the following:

Corollary 3.2. A (2n+ 1)-dimensional (n >1)quasi-conformally flat Sasakian manifold is of constant curvature.

The above result has been proved in the paper [10].

In view of Remark 2.1 and Theorem 3.2, we get the following:

Corollary 3.3. A (2n+ 1)-dimensional (n >1)quasi-conformally flat (k, µ)-contact metric manifold, which is not conformally flat, is a ψ(F)2n+1.

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9 On the quasi-conformal curvature tensor of a (k, µ)-contact metric manifold 123 4. (k, µ)-CONTACT METRIC MANIFOLDS WITH VANISHING

EXTENDED QUASI-CONFORMAL CURVATURE TENSOR

The object of the present section is to study a (k, µ)-contact metric manifold with vanishing extended quasi-conformal curvature tensor. In this connection it should be mentioned that the extended pseudo projective curvature tensor of a contact metric manifold has been studied in the paper [1].

Theorem 4.1. A(2n+ 1)-dimensional(n >1) (k, µ)-contact metric manifold with vanishing extended quasi-conformal curvature tensor is a Sasakian manifold.

Proof. The extended form of the quasi-conformal curvature tensor can be written as

C˜_e(X, Y)Z =aR(X, Y)Z+bh

S(Y, Z)X−S(X, Z)Y +g(Y, Z)QX− (4.1)

−g(X, Z)QYi

− r 2n+ 1

h a 2n+ 2bi

[g(Y, Z)X−g(X, Z)Y]−

−η(X) ˜C(ξ, Y)Z−η(Y) ˜C(X, ξ)Z−η(Z) ˜C(X, Y)ξ.

Putting Y =Z =ξ, and supposing that the extended quasi-conformal curvature tensor vanishes, we get from (4.1)

aR(X, ξ)ξ+b[S(ξ, ξ)X−S(X, ξ)ξ+QX −η(X)Qξ]−

(4.2)

− r 2n+ 1

h a 2n + 2bi

[X−η(X)ξ]−η(X) ˜C(ξ, ξ)ξ−C(X, ξ)ξ˜ −C(X, ξ˜ )ξ= 0.

In view of (2.11), and (2.15) we have, from above

a[k(X−η(X)ξ) +µhX] +b[2nk(X−η(X)ξ) +QX −η(X)Qξ]−

− r 2n+ 1

h a 2n+ 2bi

[X−η(X)ξ]−2h

a k(X−η(X)ξ+µhX)+

+b(2nk(X−η(X)ξ)+QX−η(X)Qξ

− r 2n+ 1

a

2n+ 2b

(X−η(X)ξ)i

= 0.

Simplifying the above equation, we obtain

ak+ 2nbk− r (2n+ 1)

a

2n + 2b

· (4.3)

·(X−η(X)ξ) +aµhX +b(QX−η(X)Qξ) = 0.

Using (2.12) in (4.3), we get

ak+ 2nbk− r (2n+ 1)

a 2n+ 2b

+ 2b(n−1)−nbµ

·

·(X−η(X)ξ) + [aµ+b(2(n−1) +µ)]hX = 0.

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The above equation yields

ak+ 2nbk− r (2n+ 1)

a 2n+ 2b

+ 2b(n−1)−nbµ

φ²X−

(4.4)

−[aµ−b(2(n−1) +µ)]hX = 0.

In view of (2.7), (4.4) takes the form

h²X= (k−1) [aµ+b(2(n−1) +µ)]hX

(ak+ 2nbk−_(2n+1)^r (_2n^a + 2b) + 2b(n−1)−nbµ). Since trh= 0,from above we get

(4.5) trh² = 0.

Now, h²X= (k−1)φ²X.Hence, we have

(4.6) trh²= 2n(1−k).

By virtue of (4.5) and (4.6), we get k = 1. Then, from (2.7), h = 0. This means that the manifold is Sasakian. Hence, the theorem is proved.

From the above theorem, we also have the following:

Corollary 4.1. A (2n+ 1)-dimensional (n > 1) N(k)-contact metric manifold with vanishing extended quasi-conformal curvature tensor is a Sasakian manifold.

The unit tangent sphere bundle T₁M(c) of constant curvature c with standard contact metric structure is a (k, µ)-contact metric manifold with k=c(2−c) and µ=−2c [4]. Then by Theorem 4.1 we obtain c= 1.Hence, we can state the following:

Corollary 4.2. A unit tangent sphere bundle with vanishing extended quasi-conformal curvature tensor is of constant curvature 1.

5. QUASI-CONFORMALLY SEMISYMMETRIC (k, µ)-CONTACT METRIC MANIFOLDS

Definition 5.1. A (2n+ 1)-dimensional (n >1) (k, µ)-contact metric manifold is said to be quasi-conformally semisymmetric if the conditionR(X, Y).

C˜ = 0 holds, for any vector fieldsX, Y on the manifold.

Theorem 5.1. A (2n+ 1)-dimensional (n > 1) quasi-conformally semisymmetric (k, µ)-contact manifold is either locally the Riemannian product Eⁿ⁺¹×Sⁿ(4), or, locally quasi-conformally flat.

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Proof. Let us suppose that the manifold satisfies R(ξ, U)·C˜ = 0.Then, it follows that

R(ξ, U) ˜C(X, Y)Z−C(R(ξ, U˜ )X, Y)Z−

(5.1)

−C(X, R(ξ, U˜ )Y)Z−C(X, Y˜ )R(ξ, U)Z = 0.

In virtue of (2.10), the above equation reduces to

k[g(U,C(X, Y˜ )Z)ξ−η( ˜C(X, Y)Z)U]+

+µ[g(hU,C(X, Y˜ )Z)ξ−η( ˜C(X, Y)Z)hU]−

−C(k[g(U, X)ξ˜ −η(X)U] +µ[g(hU, X)ξ−η(X)hU], Y)Z−

−C(X, k[g(U, Y˜ )ξ−η(Y)U] +µ[g(hU, Y)ξ−η(Y)hU])Z−

−C(X, Y˜ )(k[g(U, Z)ξ−η(Z)U] +µ[g(hU, Z)ξ−η(Z)hU]) = 0, or,

k[g(U,C(X, Y˜ )Z)ξ−η( ˜C(X, Y)Z)U−g(X, U) ˜C(ξ, Y)Z+

(5.2)

+η(X) ˜C(U, Y)Z−g(U, Y) ˜C(X, ξ)Z+η(Y) ˜C(X, U)Z−

−g(U, Z) ˜C(X, Y)ξ+η(Z) ˜C(X, Y)U]+

+µ[g(hU,C(X, Y˜ )Z)ξ−η( ˜C(X, Y)Z)hU−g(hU, X) ˜C(ξ, Y)Z+ +η(X) ˜C(hU, Y)Z−g(hU, Y) ˜C(X, ξ)Z+η(Y) ˜C(X, hU)Z−

−g(hU, Z) ˜C(X, Y)ξ+η(Z) ˜C(X, Y)hU] = 0.

Let us consider the following cases:

Case 1. k= 0 =µ.

Case 2. k= 0, µ6= 0.

Case 3. k6= 0, µ= 0.

Case 4. k6= 0, µ6= 0.

For Case 1, we observe that R(X, Y)ξ = 0,for all X, Y. Hence, by Lem- ma 2.1, the manifold is locally the Riemannian product Eⁿ⁺¹×Sⁿ(4).

For Case 2, we get from (5.2)

[g(hU,C(X, Y˜ )Z)ξ−η( ˜C(X, Y)Z)hU−g(hU, X) ˜C(ξ, Y)Z+

(5.3)

+η(X) ˜C(hU, Y)Z−g(hU, Y) ˜C(X, ξ)Z+η(Y) ˜C(X, hU)Z−

−g(hU, Z) ˜C(X, Y)ξ+η(Z) ˜C(X, Y)hU] = 0.

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In this case, in view of (2.7), h² =−φ².Hence, replacing U byhU,we obtain, from (5.3)

[g(η(U)ξ−U,C(X, Y˜ )Z)ξ−η( ˜C(X, Y)Z)(η(U)ξ−U)−

(5.4)

−g(η(U)ξ−U, X) ˜C(ξ, Y)Z+η(X) ˜C(η(U)ξ−U, Y)Z−

−g(η(U)ξ−U, Y) ˜C(X, ξ)Z+η(Y) ˜C(X, η(U)ξ−U)Z−

−g(η(U)ξ−U, Z) ˜C(X, Y)ξ+η(Z) ˜C(X, Y)(η(U)ξ−U)] = 0.

Choosing X, Y, Z orthogonal toξ and using (2.16), we have g(U,C(X, Y˜ )Z)ξ= 0.

The above equation implies

C(X, Y˜ )Z = 0.

For Case 3, changingU by h²U in (5.2), we shall get the same result as in Case 2.

In Case 4, changing U by hU, and using (2.7) in (5.2), we obtain after simplification

k[g(hU,C(X, Y˜ )Z)ξ−η( ˜C(X, Y)Z)hU−g(X, hU) ˜C(ξ, Y)Z+

(5.5)

+η(X) ˜C(hU, Y)Z−g(hU, Y) ˜C(X, ξ)Z+η(Y) ˜C(X, hU)Z−

−g(hU, Z) ˜C(X, Y)ξ+η(Z) ˜C(X, Y)hU]+

+µ(k−1)[g(U,C(X, Y˜ )Z)ξ−η( ˜C(X, Y)Z)U−

−g(U, X) ˜C(ξ, Y)Z+η(X) ˜C(U, Y)Z−g(U, Y) ˜C(X, ξ)Z+

+η(Y) ˜C(X, U)Z−g(U, Z) ˜C(X, Y)ξ+η(Z) ˜C(X, Y)U] = 0.

Eliminating hU from (5.2) and the above equation, we obtain [k²−µ²(k−1)][g(U,C(X, Y˜ )Z)ξ−η( ˜C(X, Y)Z)U− (5.6)

−g(U, X) ˜C(ξ, Y)Z+η(X) ˜C(U, Y)Z−g(U, Y) ˜C(X, ξ)Z+

+η(Y) ˜C(X, U)Z−g(U, Z) ˜C(X, Y)ξ+η(Z) ˜C(X, Y)U] = 0.

Hence, either

(5.7) k²−µ²(k−1) = 0,

or,

g(U,C(X, Y˜ )Z)ξ−η( ˜C(X, Y)Z)U −g(U, X) ˜C(ξ, Y)Z+

(5.8)

+η(X) ˜C(U, Y)Z−g(U, Y) ˜C(X, ξ)Z+η(Y) ˜C(X, U)Z−

−g(U, Z) ˜C(X, Y)ξ+η(Z) ˜C(X, Y)U = 0.

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We assert thatk²−µ²(k−1)6= 0.This means that the sectional curvature of the plane section containing ξ are non vanishing. But Lemma 2.2 tells us that the sectional curvature of the plane section, containing ξ,equal to zero if and only if k+λµ= 0 forX ∈D(λ), andk−λµ= 0 for X ∈D(−λ),where D(λ) is the distribution defined by the vector field hX, λ=√

1−k,and T M = [ξ]⊕[D(λ)]⊕[D(−λ)].

In this case, we have k+µ√

1−k= 0 and k−µ√

1−k= 0.These givek= µ= 0.But we have assumedk6= 0, µ6= 0.Consequently, k²−µ²(k−1)6= 0.

Hence, our assertion is true.

Therefore, for the remaining possibility, as Case 3, the manifold is quasi- conformally flat, provided X, Y, Z are orthogonal to ξ. Now, from the above discussion, we get the theorem.

By virtue of the above theorem, we obtain the following:

Corollary 5.1. A (2n+ 1)-dimensional (n > 1) quasi-conformally semisymmetric N(k)-contact manifold is either locally the Riemannian product Eⁿ⁺¹×Sⁿ(4), or, locally quasi-conformally flat.

6. QUASI-CONFORMALLY RECURRENT (k, µ)-CONTACT METRIC MANIFOLDS

Definition 6.1. A (2n+ 1)-dimensional (n > 1), (k, µ)-contact metric manifold is called quasi-conformally recurrent if it satisfies

(6.1) (∇_UC)(X, Y˜ )Z =A(U) ˜C(X, Y)Z, for certain non-zero 1-form A.

Theorem 6.1. A (2n+ 1)-dimensional(n >1)quasi-conformally recurrent (k, µ)-contact metric manifold is quasi-conformally semisymmetric.

Proof. Let us suppose that ˜C 6= 0.We now define a function by f² =g( ˜C,C).˜

By the fact that ∇g= 0, it follows from above 2f(U f) = 2f²(A(U)), or,

(6.2) U f =f(A(U)),

because f 6= 0.From (6.2), we get V(U f) = 1

f(V f)(U f) + (V A(U))f.

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Hence

V(U f)−U(V f) = [V A(U)−U A(V)]f.

Therefore,

(∇_V∇_U− ∇_U∇_V − ∇_[V,U])f = (6.3)

= [V A(U)−U A(V)−A([V, U])]f = 2[dA(V, U)]f.

Since the left hand side of (6.3) is zero, and f 6= 0, it follows from (6.3) that dA(V, U) = 0. This means that the 1-form A is closed. Now, from (6.1), we have

(∇_V∇_UC)(X, Y˜ )Z = [V A(U) +A(V)A(U)] ˜C(X, Y)Z.

Hence

(∇_V∇_UC)(X, Y˜ )Z−(∇_U∇_VC)(X, Y˜ )Z−(∇_[U,V_]C)(X, Y˜ )Z =

= 2dA(V, U) ˜C(X, Y)Z = 0.

Therefore, we have from above, R(V, U)·C˜ = 0, whereR(V, U) is considered as a derivation of tensor algebra at each point of the manifold for the tangent vectors V, U. This completes the proof.

In view of Theorem 5.1 and Theorem 6.1 we state

Theorem 6.2. A (2n+ 1)-dimensional(n >1)quasi-conformally recurrent (k, µ)-contact metric manifold is either locally the Riemannian product Eⁿ⁺¹×Sⁿ(4), or, locally quasi-conformally flat.

Theorem 6.2. leads us to state the following:

Corollary 6.2.A (2n+ 1)-dimensional (n >1) quasi-conformally recurrent N(k)-contact metric manifold is either locally the Riemannian product Eⁿ⁺¹×Sⁿ(4), or, locally quasi-conformally flat.

REFERENCES

[1] C.S. Bagewadi, D.G. Prakasha and Venkatesha,On pseudo projective curvature tensor of a contact metric manifold. SUT J. Math.43(2007), 115–126.

[2] D.E. Blair, Contact manifolds in Riemannian geometry. Lecture Notes in Math. 509.

Springer Verlag, New York, 1973.

[3] D.E. Blair, Two remarks on contact metric structures. Tohoku Math. J. 29 (1977), 319–324.

[4] D.E. Blair, T. Koufogiorgos and B.J. Papantoniou,Contact metric manifolds satisfying a nullity condition.Israel J. Math.19(1995), 189–214.

[5] E. Boeckx, A full classification of contact metric (k, µ)-spaces. Illinois J. Math. 44 (2000), 212–219.

[6] B.Y. Chen and K. Yano, Hypersurfaces of a conformally flat space. Tensor (N.S.)26 (1972), 318–322.

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15 On the quasi-conformal curvature tensor of a (k, µ)-contact metric manifold 129 [7] S.S. Chern,On the curvature and characteristic classes of a Riemannian manifold. Abh.

Math. Semin. Univ. Hambg.20(1956), 117–126.

[8] J.T. Cho,A conformally flat(k, µ)-space. Indian J. Pure Appl. Math.32(2001), 501–

508.

[9] U.C. De, Y.H. Kim and A.A. Shaikh, Contact metric manifolds with ξ belonging to (k, µ)-nullity distribution. Indian J. Math.47(2005), 1–10.

[10] U.C. De, J.B. Jun and A.K. Gazi, Sasakian manifolds with quasi-conformal curvature tensor. Bull. Korean Math. Soc.45(2008), 313–319.

[11] U.C. De and Y. Matsuyama,Quasi-conformally flat manifolds satisfying certain condition on the Ricci tensor. SUT J. Math.42(2006), 295–303.

[12] H. Endo,On K-contact Riemannian manifolds with vanishingE-contact Bochner curvature tensor. Colloq. Math.62(1991), 293–297.

[13] H. Endo,On the curvature tensor fields of a type of contact metric manifolds and of its certain submanifolds. Publ. Math. Debrecen48(1996), 53–269.

[14] A. Ghosh, T. Koufogiorgos and R. Sharma,Conformally flat contact metric manifolds.

J. Geom.70(2001), 66–76.

[15] A. Ghosh, R. Sharma and J.T. Cho, Contact metric manifolds withη-parallel torsion tensor. Ann. Global Anal. Geom.34(2008), 287–299.

[16] T. Koufogiorgos, Contact Riemannian manifolds with constant φ-sectional curvature.

Tokyo J. Math.20(1997), 13–22.

[17] B. O’Neill,Semi-Riemannian Geometry. Academic Press, New York, 1983.

[18] D. Perrone, Homogeneous contact Riemannian three-manifolds. Illinois J. Math. 42 (1998), 243–258.

[19] D. Perrone,Contact Riemannian manifolds satisfyingR(X, ξ).R= 0. Yokohama Math.

J.39(1992), 141–149.

[20] B.J. Papantoniou, Contact Riemannian manifolds satisfying R(X, ξ).R = 0 and ξ ∈ (k, µ)-nullity distribution. Yokohama Math. J.40(1993), 149–161.

[21] R. Sharma,Certain results onK-contact and(k, µ)-contact metric manifolds. J. Geom.

89(2008), 138–147.

[22] S. Tanno, Ricci curvatures of contact Riemannian manifolds. Tohoku Math. J. 40 (1988), 441–448.

[23] K. Yano and S. Sawaki, Riemannian manifolds admitting a conformal transformation group. J. Differential Geom.2(1968), 161–184.

Received 27 September 2010 University of Calcutta Department of Pure Mathematics

35,Ballygunje Circular Road Kolkata 700019, West Bengal, India

uc de@yahoo.com University of Burdwan Department of Mathematics

Golapbag

Burdwan 713104, West Bengal, India avjaj@yahoo.co.in

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OPTIMIZATION OF SOME RISK MEASURES IN STOP-LOSS REINSURANCE

WITH MULTIPLE RETENTION LEVELS

SILVIA DEDU

In this paper we propose a new stop-loss reinsurance model with multiple retention levels. We determine the analytical form of the stop-loss reinsurance premium with multiple retention levels and derive the Value-at-Risk and Conditional Tail Expectation measures corresponding to the aggregate loss. We build optimization problems based on minimizing some risk measures in stop-loss reinsurance with multiple retention levels and study the existence of the optimal solution.

We provide necessary and sufficient conditions for the existence of the optimal retention.

AMS 2010 Subject Classification: 62G32, 65C50, 90B50.

Key words: optimization, risk measure, stop-loss reinsurance, retention, Value-at- Risk, Conditional Tail Expectation.

1. INTRODUCTION

Recently the study of various optimization problems with applications in insurance has received a special interest among researchers: Cai and Tan [1], Dedu and Ciumara [2], Krokhmal, Palmquist and Uryasev [5], Preda and Ciumara [8]. Reinsurance is a risk management technique used by an insurance company to protect itself against the risk of losses, transferring the risk to a second insurance carrier. The former part is called the cedent or the insurer and the latter is the reinsurer. For a fixed reinsurance premium, stop-loss is the optimal solution among a wide arrays of reinsurance, in the sense that it gives the smallest variance of the insurer’s retained risk. The stop-loss agreement states that reinsurer will pay the cedent’s losses to the extent that losses exceed a specified amount, called retention. The problem of finding the optimal retention plays an important role in reinsurance.

The paper is organized as follows. In Section 2 we introduce the stop- loss reinsurance model with multiple retention levels and derive the stop-loss reinsurance premium corresponding to this model. In Section 3 we provide the representation for the survival function of the loss random variable in

MATH. REPORTS14(64),2 (2012), 131–139

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132 Silvia Dedu 2

a stop-loss reinsurance with multiple retention levels. We derive the Value- at-Risk (VaR) measure corresponding to the total cost of the insurer in a stop-loss reinsurance with multiple retention levels. We define a VaR-based optimization problem for this model and we find necessary and sufficient conditions for the existence of the optimal solution. In Section 4 we derive the Conditional Tail Expectation (CTE) measure corresponding to the total cost of the insurer in a stop-loss reinsurance with multiple retention levels. We define a CTE-based optimization problem for this model and find necessary and sufficient conditions for the existence of the optimal solution. Section 5 summarizes the conclusions.

2. THE STOP-LOSS REINSURANCE MODEL WITH MULTIPLE RETENTION LEVELS

We will model the aggregate loss of an insurance portfolio using a nonne- gative random variable X, with cumulative distribution functionFX, survival functionS_X and mean E(X).We assume that X has a one-to-one continuous distribution function on [0,∞).LetX_I andX_R be, respectively, the loss random variables corresponding to the insurer and to the reinsurer in the presence of a stop-loss reinsurance model. Under stop-loss agreement with multiple retention levels, the reinsurer pays the part of X that exceeds the retention limit, while the insurer is effectively protected from a potential large loss by limiting the liability to the retention levels. Let di >0,i= 1, n,n∈N,n≥2 be the retention levels, 0< di < di+1,i= 1, n−1. The random variablesX_I, X_R and X can be expressed as

X_I =







X, X≤d₁,

d_i, d_i ≤X < d_i+1, i= 1, n−1, d_n, X > d_n,

and

XR=







0, X≤d1,

X−di, di ≤X < di+1, i= 1, n−1, X−dn, X > dn.

Theorem 2.1. The stop-loss reinsurance premium with multiple retention levels has the representation

(2.1) π(d₁, d₂, . . . , d_n) = Z ∞

d1

S_X(x)dx−

n−1

X

i=1

(d_i+1−d_i)S_X(d_i+1).

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3 Optimization of some risk measures in stop-loss reinsurance 133

Proof. Using the mean value principle, we have π(d1, d2, . . . , dn) =E(XR) =

=

n−1

X

i=1

Z di+1

di

(x−d_i) dF_X(x) + Z ∞

dn

(x−d_n) dF_X(x) =

=−

n−1

X

i=1

Z di+1

di

(x−di) dSX(x)− Z ∞

dn

(x−dn) dSX(x) =

=−

n−1

X

i=1

(x−d_i)S_X(x)

di+1

di

−(x−d_n)S_X(x)

di+1

di

+

n−1

X

i=1

Z di+1

di

S_X(x)dx+

+ Z ∞

dn

SX(x)dx= Z ∞

d1

SX(x)dx−

n−1

X

i=1

(di+1−di)SX(di+1).

Corollary 2.1.The total stop-loss reinsurance premium with multiple retention, levels can be expressed as

(2.2) δ(d1, d2, . . . , dn) = (1 +ρ)

"

Z ∞ d1

SX(x)dx−

n−1

X

i=1

(di+1−di)SX(di+1)

# ,

where ρ >0 represents the safety loading coefficient.

3. OPTIMAL RETENTION USING VALUE-AT-RISK MEASURE

First, we recall the definiton of Value-at-Risk measure. LetX be a random variable with cumulative distribution function F_X and survival function S_X.Letα∈(0,1).

Definition 3.1. The α-Value-at-Risk (α-VaR) corresponding to the random variable X and the probability levelα is defined by

VaR_X(α) = inf{x∈R|S_X(x)≤α}.

Remark 3.1. If X has a one-to-one continuous distribution function on [0,∞),then VaRX(α) is the unique solution of the equation

VaR_X(α) =S_X⁻¹(α).

We model the total cost of the insurer in a stop-loss reinsurance with multiple retention levels using a random variable T. The total costT has two components: the loss random variableX_I corresponding to the insurer and the reinsurance premium δ(d1, d2, . . . , dn):

T =XI+δ(d1, d2, . . . , dn).

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134 Silvia Dedu 4

Let us denote by VaRT(d1, d2, . . . , dn, α) the Value-at-Risk of the total cost of the insurer, corresponding to the retention levels (d_i)_i=1,n and to the probability level α. Our goal is to determine the optimal retention levels d^∗₁, d^∗₂, . . . , d^∗_nthat minimize the measure VaRT(d1, d2, . . . , dn, α), which means solving the optimization problem

(3.1) VaR_T(d^∗₁, d^∗₂, . . . , d^∗_n, α) = min

0<d1<d2<...<dn

{VaR_T(d₁, d₂, . . . , d_n, α)}.

Next, we will derive the analytical form of the VaR_T(d₁, d₂, . . . , d_n, α) measure corresponding to the total cost of the insurer.

Proposition 3.1. The survival function corresponding to the loss random variable of the reinsurer in a stop-loss reinsurance with multiple retention levels can be expressed as

(3.2) SXI(x) =

SX(x) 0≤x < d1, 0 x≥d₁.

Theorem 3.1.The Value-at-Risk measure corresponding to the total cost of the insurer in a stop-loss reinsurance with multiple retention levels has the representation

VaR_T(d₁, . . . , d_n, α) = (3.3)

=d1+ (1 +ρ)

"

Z ∞ d1

S_X(x)dx−

n−1

X

i=1

(di+1−di)S_X(di+1)

#

for 0< d₁ ≤S_X⁻¹(α) and, respectively,

VaRT(d1, . . . , dn, α) = (3.4)

=S_X⁻¹(α) + (1 +ρ)

"

Z ∞ d1

S_X(x)dx−

n−1

X

i=1

(d_i+1−d_i)S_X(d_i+1)

#

for d1 > S_X⁻¹(α).

Proof. Since the total costT of the insurer isT =X_I+δ(d₁, d₂, . . . , d_n), we have

(3.5) VaR_T(d₁, d₂, . . . , d_n, α) = VaR_X_I(d₁, d₂, . . . , d_n, α) +δ(d₁, d₂, . . . , d_n).

For every x ≥ d1 we have SXI(x) = 0. Consequently, if SXI(d1) ≥ α > 0 or d₁ ≤ S_X⁻¹(α), then for every x < d₁ we have S_X_I(x) > S_X_I(d₁) ≥ α and it follows VaRXI(d1, d2, . . . , dn, α) =d1.

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IfSX(d1) < α ord1 > S_X⁻¹(α), then for every x < d1 we haveSXI(x) = SX(x) and, consequently, VaRX_I(d1, d2, . . . , dn, α) = S_X⁻¹(α). We have obtained

(3.6) VaR_X_I(d₁, d₂, . . . , d_n, α) =

( d₁ 0< d₁ ≤S_X⁻¹(α), S_X⁻¹(α) d1 > S_X⁻¹(α).

Using (3.5), (3.6) and (2.2) the conclusion follows.

Theorem 3.2. The solution of the optimization problem (4.1) exists if and only if the system

(3.7)











1−(1 +ρ)S_X(d₁)−f_X(d₁) = 0;

−S_X(di) +fX(di) +SX(di+1) = 0, i= 2, n−1;

−S_X(d_n) +f_X(d_n) = 0;

0< d1 < S_X⁻¹(α);

d_i < d_i+1, i= 1, n−1 has a solution de= de₁,de₂, . . . ,de_n

and desatisfies the inequalities

(1 +ρ)fX(de1)−f_X⁰ (de1)>0;

(3.8)

fX(dei)−f_X⁰ (dei)>0, ∀i= 2, n.

(3.9)

Proof. i) Let 0< d₁< S_X⁻¹(α).From the conditions ^∂VaR^T^(d_∂d¹^,...,dⁿ^,α)

i = 0,

∀i= 1, n we obtain the system







1−(1 +ρ)SX(d1)−fX(d1) = 0;

−S_X(di) +f_X(di) +S_X(di+1) = 0, i= 2, n−1;

−S_X(dn) +fX(dn) = 0.

Let we denote

H_VaR_T(d1, d2, . . . , dn, α) =

∂²VaRT(d1, . . . , dn, α)

∂d_i∂d_j

i,j=1,n

= (hij)_i,j=1,n. Then we have h₁₁= (1 +ρ)f_X(d₁)−f_X⁰ (d₁);h_ii=f_X(d₂)−f_X⁰ (d₂),∀i= 2, n;

h_i,i+1 = −f_X(d_i+1), ∀i = 2, n−1; h_ij = 0, ∀i > j, i = 2, n, j = 1, n−1;

h₁₂ = 0; h_ij = 0, ∀i < j −1, i = 1, n−2, j = 3, n. In case the system (3.7) has a solutionde= de1,de2, . . . ,den

and conditions (3.8) and (3.9) are fulfilled, de₁,de₂, . . . ,de_n

is a minimum point of the function VaR_T(d₁, . . . , d_n, α), consequently, the optimization problem (4.1) has a solution.

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136 Silvia Dedu 6

ii) Letd1 > S_X⁻¹(α).From the conditions ^∂VaR^T^(d_∂d¹^,...,dⁿ^,α)

i = 0, ∀i= 1, n, we get the system

(3.10)







−(1 +ρ)SX(d1)−fX(d1) = 0;

−S_X(di) +fX(di) +SX(di+1) = 0, i= 2, n−1;

−S_X(dn) +fX(dn) = 0.

Since the first equation in (3.10) has no solution, there is no solution for the optimization problem (4.1) in the case d1 > S_X⁻¹(α). From i) and ii) the conclusion follows.

4. OPTIMAL RETENTION USING

CONDITIONAL TAIL EXPECTATION MEASURE

We recall the definiton of Conditional Tail Expectation measure. LetX be a random variable with cumulative distribution function F_X and survival function S_X.Letα∈(0,1).

Definition 4.1. The α-Conditional Tail Expectation (α-CTE) measure corresponding to the random variable X and the probability level α is defined by

CTEα(X) = E [X|X ≥VaRα(X)].

Let us denote by CTET(d1, d2, . . . , dn, α) the Conditional Tail Expectation of the total cost of the insurer, corresponding to the retention levels (di)_i=1,n and to the probability level α. We want to determine the optimal retention levels d^∗₁, d^∗₂, . . . , d^∗_n that minimize CTE_T(d₁, d₂, . . . , d_n, α) measure, which means solving the optimization problem

(4.1) CTET(d^∗₁, d^∗₂, . . . , d^∗_n, α) = min

0<d1<d2<...<dn

{CTE_T(d1, d2, . . . , dn, α)}.

We will derive an analytical expression of CTET(d1, d2, . . . , dn, α) measure.

Theorem 4.1. For every d >0 and 0< α < S_X(0), we have

CTE_T(d₁, . . . , d_n, α) =d₁+ (1 +ρ) Z ∞

d1

S_X(x)dx−

n−1

X

i=1

(d_i+1−d_i)S_X(d_i+1)

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for 0< d1 ≤S_X⁻¹(α) and

CTE_T(d₁, . . . , d_n, α) =S_X⁻¹(α) + (1 +ρ) Z ∞

d1

S_X(x)dx−

−

n−1

X

i=1

(d_i+1−d_i)S_X(d_i+1)

+ 1 α

Z d1

S_X⁻¹(α)

S_X(x)dx for d1 > S_X⁻¹(α).

Theorem 4.2. The optimization problem (4.1)has solution if and only if one of the following conditions is fulfilled:

i)the system

(4.2)











1−(1 +ρ)S_X(d₁)−f_X(d₁) = 0;

−S_X(di) +fX(di) +SX(di+1) = 0, i= 2, n−1;

−S_X(dn) +fX(dn) = 0;

0< d₁ < S_X⁻¹(α);

di < di+1, i= 1, n−1 has a solution de= de₁,de₂, . . . ,de_n

and desatisfies the inequalities (4.3) (1 +ρ)f_X(de1)−f_X⁰ (de1)>0

and

(4.4) fX(dei)−f_X⁰ (dei)>0, ∀i= 2, n or

ii) the system

(4.5)









 ₁

α −(1 +ρ)

SX(d1)−fX(d1) = 0;

−S_X(di) +fX(di) +SX(di+1) = 0, i= 2, n−1;

−S_X(dn) +fX(dn) = 0;

0< d1 < S_X⁻¹(α);

di < di+1, i= 1, n−1 has a solution de= de1,de2, . . . ,den

and desatisfies the inequalities

(4.6)

(1 +ρ)− _α¹

fX(d1)−f_X⁰ (d1)>0 and

(4.7) fX(di)−f_X⁰ (di)>0, ∀i= 2, n.

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138 Silvia Dedu 8

Proof. Let

H_VaR_T(d₁, d₂, . . . , d_n, α) =

∂²VaR_T(d₁, . . . , d_n, α)

∂d_i∂d_j

i,j=1,n

= (h_ij)_i,j=1,n. i) For 0< d1 < S_X⁻¹(α) we have

h11= (1 +ρ)fX(d1)−f_X⁰ (d1);

h_ii=f_X(d₂)−f_X⁰ (d₂), ∀i= 2, n;

h_i,i+1 =−f_X(d_i+1), ∀i= 2, n−1;

hij = 0, ∀i > j, i= 2, n, j = 1, n−1;

h₁₂= 0;

h_ij = 0, ∀i < j−1, i= 1, n−2, j = 3, n.

If the system (4.2) has a solution de₁,de₂, . . . ,de_n

and conditions (4.3) and (4.4) are fulfilled, de1,de2, . . . ,den

is a minimum point for the function CTET

(d₁, . . . , d_n, α),consequently, the optimization problem (4.1) has solution.

ii) Ford₁> S_X⁻¹(α) we have

h11= (1 +ρ)fX(d1)−f_X⁰ (d1);

h_ii=f_X(d₂)−f_X⁰ (d₂), ∀i= 2, n;

h_i,i+1 =−f_X(d_i+1), ∀i= 2, n−1;

hij = 0, ∀i > j, i= 2, n, j = 1, n−1;

h₁₂= 0;

hij = 0, ∀i < j−1, i= 1, n−2, j = 3, n.

If the system (4.5) has a solution (de₁,de₂, . . . ,de_n) and conditions (4.6) and (4.7) are fulfilled, (de1,de2, . . . ,den) is a minimum point for the function CTE_T(d1, . . . , d_n, α), consequently, the optimization problem (4.1) has solution.

5. CONCLUSIONS

In this paper we propose a new model, based on stop-loss reinsurance with multiple retention levels and we derive the stop-loss reinsurance premium corresponding to this model. We provide the analytical form of Value-at-Risk and Conditional Tail Expectation measures corresponding to the aggregated loss and build optimization problems based on these risk measures. We estab- lish necessary and sufficient conditions for the existence of the optimal solution and we find out these conditions depend on the distribution of the loss random variable, the probability level and the safety loading coefficient.

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9 Optimization of some risk measures in stop-loss reinsurance 139 Acknowledgement.This research work was supported by the National University Research Council CNCSIS-UEFISCSU, Project number 844 PN II–IDEI, code ID 1778/2008.

REFERENCES

[1] J. Cai and K.S. Tan,Optimal retention for a Stop-Loss reinsurance under the VaR and CTE risk measures. Astin Bull.37(2007),1, 93–112.

[2] S. Dedu and R. Ciumara, Restricted optimal retention in Stop-Loss reinsurance. Proc.

Rom. Acad. Ser A Math. Phys. Tech. Sci. Ing. Sci.11(2010),3, 213–217.

[3] R. Kaas, M. Goovaerts, M. Denuit and J. Dhaene, Modern Actuarial Risk Theory.

Kluwer, Boston, 2001.

[4] S.A. Klugman, H.H. Panjer and G.E. Willmot. Loss Models: from Data to Decisions (2nd Edition). Wiley, New York, 2004.

[5] P. Krokhmal, J. Palmquist and S. Uryasev, Portfolio optimization with Conditional Value-at-Risk objective and constraints. J. Risk Uncertain4(2002),2, 11–27.

[6] N. Miller and A. Ruszczynski,Risk-adjusted probability measures in portfolio optimization with coherent measures of risk. European J. Oper. Res.191(2008),1, 193–206.

[7] G. Pflug, Some remarks on the Value-at-Risk and the Conditional Value-at-Risk. In:

S. Uryasev (Ed.), Probabilistic Constrained Optimization: Methodology and Applica- tions, Kluwer, 2000.

[8] V. Preda and R. Ciumara, On insurer portfolio optimization. An underwriting risk model. Romanian J. Econom. Forecasting9(2008),1, 102–112.

[9] V. Preda, S. Dedu and R. Ciumara. Restricted optimal retention in Stop-Loss reinsurance under VaR risk measure. Proc. 12th WSEAS Math. Methods, Comput. Tech- niques, Intelligent Systems Internat. Conference, 2010, pp. 143–146.

Received 5 July 2010 Academy of Economic Studies

Department of Mathematics 6 Piata Romana 010374 Bucharest, Romania

silvia.dedu@csie.ase.ro

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FIXED POINT THEOREMS FOR EXPANDING MAPPINGS IN CONE METRIC SPACES

XIANJIU HUANG, CHUANXI ZHU and XI WEN

In this paper, we define expanding mappings in the setting of cone metric spaces analogous to expanding mappings in metric spaces. We also obtain some results for two mappings to the setting of cone metric spaces.

AMS 2010 Subject Classification: 47H10, 54H25.

Key words: expanding mappings, fixed point theorem, cone metric spaces.

1. INTRODUCTION AND PRELIMINARIES

The existing literature of fixed point theory contains many results enun- ciating fixed point theorems for self-mappings in metric and Banach spaces.

Recently, Huang and Zhang [4] introduced the concept of cone metric spaces which generalized the concept of the metric spaces, replacing the set of real numbers by an ordered Banach space, and obtained some fixed point theorems for mapping satisfying different contractive conditions. The study of fixed point theorems in such spaces is followed by some other mathematicians, see [1–2, 5–6, 8–10, 12]. In 1984, Wang et. al. [11] introduced the concept of expanding mappings and proved some fixed point theorems in complete metric spaces. In 1992, Daffer and Kaneko [3] defined an expanding condition for a pair of mappings and proved some common fixed point theorems for two mappings in complete metric spaces.

In this paper, we define expanding mappings in the setting of cone metric spaces analogous to expanding mappings in complete metric spaces. We also extend a result of Daffer and Kaneko [3] for two mappings to the setting of cone metric spaces.

Consistent with Huang and Zhang [4], the following definitions and results will be needed in the sequel.

Let E be a real Banach space. A subset P of E is called a cone if and only if:

(a)P is closed, nonempty and P 6={θ};

(b)a, b∈R, a, b≥0, x, y∈P impliesax+by ∈P; (c)P ∩(−P) ={θ}.

MATH. REPORTS14(64),2 (2012), 141–148

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142 Xianjiu Huang, Chuanxi Zhu and Xi Wen 2

Given a coneP ⊂E, we define a partial ordering≤with respect toP by x≤yif and only if y−x∈P. A coneP is called normal if there is a number K >0 such that for all x, y∈E,

θ≤x≤y implieskxk ≤Kkyk.

The least positive number satisfying the above inequality is called the normal constant of P, while xy stands for y−x∈intP (interior of P).

Definition 1.1. Let X be a nonempty set. Suppose that the mapping d:X×X →E satisfies:

(d1)θ≤d(x, y) for all x, y∈X and d(x, y) =θ if and only if x=y;

(d2)d(x, y) =d(y, x) for allx, y∈X;

(d3)d(x, y)≤d(x, z) +d(z, y) for allx, y, z ∈X.

Then d is called a cone metric on X and (X, d) is called a cone metric space.

The concept of a cone metric space is more general than that of a metric space.

Example1.1 ([4]).LetE =R²,P ={(x, y)∈E |x, y≥0},X =R and d :X×X → E be such that d(x, y) = (|x−y|, α|x−y|), where α ≥0 is a constant. Then (X, d) is a cone metric space.

Definition1.2 ([4]).Let (X, d) be a cone metric space. We say that{x_n}is:

(e) a Cauchy sequence if for everyc∈E withθc, there is anN such that for all n, m > N,d(xn, xm)c;

(f) a convergent sequence if for every c∈ E with θ c, there is an N such that for all n > N,d(x_n, x)c for some fixedx∈X.

(g) (X, d) is a complete cone metric space if every Cauchy sequence is convergent.

The continuity of the self-maps in the cone metric spaces is, in fact, the sequential continuity. Iff :X→X, where (X, d) is a cone metric space, then f is continuous at the point a ∈ X if, for every sequence {x_n} ∈ X, which converges in the cone metric dtoa, the sequencef x_n converges tof a, i.e.,

d(x_n, a)c⇒d(f x_n, f a)c.

In the rest of this paper, we always suppose thatEis a real Banach space, P ⊆E is a cone with intP 6=∅ and ≤ is partial ordering with respect to P. We also note that the relations intP+intP ⊆intP andλintP ⊆intP(λ >0) always hold true.

Definition1.3. Let (X, d) be a cone metric space and T :X→X. Then T is called a expanding mapping, if for every x, y∈X there exists a number k >1 such thatd(T x, T y)≥kd(x, y).

Definition 1.4 ([7]). Two self mappings f and g of a cone metric space (X, d) are said to be commuting iff gx=gf xfor all x∈X.