Bulletin T.CXXXVII de l’Académie serbe des sciences et des arts − 2007 Classe des Sciences mathématiques et naturelles Sciences mathématiques, N\underline{o} 33

ON RICCI H-PSEUDOSYMMETRIC H-HYPERSURFACES OF SOME ANTI-KÄHLER MANIFOLDS

R. DESZCZ, MILEVA PRVANOVIĆ

(Presented at the 1st Meeting, held on February 29, 2008)

A b s t r a c t. We adopt the notion of the pseudosymmetry and Ricci pseudosymmetry to the case of the anti-Kähler manifolds and then we extend the results of the paper [1] to the h-hypersurfaces of the anti-Kähler manifolds of the constant totally real sectional curvatures.

AMS Mathematics Subject Classification (2000): 53C56

Key Words: anti-Kähler manifold, h-hypersurface, h-pseudosymmetry, Ricci h-pseudosymmetry

1. The object of the paper

Let (M, g) be a semi-Riemannian manifold of dimension ≥ 3. The manifold (M, g) is locally symmetric if ∇R = 0, on M, where ∇ is its Levi-Civita connection and R the curvature tensor. The proper generalization of locally symmetric manifolds form semi-symmetric manifolds. They are characterized by the condition

R · R = 0,

which holds on M, where R acts as a derivation. Some of the investigations of such manifolds gave rise to the next generalization, namely to the pseudosymmetric manifolds, i.e., manifolds satisfying on some set U ⊂ M the condition

R · R = L Q(g, R),  (1.1)

where L is a function on U and Q is a special operator (see section 2).

A manifold (M, g), dim M ≥ 3, is said to be Ricci pseudosymmetric, resp. Ricci semi-symmetric, if

R · ρ = L Q(g, ρ), resp. R · ρ = 0,  (1.2)

holds on the appropiate set U ⊂ M, and ρ is the Ricci tensor.

For a survey of results on different aspects of pseudosymmetric manifolds, we refer to [3]; see also [2], [10], [11], [14]. Among other problems there were studied the extrinsic characterizations of Ricci pseudosymmetric hypersurfaces of semi-Riemannian spaces of constant curvature in terms of the shape operator. Namely, in [1] (see Theorems 3.1 and 3.2) the following result is proved

Let M be a hypersurface of a semi-Reimannian space of constant curvature and dimension n ≥ 3. Then M is Ricci pseudosymmetric if and only if at every point p ∈ M, the second fundamental form h satisfies one of the following conditions

h^{2} = αh + βq,  α, β ∈ R,  (1.3)

or

h^{3} = tr h h^{2} + λh, λ ∈ R.

In practicular, for semi-Euclidean space, the previous result imply

A hypersurface M of semi-Euclidean space of dimension n ≥ 3 is Ricci pseudosymmetric if and only if for every point p ∈ M the tensor R · ρ vanishes at p, or (1.3) holds.

In section 4 of the present paper, we adopt the notion of pseudosymmetry and Ricci pseudosymmetry to the complex structure of the anti-Kähler manifolds and then we extend the above theorems for the h-hypersurface of anti-Kähler manifold of constant totally real sectional curvature. To do this, we use two formulas proved in section 3, valid for h-hypersurface of the anti-Kähler manifold of constant totally real sectional curvature. In section 2, we explain notations used in the paper.

2. Preliminaries

Let M̃ be a connected differentiable manifold endoved with pseudo-Riemannian metric G and a (1, 1) tensor field F such that, with respect to the local coordinates, holds

F_{B}^{A}F_{C}^{B} = −δ_{C}^{A}, F_{A}^{E}F_{B}^{D}G_{ED} = −G_{AB}, ˜∇_{D}F_{B}^{A} = 0.  (2.1)

Here ˜∇ is the Levi-Civita connection of (M̃, G) and A, B, C, D ∈ {1, 2, …, 2m}, 2m = dim M̃. The manifold (M̃, G, F) is said to be anti-Kähler manifold [12]. In some papers (M̃, G, F) is named B-manifold ([6],[7],[13]) and in some others - the Kähler manifold with the Norden metric ([8],[9]).

The manifold (M̃, G, F) is orientable and evendimensional. The metric G of such a manifold is indefinite and the signature is (m, m). Also, tr F = 0.

We denote by

R̃_{ABCD}  - the Riemannian curvature tensor,
ρ̃_{AB} = R̃_{ABC}^{C} - the Ricci tensor,
ρ̃ *_{AB} = F_{A}^{D}ρ̃_{DB} - the second Ricci tensor,
κ̃ = G^{AB}ρ̃_{AB}  - the scalar curvature,
κ̃ * = G^{AB} ρ̃ *_{AB} - the second scalar curvature.

Since ˜∇F = 0, the curvature tensor and the Ricci tensors satisfy

\begin{matrix}F_{A}^{L}F_{B}^{M} R̃_{LMCD} = −R̃_{ABCD} , \\ F_{A}^{L}F_{B}^{M} ρ̃_{LM} = −ρ̃_{AB} , \\ F_{A}^{L}F_{B}^{M} ρ̃ *_{LM} = − ρ̃ *_{AB} . \end{matrix} \}  (2.2)

The manifold (M̃, G, F) is of pointwise constant totally real sectional curvature if at p ∈ M, ([6], [7]):

R̃_{ABCD} =

\frac{κ̃(p)}{4m(m − 1)}(G_{AD}G_{BC} − G_{AC}G_{BD} − F_{A}^{L}G_{LD}F_{B}^{M}G_{MC} + F_{A}^{L}G_{LC}F_{B}^{M}G_{MD})

−\frac{κ̃ *(p)}{4m(m − 1)}(G_{AD}F_{B}^{L}G_{LC} + G_{BC}F_{A}^{L}G_{LD} − G_{AC}F_{B}^{L}G_{LD} − G_{BD}F_{A}^{L}G_{LC}).

(2.3)

If m ≥ 3, both functions κ̃ and κ̃ * are constants.

Now, we consider a differentiable submanifold M of M̃, dim M = 2n, n = m − 1. Suppose that M is expressed in each neighbourhood Ũ of M̃ by the equations

x^{A} = x^{A}(u^{a}) ,

where x^{A} are the local coordinates of M̃ in Ũ and u^{a} are the local coordinates in U = Ũ ⋂ M. Lowercase Latin indices a, b, c, …, i, j, k, … run over the range \{1, 2, …, 2n\}. M is said to be a \underline{h-hypersurface} (\underline{holomorphic hypersur}- \underline{face}) of M̃ if the restriction g of G on M has the maximal rank and the complex structure F leaves invariant the tangent space of M at each point p ∈ M. F induces on M the complex structure f such that (M, g, f) itself is an anti-Kähler manifold [4]. Similarly to (2.1) and (2.2), we have

f_{i}^{a}f_{a}^{j} = −δ_{i}^{j}, f_{i}^{a}f_{j}^{b}g_{ab} = −g_{ij}, ∇_{i}f_{j}^{k} = 0 ,

\begin{matrix}f_{i}^{a}f_{j}^{b}R_{ablm} = −R_{ijlm}, & ρ *_{ij} = f_{i}^{a}ρ_{aj} , \\ f_{i}^{a}f_{j}^{b}ρ_{ab} = −ρ_{ij} , & f_{i}^{a}f_{j}^{b} ρ *_{ab} = − ρ *_{ij}, \end{matrix} \}  (2.4)

where ∇ is the Levi-Civita connection with respect to the metric g, and R_{ijlm}, ρ_{ij} and ρ *_{ij} denote the local components of the Riemannian curvature tensor, Ricci tensor and the second Ricci tensor, respectively. We denote by κ and κ * the scalar curvature and the second scalar curvature of (M, g, f).

Because F leaves invariant the tangent space of M, it leaves invariant the normal space, too. There exist locally vector fields N_{1|} and N_{2|} normal to M, such that ([4]):

\begin{matrix}G_{AB}N_{1|}^{A}N_{1|}^{B} = −G_{AB}N_{2|}^{A}N_{2|}^{B} = 1, G_{AB}N_{1|}^{A}N_{2|}^{B} = 0, \\ F_{B}^{A}N_{1|}^{B} = −N_{2|}^{A}, F_{B}^{A}N_{2|}^{B} = N_{1|}^{A}. \end{matrix}

Denoting by h and k the second fundamental forms corresponding to N_{1|} and N_{2|} respectively, we have

h_{ij} = f_{i}^{a}k_{aj},  k_{ij} = −f_{i}^{a}h_{aj} .  (2.5)

Also, we shall use

h_{ij}^{2} = h_{i}^{a}h_{aj}, h_{ij}^{3} = h_{i}^{a}h_{aj}^{2}.

It is easy to see that the following conditions are satisfied

\begin{matrix}f_{i}^{a}f_{j}^{b}h_{ab} = −h_{ij},  f_{i}^{a}f_{j}^{b}k_{ab} = −k_{ij} , \\ f_{i}^{a}h_{aj} = f_{j}^{a}h_{ai},  f_{i}^{a}k_{aj} = f_{j}^{a}k_{ia}, \\ h_{ij}^{2} = h_{ji}^{2}, f_{i}^{a}f_{j}^{b}h_{ab}^{2} = −h_{ij}^{2}, f_{i}^{a}h_{aj}^{2} = f_{j}^{a}h_{ia}^{2}, \\ h_{ij}^{3} = h_{ji}^{3}, f_{i}^{a}f_{j}^{b}h_{ab}^{3} = −h_{ij}^{3}, f_{i}^{a}h_{aj}^{3} = f_{j}^{a}h_{ia}^{3}. \end{matrix}  \}  (2.6)

Let at p ∈ M, A and D be two symmetric (0, 2) tensors and B the curvature like tensor, satisfying

f_{i}^{a}f_{j}^{b}A_{ab} = −A_{ij},  f_{i}^{a}f_{j}^{b}D_{ab} = −D_{ij} ,  (2.7)
f_{i}^{a}f_{j}^{b}B_{ablm} = −B_{ijlm}  (2.8)

Let T be a (0, 4) tensor. We define the tensors B · A, B · T, Q(A, D), Q(A, B) by the formulas

(B · A)_{rsij} = A_{aj}B_{irs}^{a} + A_{ia}B_{jrs}^{a},  (2.9)
(B · T)_{rsijlm} = T_{ajlm}B_{irs}^{a} + T_{ialm}B_{jrs}^{a} + T_{ijam}B_{lrs}^{a} + T_{ijla}B_{mrs}^{a}, (2.10)
\begin{aligned}Q(A, D)_{rsij} = & A_{ri}D_{sj} + A_{rj}D_{si} − A_{si}D_{rj} − A_{sj}D_{ri} \\ & −f_{r}^{a}f_{s}^{b}(A_{ai}D_{bj} + A_{aj}D_{bi} − A_{bi}D_{aj} − A_{bj}D_{ai}), \end{aligned} (2.11)

\begin{matrix}Q(A, B)_{rsijlm} = & A_{ri}B_{sjlm} + A_{rj}B_{islm} + A_{rl}B_{ijsm} + A_{rm}B_{ijls} \\ −A_{si}B_{rjlm} − A_{sj}B_{irlm} − A_{sl}B_{ijrm} − A_{sm}B_{ijlr} \\ −f_{r}^{a}f_{s}^{b} (A_{ai}B_{bjlm} + A_{aj}B_{iblm} + A_{al}B_{ijbm} + A_{am}B_{ijlb} \\ −A_{bi}B_{ajlm} − A_{bj}B_{ialm} − A_{bl}B_{ijam} − A_{bm}B_{ijla}) . \end{matrix} (2.12)

Remark. The operator Q of a semi-Riemannian manifold (M, g) is defined in the following way (e.g. see [1],[2],[3]):

Q(A, D)_{rsij} = A_{ri}D_{sj} + A_{rj}D_{si} − A_{si}D_{rj} − A_{sj}D_{ri},

Q(A, B)_{rsijlm} = A_{ri}B_{sjlm} + A_{rj}B_{islm} + A_{rl}B_{ijsm} + A_{rm}B_{ijls}

−A_{si}B_{rjlm} − A_{sj}B_{irlm} − A_{sl}B_{ijrm} − A_{sm}B_{ijlr}.

Thus, (2.11) and (2.12) are the same operators, but adopted to the complex structure of the manifold.

We note that

\begin{matrix}Q(A, D) = −Q(D, A) and therefore Q(A, A) = 0, \\ Q(fA, fD) = −Q(A, D) and therefore Q(fA, D) = Q(A, fD), \\ Q(A, fA) = 0, Q(fD, B) = Q(D, fB). \end{matrix} \} (2.13)

For the latter use, we present

Lemma 2.1 ([4]) Let as a point p ∈ M, A and D be two symmetric (0, 2) tensors satisfying (2.7). If

Q(A, D) = 0 ,  (2.14)

then

D = δA + δ̄fA,  δ, δ̄ ∈ R.  (2.15)

P r o o f. Let X be a vector such that

X^{a}X^{b}A_{ab} = ω ≠ 0, X^{a}X̄^{b}A_{ab} = ω̄ ≠ 0,

where X̄^{i} = f_{a}^{i}X^{a}. We put

η = X^{a}X^{b}D_{ab},  η̄ = X^{a}X̄^{b}D_{ab}.

Transvecting (2.14) with X^{i}X^{r}, and symmetrizing the resulting equality, we get

ωD_{sj} − ηA_{sj} − ω̄f_{s}^{a}D_{aj} + η̄f_{s}^{a}A_{aj} = 0 ,

from which it follows that

ωf_{i}^{a}D_{aj} − ηf_{i}^{a}A_{aj} + ω̄D_{ij} − η̄A_{ij} = 0 .

These two relations imply

D_{ij} = \frac{ωη + ω̄η̄}{ω^{2} + ω̄^{2}}A_{ij} − \frac{ωη̄ − ω̄η}{ω^{2} + ω̄^{2}} f_{i}^{a}A_{aj} .

But this is just the relation (2.15). □

3. H-hypersurface of an anti-Kähler manifold of constant totally real sectional curvatures

The Gauss equation for an h-hypersurface (M, g, f) reads

R̃_{ABCD} \frac{∂x^{A}}{∂u^{i}} \frac{∂x^{B}}{∂u^{j}} \frac{∂x^{C}}{∂u^{l}} \frac{∂x^{D}}{∂u^{m}} = R_{ijlm} − (h_{im}h_{jl} − h_{il}h_{jm}) + (k_{im}k_{jl} − k_{il}k_{jm}).

Now, we suppose that the ambient manifold (M̃, G, F) is a manifold of constant totally real sectional curvatures. Then, substituting (2.3) into above Gauss equation, and taking into account that m = n + 1, we get

R_{ijlm} = K G_{ijlm} + K * f_{i}^{a}G_{ajlm} + E_{ijlm}  (3.1)

where

G_{ijlm} = g_{im}g_{jl} − g_{il}g_{jm} − f_{im}f_{jl} + f_{il}f_{jm} ,  (3.2)
E_{ijlm} = h_{im}h_{jl} − h_{il}h_{jm} − k_{im}k_{jl} + k_{il}k_{jm}  (3.3)
K = \frac{κ̃}{4n(n + 1)} ,  K * = −\frac{κ̃ *}{4n(n + 1)} ,  (3.4)

and f_{ij} = f_{i}^{a}g_{aj}.

The relation (3.1) yields

\begin{matrix}ρ_{im} = 2(n − 1)(Kg_{im} + K * f_{im}) + tr h h_{im} + tr k f_{i}^{a}h_{am} − 2h_{im}^{2} , \\ ρ *_{im} = 2(n − 1)(Kf_{im} − K * g_{im}) + tr h f_{i}^{a}h_{am} − tr k h_{im} − 2f_{i}^{a}h_{am}^{2} , \end{matrix} \} (3.5)

and therefore

\begin{matrix}κ = 4n(n − 1)K + (tr h)^{2} − (tr k)^{2} − 2tr (h^{2}), \\ κ * = −4n(n − 1) K * − 2tr h tr k − 2tr (fh^{2}). \end{matrix} \}  (3.6)

We note that, because of k_{ij} = −f_{i}^{a}h_{aj}, we have tr k = −tr (fh). In view of (3.1), we have

R · R = K G · R + K * (fG) · R + E · R .

Using (2.12), we can easy to see that

G · R = Q(g, R), (fG) · R = Q(fg, R) .

50 R. Deszcz, Mileva Prvanović

Therefore

R · R = K Q(g, R) + K * Q(fg, R) + E · R .  (3.7)

On the other hand

E · R = K (E · G) + K * (E · fG) + E · E .

But (E · G)_{rsijlm} =

= G_{ajlm}E_{irs}^{a} + G_{ialm}E_{jrs}^{a} + G_{ijam}E_{lrs}^{a} + G_{ijla}E_{mrs}^{a}

= g_{jl} (E_{mirs} + E_{imrs}) − g_{jm} (E_{lirs} + E_{ilrs})
+g_{im} (E_{ljrs} + E_{jlrs}) − g_{il} (E_{mjrs} + E_{jmrs})
−f_{im} (f_{l}^{a}E_{ajrs} + f_{j}^{a}E_{alrs}) + f_{il} (f_{m}^{a}E_{ajrs} + f_{j}^{a}E_{amrs})
−f_{jl} (f_{m}^{a}E_{airs} + f_{i}^{a}E_{amrs}) + f_{jm} (f_{l}^{a}E_{airs} + f_{i}^{a}E_{alrs}) = 0 ,

because of

E_{mirs} = −E_{mirs} and f_{m}^{a}E_{airs} = −f_{i}^{a}E_{amrs} .

Similary we have E · fG = 0, and therefore (3.7) reduces to

R · R = K Q(g, R) + K * Q(fg, R) + E · E .

Finally

(E · E)_{rsijlm} =

= − [h_{ri}^{2}E_{sjlm} + h_{rj}^{2}E_{islm} + h_{rl}^{2}E_{ijsm} + h_{rm}^{2}E_{ijls}

−h_{si}^{2}E_{rjlm} − h_{sj}^{2}E_{irlm} − h_{sl}^{2}E_{ijrm} − h_{sm}^{2}E_{ijlr}
−f_{r}^{a}f_{s}^{b} (h_{ai}^{2}E_{bjlm} + h_{aj}^{2}E_{iblm} + h_{al}^{2}E_{ijbm} + h_{am}^{2}E_{ijlb}
−h_{bi}^{2}E_{ajlm} − h_{bj}^{2}E_{ialm} − h_{bl}^{2}E_{ijam} − h_{bm}^{2}E_{ijla})]
= −Q(h^{2}, E)_{rsijlm}.

Thus, we can state Proposition 3.1. The relation

(R · R)_{rsijlm} = KQ(g, R)_{rsijlm} + K * Q(fg, R)_{rsijlm} − Q(h^{2}, E)_{rsijlm} (3.8)

holds good for any h-hypersurface of an anti-Kähler manifold of constant totally real sectional curvatures.

Transvecting (3.8) with g^{jl} we get

R · ρ = KQ(g, ρ) + K * (fg, ρ) + Q(h, tr h h^{2} + tr k fh^{2} − 2h^{3}). (3.9)

Thus, we have

Proposition 3.2. The relation (3.9) holds good for any h-hypersurface of an anti-Kähler manifold of constant totally real sectional curvatures.

4. H-pseudosymmetry

In the case of anti-Kähler manifolds, we adopt the conditions (1.1) and (1.2) to the complex structure of the manifold introducing the following

Definition. The anti-Kähler manifold (M, g, f) is said to be \underline{h-pseudo}- \underline{symmetric} if the condition

R · R = L_{1}Q(g, R) + L_{2}Q(fg, R)  (4.1)

is satisfied on some set U ⊂ M, where L_{1} and L_{2} are some scalar function on U.

The manifold (M, g, f) is said to be \underline{Ricci h-pseudosymmetric} if the condition

R · ρ = L_{1}Q(g, ρ) + L_{2}Q(fg, ρ)  (4.2)

is satisfied on U.

Now, we consider h-hypersurface (M, g, f) of the anti-Kähler manifold of constant totally real sectional curvatures. Then, according to the Proposition 3.2, the relation (3.9) holds good. Thus, if (M, g, f) is also Ricci h-pseudosymmetric, then we have

(L_{1} −K)Q(g, ρ)+(L_{2} − K * )Q(fg, ρ) = Q(h, tr h h^{2} +tr k fh^{2} −2h^{3}). (4.3)

We shall examine two cases.

\underline{Case (1).} If

L_{1} = K  and  L_{2} = K * ,  (4.4)

then (4.3) reduces to

Q(h, tr h h^{2} + tr k fh^{2} − 2h^{3}) = 0

and, in view of Lemma 2.1, we have

h^{3} = \frac{1}{2}tr h h^{2} + \frac{1}{2}tr k fh^{2} + δh + δ̄fh .  (4.5)

Conversely, if (4.5) holds, then

\begin{matrix}Q(h, tr h h^{2} + tr k fh^{2} − 2h^{3}) = \\ = −2δQ(h, h) − 2δ̄Q(h, fh) = 0 , \end{matrix}

and (3.9) reduces to

R · ρ = KQ(g, ρ) + K * Q(fg, ρ) ,

i.e., (4.4) holds.

Thus, we can state

Theorem 4.1. Let (M, g, f) be h-hypersurface of the anti-Kähler manifold (M, G, F) of constant totally real sectional curvatures. Then (4.5) is the necessary and the sufficient condition for (M, g, f) to be Ricci hpseudosymmetric on the appropriate set U ⊂ M such that (4.4) holds.

Remark. According to (3.4), (4.4) turns into

L_{1} = \frac{κ̃}{4n(n + 1)} , L_{2} = −\frac{κ̃ *}{4n(n + 1)} ,

where κ̃ and κ̃ * are the first and the second scalar curvatures of M̃ and dim M = 2n.

Corollary. Let (M, g, f) be h-hypersurface of the flat anti-Kähler manifold. Then (4.5) is the necessary and the sufficient condition for (M, g, f) to be Ricci semisymmetric.

\underline{Case (2)} If

λ_{1} = L_{1} − \frac{κ̃}{4n(n + 1)} ≠ 0 and λ_{2} = L_{2} + \frac{κ̃ *}{4n(n + 1)} ≠ 0 ,

On Ricci H-pseudosymmetric H-hypersurfaces of anti-Kähler manifolds 53

then (4.3) gives

λ_{1}Q(g, ρ) + λ_{2}Q(fg, ρ) = Q(h, tr h h^{2} + tr k fh^{2} − 2h^{3}) ,  (4.6)

from which it follows that

λ_{1}Q(fg, ρ) − λ_{2}Q(g, ρ) = −Q(h, tr k h^{2} − tr h fh^{2} + 2fh^{3}) .  (4.7)

In the local coordinates, the left hand side of (4.6) is the following

\begin{matrix}λ_{1}(g_{ri}ρ_{sj} + g_{rj}ρ_{si} − g_{si}ρ_{rj} − g_{sj}ρ_{ri} − f_{ri} ρ *_{sj} − f_{rj} ρ *_{si} + f_{si} ρ *_{rj} + f_{sj} ρ *_{ri}) \\ +λ_{2}(f_{ri}ρ_{sj} + f_{rj}ρ_{si} − f_{si}ρ_{rj} − f_{sj}ρ_{ri} + g_{ri} ρ *_{sj} + g_{rj} ρ *_{si} − g_{si} ρ *_{rj} − g_{sj} ρ *_{ri}), \end{matrix}

from which, by transvection with g^{ri} we get

λ_{1}(2nρ − κg + κ *fg) + λ_{2}(2n ρ * − κfg − κ *g).

In the similar way, we obtain from

Q(h, tr h h^{2} + tr k fh^{2} − 2h^{3})

the following expression − [tr h tr h^{2} + tr k tr(fh^{2}) − 2tr h^{3}] h + [tr h tr(fh^{2}) − tr k tr h^{2} − 2tr(fh^{3})] fh + [(tr h)^{2} − (tr k)^{2}] h^{2} + 2tr h tr k fh^{2} − 2tr h h^{3} − 2tr k fh^{3} .

Thus, as a consequence of (4.6), we have λ_{1}Q(2nρ − κg + κ *fg, h) + λ_{2}Q(2n ρ * − κfg − κ *g, h)

= − [(trh)^{2} − (trk)^{2}] Q(h, h^{2}) − 2trh trk Q(h, fh^{2})
+2trh Q(h, h^{3}) + 2trk Q(h, fh^{3}) .

(4.8)

But, the right hand side of (4.8) can be written in the form [−(trh)^{2} + (trk)^{2}] Q(h, h^{2}) − 2trh trk Q(h, fh^{2})

+2trh Q(h, h^{3}) + 2trk Q(h, fh^{3})

= −trh [trh Q(h, h^{2}) + trk Q(h, fh^{2}) − 2Q(h, h^{3})]

+trk [trk Q(h, h^{2}) − trh Q(h, fh^{2}) + 2Q(h, fh^{3})] = −trh Q(h, trh h^{2} + trk fh^{2} − 2h^{3}) + trk Q(h, trk h^{2} − trh fh^{2} + 2fh^{3}) = −trh [λ_{1}Q(g, ρ) + λ_{2}Q(fg, ρ)] + trk [−λ_{1}Q(fg, ρ) + λ_{2}Q(g, ρ)] ,

54 R. Deszcz, Mileva Prvanović

because of (4.6) and (4.7). Thus, (4.8) is of the form λ_{1}Q(2nρ − κg + κ *fg, h) + λ_{2}Q(2n ρ * − κfg − κ *g, h) = −λ_{1} [trh Q(g, ρ) + trk Q(fg, ρ)] + λ_{2} [trk Q(g, ρ) − trh Q(fg, ρ)] .

(4.9)

On the other hand, using (3.5), we have

\begin{matrix}Q(2nρ − κg + κ *fg, h) \\ = [4n(n − 1)K − κ] Q(g, h) + [4n(n − 1) K * + κ *] Q(fg, h) + 4nQ(h, h^{2}) . \end{matrix}

Similarly

\begin{matrix}Q(2n ρ * − κfg − κ *g, h) \\ = [4n(n − 1)K − κ] Q(fg, h) − [4n(n − 1) K * + κ *] Q(g, h) + 4nQ(h, fh^{2}) , \end{matrix}

such that (4.9) becomes

λ_{1}{[4n(n − 1)K − κ]Q(g, h) + [4n(n − 1) K * + κ *]Q(fg, h)

+4nQ(h, h^{2}) + trh Q(g, ρ) + trk Q(fg, ρ)}
+λ_{2}{[4n(n − 1)K − κ]Q(fg, h) − [4n(n − 1) K * + κ *]Q(g, h)
+4nQ(h, fh^{2}) − trk Q(g, ρ) + trh Q(fg, ρ)} = 0 .

(4.10)

If we set

\begin{aligned}P = & [4n(n − 1)K − κ]Q(g, h) + [4n(n − 1) K * + κ *]Q(fg, h) \\ & +4nQ(h, h^{2}) + trh Q(g, ρ) + trk Q(fg, ρ) , \end{aligned}

then

\begin{aligned}fP = & [4n(n − 1)K − κ]Q(fg, h) − [4n(n − 1) K * − κ *]Q(g, h) \\ & +4nQ(fh, h^{2}) + trh Q(fg, ρ) − trk Q(g, ρ) . \end{aligned}

But

Q(h, fh^{2}) = Q(fh, h^{2}) .

This means that (4.10) can be expressed in the form λ_{1}P + λ_{2}fP = 0 .

On Ricci H-pseudosymmetric H-hypersurfaces of anti-Kähler manifolds 55

This relation, together with

−λ_{2}P + λ_{1}fP = 0 ,

yields

(λ_{1}^{2} + λ_{2}^{2})P = 0 ,

and P = 0 if at last one of the conditions λ_{1} ≠ 0 and λ_{2} ≠ 0 is satisfied. Thus we have

\begin{matrix}[4n(n − 1)K − κ]Q(g, h) + [4n(n − 1) K * + κ *]Q(fg, h) \\ +4nQ(h, h^{2}) + trh Q(g, ρ) + trk Q(fg, ρ) = 0 , \end{matrix}

or

[4n(n − 1)K − κ + (trh)^{2} − (trk)^{2}]Q(g, h)
+[4n(n − 1) K * + κ * + 2trh trk]Q(fg, h)
−2trh Q(g, h^{2}) − 2trk Q(g, fh^{2}) + 4nQ(h, h^{2}) = 0 ,

(4.11)

because of

Q(g, ρ) = trh Q(g, h) + trk Q(g, fh) − 2Q(g, h^{2}) ,
Q(fg, ρ) = trh Q(fg, h) + trk Q(g, h) − 2Q(g, fh^{2}) .

Finally, according to (3.6), 4n(n − 1)K − κ + (trh)^{2} − (trk)^{2} = 2tr(h^{2}),

4n(n − 1) K * + κ * + 2trh trk = −2tr(fh^{2}) ,

and (4.11) becomes trh^{2}Q(g, h) − tr(fh^{2})Q(fg, h)

−trh Q(g, h^{2}) − trk Q(g, fh^{2}) + 2nQ(h, h^{2}) = 0 .

(4.12)

On the other hand,

Q(h − \frac{trh}{2n}g + \frac{tr(fh)}{2n}fg, h^{2} − \frac{trh^{2}}{2n} g + \frac{tr(fh^{2})}{2n}fg)

= Q(h, h^{2}) + Q(g, \frac{trh^{2}}{2n} h) − Q(g, \frac{tr(fh^{2})}{2n}fh)

−Q(g, \frac{trh}{2n}h^{2}) + Q(g, \frac{trfh}{2n}fh^{2}) .

56 R. Deszcz, Mileva Prvanović

This, in view of (4.12), means that

Q(h − \frac{trh}{2n}g + \frac{tr(fh)}{2n}fg, h^{2} − \frac{trh^{2}}{2n} g + \frac{tr(fh^{2})}{2n}fg) = 0

from which, applying Lemma 2.1, we get

h^{2} − \frac{trh^{2}}{2n} g + \frac{tr(fh^{2})}{2n}fg = δ(h − \frac{trh}{2n}g + \frac{tr(fh)}{2n}fg)

+δ̄(fh − \frac{trh}{2n}fg − \frac{tr(fh)}{2n}g) ,

or

h^{2} = δh + δ̄fh + μg + μ̄fg ,  (4.13)

where

μ = \frac{trh^{2}}{2n} − δ \frac{trh}{2n} − δ̄\frac{tr(fh)}{2n} ,
μ̄ = −\frac{tr(fh^{2})}{2n} + δ \frac{tr(fh)}{2n} − δ̄\frac{trh}{2n} .

Conversely, if (4.13) holds, then

\begin{matrix}Q(h^{2}, E) = & Q(δh + δ̄fh + μg + μ̄fg, E) \\ = & δQ(h, E) + δ̄Q(fh, E) + μQ(g, E) + μ̄Q(fg, E). \end{matrix}

But

Q(h, E) = Q(fh, E) = 0 ,

and therefore

Q(h^{2}, E) = μQ(g, E) + μ̄Q(fg, E) .  (4.14)

On the other hand, in view of Q(g, G) = Q(fg, G) = 0, we have

μQ(g, KG + K * fG) = 0 , μ̄Q(fg, KG + K * fG) = 0 , that is, the relation (4.14) is equivalent to

Q(h^{2}, E) = μQ(g, KG + K * fG + E) + μ̄Q(fg, KG + K * fG + E) . In the other words

Q(h^{2}, E) = μQ(g, R) + μ̄Q(fg, R) .  (4.15)

According to the Proposition 3.1, for any h-hypersurface of the anti-Kähler manifold of constant totally real sectional curvatures, the relation (3.8) holds, which, in view of (4.15) becomes

R · R = (K − μ)Q(G, R) + ( K * − μ̄)Q(fg, R) .

This means that if (4.15) holds, then (M, g, f) is h-pseudosymmetric. But h-pseudosymmetric manifold is Ricci h-pseudosymmetric, too. Thus, we can state

Theorem 4.2. Let (M, g, f), dim M = 2n, be a h-hypersurface of the anti-Kähler manifold (M̃, G, F) of constant totally real sectional curvatures.

Let κ̃ and κ̃ * be the first and the second scalar curvatures of (M̃, G, F). Then (4.13) is the necessary and the sufficient condition for (M, g, f) to be, on the appropriate set U ⊂ M, Ricci h-pseudosymmetric such that at least one of the relations

L_{1} ≠ \frac{κ̃}{4n(n + 1)} , L_{2} ≠ \frac{κ̃ *}{4n(n + 1)}

is satisfied.

5. Remark. H-pseudosymmetry is also considered in [5]. In that paper it is proved that every anti-Kähler manifold satisfying the Roter type equation

\begin{aligned}R(X, Y, Z, W) = & N_{1}Γ(X, Y, Z, W) + N_{2}Γ(fX, Y, Z, W) \\ & +N_{3}G(X, Y, Z, W) + N_{4}G(fX, Y, Z, W) , \end{aligned}

on some set U ⊂ M, is h-pseudosymmetric, where

\begin{aligned}Γ(X, Y, Z, W) = & ρ(X, W)ρ(Y, Z) − ρ(X, Z)ρ(Y, W) \\ & − ρ *(X, W) ρ *(Y, Z) + ρ *(X, Z) ρ *(Y, W) , \end{aligned}

and N_{1}, …, N_{4} are some scalar functions on U.

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Department of Mathematics Wroclaw University of Environmental and Life Sciences Grundwaldzka 53 50-357 Wroclaw, Poland

e-mail: rysz@ozi.ar.wroc.pl

Mathematical Institute SANU Kneza Mihaila 35 11001 Belgrade Serbia