Bulletin T.CXXXIV de l’Académie serbe des sciences et des arts − 2007 Classe des Sciences mathématiques et naturelles Sciences mathématiques, N\underline{o} 32

MINIMAL ANTI-KÄHLER HOLOMORPHIC HYPERSURFACES

MILEVA PRVANOVIĆ

(Presented at the 4th Meeting, held on May 25, 2007)

A b s t r a c t. M. Matsumoto examined in [10] the intrinsic properties of minimal hypersurfaces in a flat space and showed that for many of them the second fundamental form can be expressed in terms of the curvature and Ricci tensors.

The aim of this paper is to generalize the investigation of Matsumoto to holomorphic hypersurfaces of an anti-Kähler manifold of constant totally real sectional curvatures.

AMS Mathematics Subject Classification (2000): 53C42, 53C56 Key Words: Anti-Kähler manifold, holomorphic hypersurface, HC-flat

anti-Kähler manifold, holomorphically Einstein anti-Kähler manifold

\underline{The object of the paper.} – M. Matsumoto examined in [10] the intrinsic properties of minimal hypersurfaces in the flat space and showed that for many of them the second fundamental form can be expressed in terms of the curvature and Ricci tensors.

The aim of this paper is to generalize the investigation of Matsumoto to the holomorphic hypersurface of the anti-Kähler manifold of constant totally real sectional curvatures. This is done in the Section 2. The Section 3 is devoted to special properties of HC-flat minimal holomorphic hypersurface. In the Section 4 we examine holomorphically Einstein manifolds. Finally, in the Section 5 we give some remarks concerning complex hypersurfaces of the Kähler manifold.

1. Anti-Kähler manifold and its holomorphic hypersurface

By an anti-Kähler manifold we mean a triple (M̃, G, F), where M̃ is a connected differentiable manifold of dimension 2m, F = (F_{B}^{A}) is a (1, 1) tensor field and G = (G_{AB}) is a pseudo-Riemannian metric on M̃ satisfying

F^{2} = −Id., trF = 0, F_{A}^{C}F_{B}^{D}G_{CD} = −G_{AB}, ˜∇F = 0,  (1.1)

where ˜∇ is the Levi-Civita connection with respect to G and A, B, C, D, … ∈ {1, 2, …, 2m}.

The manifold (M̃, G, F) is orientable and evendimensional. The metric G is indefinite and the signature is (m, m).

The anti-Kähler manifolds are investigated by many authors (for exam. [1]-[9], [11]-[13], [15]).

Let T_{P} (M) be the tangent vector space of M̃ at the point P ∈ M̃. We denote by R̃(X, Y, Z, W), X, Y, Z, W ∈ T_{P} (M̃), the Riemannian curvature tensor of M̃. Because of ˜∇F = 0, it satisfies the condition

F_{A}^{P} F_{B}^{Q}R̃_{P QCD} = −R̃_{ABCD}.  (1.2)

The anti-Kähler manifold is of constant totally real sectional curvatures if ([5],[6])

\begin{aligned}R̃_{ABCD} & = k_{1}(G_{AD}G_{BC} − G_{AC}G_{BD} − F_{A}^{P} G_{P D}F_{B}^{Q}G_{QC} + F_{A}^{P} G_{P C}F_{B}^{Q}G_{QD}) \\ & +k_{2}(G_{AD}F_{B}^{P} G_{P C} + G_{BC}F_{A}^{P} G_{P D} − G_{AC}F_{B}^{P} G_{P D} − G_{BD}F_{A}^{P} G_{P C}). \end{aligned}

(1.3)

If m ≥ 3, both functions k_{1} and k_{2} are constants.

Now we consider a differentiable submanifold M of M̃, dim M = 2n, n = m − 1. Let the equation

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

be the local parametric expression of M in (M̃, G), where u^{a} are the local coordinates in M, a, b, c, …, i, j, k, … ∈ {1, 2, …, 2n}. A submanifold M is said to be holomorphic hypersurface of M̃ if the restriction of G on M has the maximal rank and F T_{p}(M) = T_{p}(M), p ∈ M. We denote the restriction of G and F on M by g and f. Then it can be proved [9] that (M, g, f) is itself an anti-Kähler manifold, i.e.

f^{2} = −Id., tr f = 0, f_{i}^{a}f_{j}^{b}g_{ab} = −g_{ij}, ∇f = 0  (1.4)
f_{i}^{a}f_{j}^{b}R_{abkl} = −R_{ijkl},  (1.5)

where ∇ is the Levi-Civita connection and R_{ijkl} are the local components of the Riemannian curvature tensor with respect to g. It follows from (1.5) that the Ricci tensor ρ_{ij} satisfies

f_{i}^{a}f_{j}^{b}ρ_{ab} = −ρ_{ij},  (1.6)

and therefore

f_{i}^{a}ρ_{aj} = f_{j}^{a}ρ_{ia}.

An anti-Kähler manifold (M, g, f) is holomorphically Einstein if its Ricci tensor has the form

ρ_{ij} = \frac{κ}{2n}g_{ij} − \frac{κ *}{2n}f_{ij}  (1.7)

(see, for ex. [15]), where κ and κ * are the first and the second scalar curvatures, and f_{ij} = f_{i}^{a}g_{aj} = f_{ji}.

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 [9]

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}.

Let h_{ij} be the components of the second fundamental form corresponding to N_{1|}. Then −f_{i}^{a}h_{aj} are those corresponding to N_{2|}, and

f_{i}^{a}f_{j}^{b}h_{ab} = −h_{ij}.  (1.8)

The relation (1.8) implies

f_{j}^{a}h_{ai} = f_{i}^{a}h_{aj} and f_{a}^{i}h_{j}^{a} = f_{j}^{a}h_{a}^{i}.

Now, using the induction, it is easy to see that

(h^{r})_{ij} = (h^{r})_{ji}, f_{i}^{a}f_{j}^{b}(h^{r})_{ab} = −(h^{r})_{ij},  (1.9)

where (h^{r}) is the fundamental form of order r and is defined as follows [14]

(h^{r})_{ij} = (h^{r−1})_{ia}h_{j}^{a}, (h^{1})_{ij} = h_{ij}, r = 2, 3, ….

Let v^{i} represent a principal direction of the holomorphic hypersurface M at P ∈ M with respect to the normal N_{1|}, i.e., an eigenvector of the matrix (h_{ij}) so that

h_{ij}v^{j} = λg_{ij}v^{j} ,  (1.10)

where λ is the corresponding eigenvalue. Then

(h^{r})_{ij}v^{j} = λ^{r}g_{ij}v^{j} ,

so that v^{i} is also an eigenvector of (h^{r}), but the corresponding eigenvalue is λ^{r}. We associate to (1.10) the equation

det (h_{ij} − λg_{ij}) = 0 ,

and denote its 2n roots by λ_{1}, λ_{2}, …, λ_{2n}.

On the other hand, if, at a fixed point P ∈ M, we choose the parameters u^{i} such that the tangents to the curves u^{i} = const. at P coincide with the principal directions of M at P, the components (h^{r})_{ij} are given by

(h^{r})_{ij} = | \begin{matrix}λ_{1}^{r} & 0 & … & 0 \\ 0 & λ_{2}^{r} & … & 0 \\ . & . & . \\ 0 & 0 & … & λ_{2n}^{r} \end{matrix} | .  (1.11)

We denote by H_{p} the p-th elementary symmetric function of λ_{1}, …, λ_{2n}, i.e.,

H_{1} = λ_{1} + λ_{2} + ⋯ + λ_{2n}
H_{2} = λ_{1}λ_{2} + λ_{1}λ_{3} + ⋯ + λ_{2n−1}λ_{2n}
⋯
H_{2n} = λ_{1}λ_{2} ⋯ λ_{2n},

and put

P_{p} = ∑_{a=1}^{2n}(λ_{a})^{p}.

According to the theory of the symmetric polynomials, by means of the Newton formula, we have

P_{p} + ∑_{q=1}^{p−1}(−1)^{q}H_{q}P_{p−q} + (−1)^{p}pH_{p} = 0,

i.e.,

P_{1} − H_{1} = 0
P_{3} − H_{1}P_{2} + H_{2}P_{1} − 3H_{3} = 0,
P_{5} − H_{1}P_{4} + H_{2}P_{3} − H_{3}P_{2} + H_{4}P_{1} − 5H_{5} = 0,

and so on. This means that If P_{1} = 0, then H_{1} = 0,

if P_{1} = P_{3} = 0, then H_{3} = 0,

..........

if P_{1} = P_{3} = ⋯ = P_{2p+1} = 0, then H_{2p+1} = 0.

But, in view of (1.11), P_{r} = tr (h^{r}). On the other hand H_{r} = 0 for r > 2n. Thus

if tr (h^{2p+1}) ≠ 0, then 2p + 1 < 2n.  (1.12)

In view of (1.10), we have

f_{i}^{a}h_{aj}v^{j} = λf_{ij}v^{j} = λf_{j}^{a}v^{j}g_{ai},

and taking into account that −f_{i}^{a}h_{aj} is the second fundamental form with respect to the normal N_{2|}, we see that if v^{i} represents a principal direction of M with respect to N_{1|} then f_{a}^{i}v^{a} represents the principal direction with respect to N_{2|}, and the corresponding eigenvalue is −λ. Putting

(fh^{2p+1})_{ij} = f_{i}^{a}(h^{2p+1})_{aj},

we conclude, in the similar way as above, that

if tr (fh^{2p+1}) ≠ 0, then 2p + 1 < 2n.  (1.13)

2. Minimal holomorphic hypersurface of an anti-Kähler manifold of constant totally real sectional curvatures

The Gauss equation for the holomorphic hypersurface (M, g, f) is

R̃_{ABCD} \frac{∂x^{A}}{∂u^{i}} \frac{∂x^{B}}{∂u^{j}} \frac{∂x^{C}}{∂u^{k}} \frac{∂x^{D}}{∂u^{l}} =

R_{ijkl} − (h_{il}h_{jk} − h_{ik}h_{jl}) + (f_{i}^{a}h_{al}f_{j}^{b}h_{bk} − f_{i}^{a}h_{ak}f_{j}^{b}h_{bl}).

Let us suppose that the ambient manifold (M̃, G, F) is a manifold of constant totally real sectional curvatures. Then, substituting (1.3) into above Gauss equation, we get

R_{ijkl} = k_{1}G_{ijkl} + k_{2}f_{i}^{a}G_{ajkl} + h_{il}h_{jk} − h_{ik}h_{jl} − f_{i}^{a}h_{al}f_{j}^{b}h_{bk} + f_{i}^{a}h_{ak}f_{j}^{b}h_{bl},

where

G_{ijkl} = g_{il}g_{jk} − g_{ik}g_{jl} − f_{il}f_{jk} + f_{ik}f_{jl}.

If we put

T_{ijkl} = R_{ijkl} − k_{1}G_{ijkl} − k_{2}f_{i}^{a}G_{ajkl},  (2.1)

the Gauss equation can be written in the form

T_{ijkl} = h_{il}h_{jk} − h_{ik}h_{jl} − f_{i}^{a}h_{al}f_{j}^{b}h_{bk} + f_{i}^{a}h_{ak}f_{j}^{b}h_{bl}.  (2.2)

Then, for

τ_{ij} = T_{iabj}g^{ab},

from (2.1), we have

τ_{il} = ρ_{il} − 2(n − 1)(k_{1}g_{il} + k_{2}f_{il}),  (2.3)

while from (2.2), it follows

τ_{il} = trhh_{il} − tr(fh)(fh)_{il} − 2(h^{2})_{il}.  (2.4)

We note that the tensor T has all algebraic properties as the curvature tensor of the anti-Kähler manifold. In particular

f_{i}^{a}f_{j}^{b}T_{abkl} = −T_{ijkl},

and therefore

f_{i}^{a}T_{ajkl} = f_{j}^{b}T_{ibkl}.  (2.5)

Now we suppose that M is the minimal holomorphic hypersurface, i.e., we suppose

tr h = tr (fh) = 0.  (2.6)

Then (2.4) reduces to

τ_{ij} = −2(h^{2})_{ij}

because of which we have

(τ^{p})_{ij} = (−2)^{p}(h^{2p})_{ij},  (2.7)

where

(τ^{p})_{ij} = (τ^{p−1})_{ia}τ_{j}^{a}, p = 1, 2, …, (τ^{0})_{ij} = g_{ij}.

Definition. Let (M, g, f) be a holomorphic hypersurface of an anti-Kähler manifold of constant totally real sectional curvatures, satisfying (2.6). (M, g, f) is said to be the minimal holomorphic hypersurface of type p if

tr(h^{2p+1}) ≠ 0, tr(fh^{2p+1}) ≠ 0,  (2.8)

while

tr(h^{2q+1}) = 0, tr(fh^{2q+1}) = 0,  (2.9)

for all q < p, p = 1, 2, ….

According to (1.12) and (1.13), (M, g, f) can be of type 1, 2, …, p such that 2p + 1 < 2n.

For (M, g, f) of type p, we can determine the second fundamental forms h_{ij} and −(fh)_{ij}. To do this, we use (2.2), (2.5) and (2.7), to get

T_{rsia}(τ^{p})_{j}^{a} + T_{rsja}(τ^{p})_{i}^{a} =

= (−2)^{p} [h_{ra}h_{si} − h_{ri}h_{sa} − f_{r}^{c}h_{ca}f_{s}^{d}h_{di} + f_{r}^{c}h_{ci}f_{s}^{d}h_{da}] (h^{2p})_{j}^{a}

+(−2)^{p} [h_{ra}h_{sj} − h_{rj}h_{sa} − f_{r}^{c}h_{ca}f_{s}^{d}h_{dj} + f_{r}^{c}h_{cj}f_{s}^{d}h_{da}] (h^{2p})_{i}^{a},

from which, contracting with g^{sj} and using (1.9) and (2.6), we find

tr (h^{2p+1})h_{ri} − tr (fh^{2p+1})(fh)_{ri} = \frac{1}{(−2)^{p}} [T_{rabi}(τ^{p})^{ab} − (τ^{p+1})_{ri}] . (2.10)

This relation, together with

tr (fh^{2p+1})h_{ri}+tr (h^{2p+1})(fh)_{ri} = \frac{1}{(−2)^{p}} f_{r}^{t} [T_{tabi}(τ^{p})^{ab} − (τ^{p+1})_{ti}] , (2.11)

implies

\{[tr (h^{2p+1})]^{2} + [tr (fh^{2p+1})]^{2}\} h_{ij} = \frac{1}{(−2)^{p}} tr (h^{2p+1}) [T_{iabj}(τ^{p})^{ab} − (τ^{p+1})_{ij}]

+\frac{1}{(−2)^{p}} tr (fh^{2p+1})f_{i}^{t} [T_{tabj}(τ^{p})^{ab} − (τ^{p+1})_{tj}] .

Therefore

\begin{matrix}h_{ij} = \frac{1}{θ_{p}} \{tr (h^{2p+1}) [T_{iabj}(τ^{p})^{ab} − (τ^{p+1})_{ij}] \\ +tr (fh^{2p+1})f_{i}^{t} [T_{tabj}(τ^{p})^{ab} − (τ^{p+1})_{tj}]\} , \end{matrix}  (2.12)

where

θ_{p} = (−2)^{p} [(tr (h^{2p+1}))^{2} + (tr (fh^{2p+1}))^{2}] .

For q < p, we have the equations similar to (2.11) and (2.12), but, in view of (2.9), they now yield

T_{iabj}(τ^{q})^{ab} − (τ^{q+1})_{ij} = 0 ,  (2.13)

for all q < p.

Conversely, if (2.13) holds, then the corresponding equations (2.10) and (2.11) imply

tr(h^{2q+1})h_{ij} − tr(fh^{2q+1})f_{i}^{a}h_{aj} = 0,
tr(fh^{2q+1})h_{ij} + tr(h^{2q+1})f_{i}^{a}h_{aj} = 0,

from which these follows

(tr (h^{2q+1}))^{2} + (tr (fh^{2q+1}))^{2} = 0,

and therefore

tr(h^{2q+1}) = tr(fh^{2q+1}) = 0 for all q < p.

Thus, for the minimal holomorphic hypersurface of type p, the conditions (2.9) and (2.13) are equivalent. This means that (M, g, f) is minimal of type p if and only if (2.13) holds for all q < p, and

T_{iabj}(τ^{p})^{ab} − (τ^{p+1})_{ij} ≠ 0.  (2.14)

If the ambient manifold (M̃, G, F) is flat, i.e. if k_{1} = k_{2} = 0, (2.1) and (2.3) reduce to T_{ijkl} = R_{ijkl} and τ_{ij} = ρ_{ij}, while (2.14) and (2.13) became

\begin{matrix}R_{iabj}(ρ^{p})^{ab} − (ρ^{p+1})_{ij} ≠ 0 \\ R_{iabj}(ρ^{q})^{ab} − (ρ^{q+1})_{ij} = 0 & for all q < p, \end{matrix}

respectively. But these relations are the intrinsic conditions of (M, g, f). Thus, if (M, g, f) is holomorphic hypersurface of a flat anti-Kähler manifold, the property of (M, g, f) to be minimal of type p is its intrinsic characteristic. Also, tr(h^{2p+1}) and tr(fh^{2p+1}) are the object of the inner geometry of (M, g, f). To prove this, we note that now, (2.7) becomes

(ρ^{p})_{ij} = (−2)^{p}(h^{2p})_{ij},  (2.15)

because of which

\begin{matrix}(−2)^{p}(h^{2p+1})_{ij} = (−2)^{p}(h^{2p})_{ia}h_{j}^{a} = (ρ^{p})_{ia}h_{j}^{a}, \\ (−2)^{p}(fh^{2p+1})_{ij} = (ρ^{p})_{ia}(fh)_{j}^{a}. \end{matrix}

Therefore

\begin{matrix}(−2)^{p}tr(h^{2p+1}) = h_{ij}(ρ^{p})^{ij} \\ (−2)^{p}tr(fh^{2p+1}) = (fh)_{ij}(ρ^{p})^{ij} \end{matrix}  (2.16)

On the other hand, (2.10) and (2.11) reduce to

tr(h^{2p+1})h_{ij} − tr(fh^{2p+1})(fh)_{ij} = \frac{1}{(−2)^{p}} [R_{iabj}(ρ^{p})^{ab} − (ρ^{p+1})_{ij}] ,

tr(fh^{2p+1})h_{ij} + tr(h^{2p+1})(fh)_{ij} = \frac{1}{(−2)^{p}} f_{i}^{t} [R_{tabj}(ρ^{p})^{ab} − (ρ^{p+1})_{tj}] ,

from which, transverting with (ρ^{p})^{ij} and using (2.6) we get

[tr(h^{2p+1})]^{2} − [tr(fh^{2p+1})]^{2} = γ,
tr(h^{2p+1}) tr(fh^{2p+1}) = δ,

where γ and δ are some functions of the inner geometry of (M, g, f). This system of equations shows that tr(h^{2p+1}) and tr(fh^{2p+1}), and therefore θ_{p} are the intrinsic properties of (M, g, f).

As for (2.12), it reduces to

\begin{aligned}h_{ij} = & \frac{1}{θ_{p}} \{tr(h^{2p+1}) [R_{iabj}(ρ^{p})^{ab} − (ρ^{p+1})_{ij}] \\ & + tr(fh^{2p+1})f_{i}^{t} [R_{tabj}(ρ^{p})^{ab} − (ρ^{p+1})_{tj}]\} . \end{aligned}  (2.17)

From the above exposed, the following theorem holds.

Theorem 1. Let (M, g, f) be a holomorphic hypersurface of an anti-Kähler manifold of constant totally real sectional curvatures. If it is the minimal hypersurface of type p, then:

1) Type p can be p = 1, 2, …, such that 2p + 1 < 2n, 2n = dim M.

2) The conditions (2.8) and (2.14) are equivalent, as well as the conditions (2.9) and (2.13).

3) The second fundamental form is given by (2.12).

In particular, if (M, g, f) is a holomorphic hypersurface of the flat anti-Kähler manifold, then

4) The property of (M, g, f) to be minimal of type p is its intrinsic characteristic.

5) The second fundamental form is given by (2.17); it is the intrinsic characteristic of (M, g, f), too.

6) Any fundamental form of even order satisfies (2.15).

3. Minimal HC-flat holomorphic hypersurface

We consider in [4] HC-flat (holomorphically conformally flat) hypersurface, (M, g, f), n > 3, dim M = 2n, of an anti-Kähler manifold of constant totally real sectional curvatures and proved that for such (M, g, f) the following hold:

1. (M, g, f) is quasi-umbilical, i.e.

h_{ij} = φg_{ij} + ψf_{ij} + τV_{ij} + σV̄_{ij},  (3.1)

where φ, ψ, τ and σ are some scalar functions,

V_{ij} = V_{i}V_{j} − V̄_{i}V̄_{j}, V̄_{ij} = f_{i}^{a}V_{aj} = V̄_{i}V_{j} + V_{i}V̄_{j},

V is a vector field and V = fV;

Minimal anti-Kähler holomorphic hypersurfaces 95

2. (M, g, f) is of quasi-constant totally real sectional curvatures, i.e., its

curvature tensor has the form R_{ijlm} = λG_{ijlm} + μf_{i}^{a}G_{ajlm} +ξ(g_{im}V_{jl} + g_{jl}V_{im} − g_{il}V_{jm} − g_{jm}V_{il} − f_{im}V̄_{jl} − f_{jl}V̄_{im} + f_{il}V̄_{jm} + f_{jm}V̄_{il}) +θ(g_{im}V̄_{jl} + g_{jl}V̄_{im} − g_{il}V̄_{jm} − g_{jm}V̄_{il} + f_{im}V_{jl} + f_{jl}V_{im} − f_{il}V_{jm} − f_{jm}V_{il}),

(3.2)

where

\begin{matrix}λ = \frac{κ̃}{4n(n + 1)} + φ^{2} − ψ^{2},  μ = −\frac{κ̃ *}{4n(n + 1)} + 2φψ, \\ ξ = τφ − σψ,  θ = σφ + τψ \end{matrix}  (3.3)

while κ̃ and κ̃ * are the scalar curvatures of the ambient manifold. We get from (3.1)

\begin{matrix}trh = 2nφ + 2τV_{a}V^{a} + 2σV_{a}V̄^{a} , \\ tr(fh) = −2nψ + 2τV_{a}V̄^{a} − 2σV_{a}V^{a} . \end{matrix}

Thus, if (M, g, f) is minimal, we have

\begin{matrix}nφ + τV_{a}V^{a} + σV_{a}V̄^{a} = 0 , \\ nψ − τV_{a}V̄^{a} + σV_{a}V^{a} = 0 . \end{matrix}  (3.4)

The relation (3.1) implies

\begin{aligned}(h^{2})_{ij} = & (φ^{2} − ψ^{2})g_{ij} + 2φψf_{ij} \\ & +V_{ij} [2(τφ − σψ) + (τ^{2} − σ^{2})V_{a}V^{a} + 2τσV_{a}V̄^{a}] \\ & +V̄_{ij} [2(σφ + τψ) − (τ^{2} − σ^{2})V_{a}V̄^{a} + 2τσV_{a}V^{a}] . \end{aligned}  (3.5)

On the other hand, (3.4) yields

\begin{matrix}n(τφ − σψ) + (τ^{2} − σ^{2})V_{a}V^{a} + 2τσV_{a}V̄^{a} = 0, \\ n(σφ + τψ) − (τ^{2} − σ^{2})V_{a}V̄^{a} + 2τσV_{a}V^{a} = 0. \end{matrix}  (3.6)

Substituting this into (3.5), we get

\begin{aligned}(h^{2})_{ij} = & (φ^{2} − ψ^{2})g_{ij} + 2φψf_{ij} \\ & −(n − 2)(τφ − σψ)V_{ij} − (n − 2)(σφ + τψ)V̄_{ij}. \end{aligned}  (3.7)

Next, we calculate (h^{3})_{ij} and, using (3.6), we find

\begin{aligned}(h^{3})_{ij} = & φ(φ^{2} − 3ψ^{2})g_{ij} − ψ(ψ^{2} − 3φ^{2})f_{ij} \\ & +(n^{2} − 3n + 3)[τ(φ^{2} − ψ^{2}) − 2σφψ]V_{ij} \\ & +(n^{2} − 3n + 3)[σ(φ^{2} − ψ^{2}) + 2τφψ]V̄_{ij}. \end{aligned}  (3.8)

This relation implies

(fh^{3})_{ij} = ψ(ψ^{2} − 3φ^{2})g_{ij} + φ(φ^{2} − 3ψ^{2})f_{ij}
−(n^{2} − 3n + 3) [σ(φ^{2} − ψ^{2}) + 2τφψ] V_{ij}
+(n^{2} − 3n + 3) [τ(φ^{2} − ψ^{2}) − 2σφψ] V̄_{ij}.

Now, we have

\begin{matrix}tr(h^{3}) = −2n(n − 1)(n − 2)φ(φ^{2} − 3ψ^{2}), \\ tr(fh^{3}) = −2n(n − 1)(n − 2)ψ(ψ^{2} − 3φ^{2}). \end{matrix}  (3.9)

If tr(h^{3}) ≠ 0 and tr(fh^{3}) ≠ 0, (M, g, f)is the minimal hypersurface of type 1.

Let us examine the case

tr(h^{3}) = tr(fh^{3}) = 0.

Then, if n > 2, (3.9) gives

φ(φ^{2} − 3ψ^{2}) = 0, ψ(ψ^{2} − 3φ^{2}) = 0,

which holds if and only if φ = ψ = 0. But then (3.3) reduces to

λ = \frac{κ̃}{4n(n + 1)}, μ = −\frac{κ̃ *}{4n(n + 1)}, ξ = 0, θ = 0,

while (3.2) becomes

R_{ijlm} = \frac{κ̃}{4n(n + 1)}G_{ijlm} − \frac{κ̃ *}{4n(n + 1)}f_{i}^{a}G_{ajlm}.

This mean that (M, g, f) is of constant totally real sectional curvatures too, and its scalar curvatures are related to the scalar curvatures of the ambient manifold in the following way:

κ = \frac{n − 1}{n + 1}κ̃, κ * = \frac{n − 1}{n + 1} κ̃ *.  (3.10)

Also, in view of (3.7), we have

(h^{2})_{ij} = (fh^{2})_{ij} = 0 .

Therefore

(h^{r})_{ij} = (fh^{r})_{ij} = 0 for all r ≥ 2

and

tr(h^{r}) = tr(fh^{r}) = 0 for all r.

But then, according to the discussion in the section 2, (2.13) holds for all q. Finally, (3.4) reduce to

\begin{matrix}τV_{a}V^{a} + σV_{a}V̄^{a} = 0 , \\ τV_{a}V̄^{a} − σV_{a}V^{a} = 0 . \end{matrix}

These equations imply V_{a}V^{a} = V_{a}V̄^{a} = 0 or τ = σ = 0. The first case means that V is a null vector and is orthogonal to fV. In the second case (3.1) reduces to h_{ij} = 0, that is (M, g, f) is totally geodesic.

Thus, we can state

Theorem 2. Let (M, g, f) be the holomorphically conformally flat holomorphic hypersurface of an anti-Kähler manifold of constant totally real sectional curvatures, and n > 3, 2n = dim M.

If (M, g, f) is minimal, then

it is minimal of type 1, or

it is itself a manifold of constant totally real sectional curvatures. In this case its scalar curvatures are related to the scalar curvatures of the ambient manifold according to (3.10), and (3.13) holds for all q = 1, 2 …. The vector V is a null vector and is orthogonal to fV, or (M, g, f) is totally geodesic.

4. Holomorphically Einstein hypersurface

First we shall prove that for any anti-Kähler holomorphically Einstein hypersurface, (2.13) is satisfied for all q = 1, 2, …. Namely, substituting (1.7) into (2.3), we find

τ_{ij} = αg_{ij} − βf_{ij},  (4.1)

where

α = \frac{κ}{2n} − 2(n − 1)k_{1}, β = \frac{κ *}{2n} + 2(n − 1)k_{2} .  (4.2)

Then

(τ^{p})_{ij} = α_{1}g_{ij} − β_{1}f_{ij}  (4.3)

where α_{1} and β_{1} are some scalar functions. Therefore

(τ^{p+1})_{ij} = (αα_{1} − ββ_{1})g_{ij} − (αβ_{1} + α_{1}β)f_{ij}.

Using (1.5) and (4.3), we obtain

\begin{aligned}R_{iabj}(τ^{p})^{ab} & = α_{1}ρ_{ij} − β_{1}f_{i}^{a}ρ_{aj} \\ & = [α_{1} \frac{κ}{2n} − β_{1} \frac{κ *}{2n}] g_{ij} − [α_{1} \frac{κ *}{2n} + β_{1} \frac{κ}{2n}] f_{ij}, \end{aligned}

and therefore

T_{iabj}(τ^{p})^{ab} − (τ^{p+1})_{ij} =

(R_{iabj} − k_{1}G_{iabj} − k_{2}f_{i}^{t}G_{tabj})(τ^{p})^{ab} − (τ^{p+1})_{ij} =

= \{α_{1} [\frac{κ}{2n} − 2(n − 1)k_{1} − α] − β_{1} [\frac{κ *}{2n} + 2(n − 1)k_{2} − β]\} g_{ij}

− \{α_{1} [\frac{κ *}{2n} + 2(n − 1)k_{2} − β] + β_{1} [\frac{κ}{2n} − 2(n − 1)k_{1} − α]\} f_{ij} = 0

because of (4.2).

We note that for any holomorphically Einstein anti-Kähler manifold, being it holomorphic hypersurface or not, we have

R_{iabj}(ρ^{p})^{ab} − (ρ^{p+1})_{ij} = 0 .

So, if (M, g, f) is minimal holomorphic hypersurface and, at the same time, holomorphically Einstein, besides (2.6) the conditions (2.13) are satisfied for all q = 1, 2, …, and we can not determine the second fundamental form using the method described in the Section 2. But, holomorphically Einstein hypersurface may not be minimal and in that case the second fundamental form can be determined.

From now on in this section we suppose that (M, g, f) is a holomorphically Einstein holomorphic non-minimal hypersurface of an anti-Kähler manifold of constant totally real sectional curvatures. Thus (2.2) and (2.4) hold, as well as (4.1) and (4.2).

Using (2.2) we can calculate

(T_{abhj}T_{stik} − 2T_{haib}T_{jskt})g^{at}g^{bs} =−4(h^{2})_{ih}(h^{2})_{jk} + 4h_{ih}(h^{3})_{jk} + 4(h^{3})_{ih}h_{jk}

+4(fh^{2})_{ih}(fh^{2})_{jk} − 4(fh)_{ih}(fh^{3})_{jk} − 4(fh^{3})_{ih}(fh)_{jk}

−2tr(h^{2}) [h_{ih}h_{jk} − f_{i}^{a}h_{ah}f_{j}^{b}h_{bk}] + 2tr(fh^{2}) [h_{ih}f_{j}^{a}h_{ak} + f_{i}^{a}h_{ah}h_{jk}].

(4.4)

In view of (4.1), (2.4) becomes

(h^{2})_{ij} = \frac{1}{2} [−αg_{ij} + βf_{ij} + trhh_{ij} − tr(fh)f_{i}^{a}h_{aj}] ,  (4.5)

from which we obtain

(fh^{2})_{ij} = \frac{1}{2} [−βg_{ij} − αf_{ij} + tr(fh)h_{ij} + trhf_{i}^{a}h_{aj}] ,  (4.6)

(h^{3})_{ij} = \frac{1}{4} [−αtrh + β tr(fh)] g_{ij} + \frac{1}{4} [αtr(fh) + β trh] f_{ij}

+\frac{1}{4} [−2α + (trh)^{2} − (tr(fh))^{2}] h_{ij} + \frac{1}{2} [β − trhtr(fh)] f_{i}^{a}h_{aj},

(fh^{3})_{ij} = −\frac{1}{4} [αtr(fh) + βtrh] g_{ij} + \frac{1}{4} [−αtrh + βtr(fh)] f_{ij}

−\frac{1}{2} [β − (trh)(tr(fh))] h_{ij} + \frac{1}{4} [−2α + (trh)^{2} − (tr(fh))^{2}] f_{i}^{a}h_{aj}.

Substituting this into (4.4), we get

(T_{abhj}T_{stik} − 2T_{haib}T_{jskt})g^{at}g^{bs}+

+(α^{2} − β^{2})(g_{ih}g_{jk} − f_{ih}f_{jk}) − 2αβ(g_{ih}f_{jk} + g_{jk}f_{ih}) =

= [−4α + (trh)^{2} − (trfh)^{2} − 2tr(h^{2})] [h_{ih}h_{jk} − f_{i}^{a}h_{ah}f_{j}^{b}h_{bk}]

+ [4β − 2trhtr(fh) + 2tr(fh^{2})] [f_{i}^{a}h_{ah}h_{jk} + h_{ih}f_{j}^{b}h_{bk}] (4.7)

100 Mileva Prvanović

According to (4.5) and (4.6)

(trh)^{2} − (trfh)^{2} − 2tr(h^{2}) = 2nα

−2trhtr(fh) + 2tr(fh^{2}) = −2nβ,

because of which (4.7) reduces to 2(n − 2)α [h_{ih}h_{jk} − f_{i}^{a}h_{ah}f_{j}^{b}h_{bk}] − 2(n − 2)β [f_{i}^{a}h_{ah}h_{jk} + h_{ih}f_{j}^{b}h_{bk}] = (T_{abhj}T_{stik} − 2T_{haib}T_{jskt})g^{at}g^{bs} + (α^{2} − β^{2})(g_{ih}g_{jk} − f_{ih}f_{jk})

−2αβ(g_{ih}f_{jk} + g_{jk}f_{ih}). Transecting this relation with g^{ih}, we get

2(n − 2) [αtrh − β tr(fh)] h_{jk} − 2(n − 2) [αtr(fh) + β trh] f_{j}^{a}h_{ak} = = −T _{j}^{abc}T_{abck} − 2T_{jabk}τ^{ab} + 2n [(α^{2} − β^{2})g_{jk} − 2αβf_{jk}] .

(4.8)

Putting

2(n − 2) [αtrh − β tr(fh)] = p ,
2(n − 2) [αtr(fh) + β trh] = q ,

and using the condition (2.13), we rewrite (4.8) in the form

\begin{aligned}ph_{ij} − qf_{i}^{a}h_{aj} = & − [T _{i}^{abc}T_{abcj} + 2(τ^{2})_{ij}] \\ & +2n [(α^{2} − β^{2})g_{ij} − 2αβf_{ij}] . \end{aligned}

This relation, together with

\begin{aligned}qh_{ij} + pf_{i}^{a}h_{aj} = & −f_{i}^{t} [T _{t}^{abc}T_{abcj} + 2(τ^{2})_{tj}] \\ & +2n [2αβg_{ij} + (α^{2} − β^{2})f_{ij}] \end{aligned}

implies

\begin{aligned}(p^{2} + q^{2})h_{ij} = & p \{− [T _{i}^{abc}T_{abcj} + 2(τ^{2})_{ij}] + 2n [(α^{2} − β^{2})g_{ij} − 2αβf_{ij}]\} \\ & +q \{−f_{i}^{t} [T _{t}^{abc}T_{abcj} + 2(τ^{2})_{tj}] + 2n [2αβg_{ij} + (α^{2} − β^{2})f_{ij}]\} . \end{aligned}

(4.9)

But if trh = tr(fh) = 0, (4.8) yields

T _{i}^{abc}T_{abcj} + 2(τ^{2})_{ij} = 2n [(α^{2} − β^{2})g_{ij} − 2αβf_{ij}] .  (4.10)

Thus, we can state

Theorem 3. Let (M, g, f) be the holomorphically Einstein hypersurface of an anti-Kähler manifold of constant totally real sectional curvatures. Then, if it is not minimal, the second fundamental form is given by (4.9). But if trh = tr(fh) = 0, then (4.10) holds.

5. Remarks on complex hypersurfaces of Kähler manifold

A differentiable manifold M̃, dim M̃ = 2m is a Kähler manifold if it is endowed with metric G and complex structure J̃ such that

J̃^{2} = −Id., J̃_{A}^{C} J̃_{B}^{D}G_{CD} = G_{AB}, ˜∇J̃ = 0 .

A differentiable submanifold M of M̃, dim M = 2n, m = n + 1, is said to be complex hypersurface of M̃, if the complex structure J̃ of M̃ leaves invariant the tangent space of M at each point P ∈ M. In this case, G and J̃ induce on M the metric g and the complex structure J such that (M, g, J) is itself a Kählar manifold [16], i.e.

J^{2} = −Id., J_{i}^{a}J_{j}^{b}g_{ab} = g_{ij}, ∇J = 0 .

If we put

F_{ij} = J_{i}^{a}g_{aj}, then F_{ij} = −F_{ji} .

Also, the complex structure J̃ leaves invariant the normal plane to M at each point P ∈ M. Thus, there exist, in each neighborhood U of P ∈ M, two local unit vector fields, N and J̃N, mutually orthogonal and normal to M. If h and k are second fundamental forms corresponding to N and JN, and h_{ij} and k_{ij} are their local components, than [16]

h_{ij} = h_{ji}, k_{ij} = k_{ji} ,
h_{ij} = J_{i}^{a}k_{aj}, k_{ij} = −J_{i}^{a}h_{aj} ,
J_{i}^{a}J_{j}^{b}h_{ab} = −h_{ij}, J_{i}^{a}J_{j}^{b}k_{ab} = −k_{ij}

Thus

trh = trk = 0 ,

and therefore \underline{any complex hypersurface of any Kähler manifold is minimal}. If (h^{r}) is fundamental form of order r, then

J_{i}^{a}J_{j}^{b}(h^{2p})_{ab} = (h^{2p})_{ij}, J_{i}^{a}J_{j}^{b}(h^{2p+1})_{ab} = −(h^{2p+1})_{ij} .  (5.1)

The second relation (5.1) shows that

tr(h^{2p+1}) = 0 for all p = 1, 2, ….  (5.2)

Similarly,

tr(Jh^{2p+1}) = 0 for all p = 0, 1, 2, ….  (5.3)

Now, let us suppose that the ambient (M̃, G, J̃) is a manifold of constant holomorphic sectional curvature c. Then the Gauss equation for (M, g, J) is

\begin{matrix}R_{ijlm} − \frac{c}{4}(g_{im}g_{jl} − g_{il}g_{jm} + F_{im}F_{jl} − F_{il}F_{jm} − 2F_{ij}F_{lm}) = \\ = h_{im}h_{jl} − h_{il}h_{jm} + k_{im}k_{jl} − k_{il}k_{jm} , \end{matrix}

where R_{ijlm} is the Riemannian curvature tensor of M.

If we put

T_{ijlm} = R_{ijlm} − \frac{c}{4}(g_{im}g_{jl} − g_{il}g_{jm} + F_{im}F_{jl} − F_{il}F_{jm} − 2F_{ij}F_{lm}) , (5.4)

the Gauss equation becomes

T_{ijlm} = h_{im}h_{jl} − h_{il}h_{jm} + k_{im}k_{jl} − k_{il}k_{jm} .  (5.5)

We remark that the tensor (5.4) has all algebraic properties as the Riemannian curvature tensor of a Kähler manifold. In particular

J_{i}^{a}J_{j}^{b}T_{abkl} = T_{ijkl} .  (5.6)

If

τ_{im} = T_{ijlm}g^{jl} ,

and if we denote by ρ_{ij} the Ricci tensor of M, from (5.4) we have

τ_{ij} = ρ_{ij} − \frac{n + 1}{2}cg_{ij}  (5.7)

On the other hand, (5.5) yields

τ_{ij} = trh h_{ij} + trk k_{ij} − (h)_{ij}^{2} − (k^{2})_{ij} .

But trh = trk = 0, (k^{2})_{ij} = (h^{2})_{ij}. Therefore

τ_{ij} = −2(h^{2})_{ij} ,

from which there follows

(τ^{p})_{ij} = (−2)^{p}(h^{2p})_{ij} .  (5.8)

Now, using (5.5) and (5.8), we find

T_{rsia}(τ^{p})_{j}^{a} + T_{rsja}(τ^{p})_{i}^{a} =

= (−2)^{p} [(h^{2p+1})_{rj}h_{si} − h_{ri}(h^{2p+1})_{sj} + (h^{2p+1})_{tj}J_{r}^{t}h_{qi}J_{s}^{q} − h_{ti}J_{r}^{t}(h^{2p+1})_{qj}J_{s}^{q}

+ (h^{2p+1})_{ri}h_{sj} − h_{rj}(h^{2p+1})_{si} + (h^{2p+1})_{ti}J_{r}^{t}h_{qj}J_{s}^{q} − h_{tj}J_{r}^{t}(h^{2p+1})_{qi}J_{s}^{q}] ,

from which, transvectin with g^{sj} and using (5.2) and (5.3), we get

\begin{matrix}−T_{rabi}(τ^{p})^{ab} + (τ^{p+1})_{ri} = \\ = (−2)^{p} [(h^{2p+1})_{tj}J_{r}^{t}h_{qi}J_{s}^{q}g^{sj} − h_{tj}J_{r}^{t}(h^{2p+1})_{qi}J_{s}^{q}g^{sj}] . \end{matrix}

But, according to the second relation (5.1)

J_{i}^{a}(h^{2p+1})_{ak} = J_{k}^{a}(h^{2p+1})_{ia} ,

because of which

(h^{2p+1})_{tj}J_{r}^{t}h_{qi}J_{s}^{q}g^{sj} = (h^{2p+1})_{tr}J_{j}^{t}h_{qi}J_{s}^{q}q^{sj} = (h^{2p+1})_{tr}h_{qi}g^{tq} = (h^{2p+2})_{ri} ,

and

−h_{tj}J_{r}^{t}(h^{2p+1})_{qi}J_{s}^{q}g^{sj} = −(h^{2p+2})_{ri} .

Therefore

T_{iabj}(τ^{p})^{ab} − (τ^{p+1})_{ij} = 0 .  (5.9)

Thus, we can state

Theorem 4. Let (M, g, J) be a complex hypersurface of a Kähler manifold of constant holomorphic sectional curvature. Then the relation (5.9) is valid for all integers p = 1, 2, ….

In the case c = 0, i.e. for a complex hypersurface of the flat Kähler space, (5.9) reduces to

R_{iabj}(ρ^{p})^{ab} − (ρ^{p+1})_{ij} = 0 ,

and is the intrinsic property of (M, g, J).

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