On the geometry of B-manifolds Текст научной статьи по специальности «Математика»

УЧЕНЫЕ ЗАПИСКИ КАЗАНСКОГО ГОСУДАРСТВЕННОГО УНИВЕРСИТЕТА

Том 151, кн. 4

Физико-математические пауки

2009

UDK 514.76

ON THE GEOMETRY OF B-MANIFOLDS

A. A. Salimov, M. Is can

Abstract

The main purpose of the present paper is to study almost B-structures (Norden structures) on 8-dimensional Walker manifolds. We discuss the problem of int.egrability, Kaliler (liolomorphic) and Einstein conditions for these structures.

Key words: Walker 8-manifold, Norden metric, liolomorphic metric. Einstein metric.

1. Introduction

Let M2n be a Riemannian manifold with neutral metric, i.e., with pseudo-Riemannian metric g of signature (n, n). We denote by 9p(M2n) the set of all tensor fields of type (p, q) on M2n. Manifolds, tensor fields and connections are always assumed to be differentiable and of class CX.

Let (M2n, y) be an almost complex manifold with almost complex structure y. Such a structure is said to be integrable if the matrix y = (yj) is reduced to constant form

in a certain holonomic natural frame in a neighborhood Ux of every point x e M2n.

y

that it be possible to introduce a torsion-free afiine connection V with respect to which the structure tensor y is covariantly constant, i.e., Vy = 0. It is also known that the integrability of y is equivalent to the vanishing of the Nijenhuis tensor Nv e ^^(M2n). If y is integrable, then y is a complex structure and, moreover, M2n is a C-holomorphic

manifold Xn(C) whose transition functions are liolomorphic mappings.

g

g(yX, yY) = -g(X,Y)

or eqnivalently

g(yX, Y)= g(X, yY)

for any X, Y e &o(M2n) • Metrics of this type have also been studied under the names: B-metrics, pure metrics and anti-Hermitian metrics [2-7]). If (M2n,y) is an almost complex manifold with Norden metric g, we say that (M2n,y,g) is an almost Norden

manifold. If y is integrable, we say that (M2n, y, g) is a Norden manifold.

*

1.2. Hoiomorphic (almost liolomorphic) tensor fields. Let t be a complex tensor field on Xn(C). The re^ model of such a tensor field is a tensor field t on M2n of ^^e same order such that the action of the structure afiinor y on t does not

ty

y

In particular, being applied to a (0, q)-tensor fie Id w, the purity means that for any Xo,... ,Xq e 9¿(M2n), the following conditions should hold:

w(yX 1, X2, . . . ,Xq) = w(Xo, yX2, . . . , Xq) = • • • = w(Xo, X2, . . . , yXq).

Wo define an operator

applied to a pure tensor field w by (see [11])

($vu)(X,Yi,Y2, ... ,Yq) = (fX)(w(Yi, Y2,..., Yq)) - X(w(fYi, Y2,..., Yq))+

+ w((LYl f )X, Y2,..., Yq) + • • • + w(Yi, Y2,..., (Ly,f )X),

where LY denotes the Lie differentiation with respect to Y.

When f is a complex structure on M2n and the tensor field vanishes, the

complex tensor field w on Xn(C) is said to be holomorphic (see [4, 6, 11]). Thus, a holomorphic tensor field w on Xn(C) is realized on M2n in the form of a pure tensor field w, such that

(Vw)(X,Yi,Y2,...,Yq) = 0

for any X, Y1;..., Yq G 9g(M2n) . Such a tensor field w on M2n is also called a holomorphic tensor field. When f is an almost complex structure on M2n, a tensor field w satisfying = 0 is said to be almost holomorphic.

1.3. Holomorphic Norden (Kahler— Norden) metrics. On a Norden manifold, a Norden metric g is called holomorphic if

($vg)(X,Y,Z) = 0 (1)

for any X,Y,Z G30(M2„).

By setting X = Y = Z = cj in equation (1), we see that the components ($vg)fcij of ^vg respect to a local coordinate system x1,..., xn can be expressed as follows:

($Vg)kij = fmdmgij - <fTdkgmj + gmj (djf™ - dfcf") + gimÖj

If (M2n,f, g) is a Norden manifold with holomorphic Norden metric, we say that (M2n,f, g) is a holomorphic Norden manifold.

In some aspects, holomorphic Norden manifolds are similar to Kahler manifolds. The following theorem is an analogue to the next known result: an almost Hermitian manifold is Kahler if and only if the almost complex structure is parallel with respect to the Levi Civita connection.

Theorem 1 [12] (For a paracomplex version see [13]). For an almost complex manifold with Norden metric g, the condition = 0 is equivalentt to Vf = 0, where V is the Levi - Civita connection of g.

A Kahler - Norden manifold can be defined as a triple (M2n, f, g) which consists of a manifold M2n endowed with an almost complex structure f and a pseudo-Riemannian

metric g such that Vf = 0, where V is the Levi - Civita connection of g and the metric g

between Kühler Norden manifolds and Norden manifolds with holomorphic metric. Recall that the Riemannian enrvatnre tensor of such a manifold is pure and holomorphic, and the scalar enrvatnre is a locally holomorphic function (see [5, 12]).

Remark 1. We know that the integrability of an almost complex structure f is equivalent to the existence of a torsion-free affine connection with respect to which the equation Vf = 0 holds. Since the Levi - Civita connection V of g is a torsion-free affine connection, we have: if = 0, then f is integrable. Thus, almost Norden manifolds with conditions = 0 Mid Nv = 0, i.e., almost holomorphic Norden manifolds (analogues of almost Kahler manifolds with closed Kahler form) do not exist.

In the present paper, we shall focus our attention on Norden manifolds of dimension eight. Using a Walker metric, we construct new Norden Walker metrics together with almost complex structures. Note that indefinite Kahler Einstein metrics on an eight-dimensional Walker manifolds have recently been investigated in [14].

2. Norden — Walker metrics

2.1. Walker metric g. A neutral metric g on an 8-manifold M8 is said to be a Walker metric if there exists a 4-dimensional null distribution D on M8 which is parallel with respect to g. By Walker's theorem [15], there is a system of coordinates

(x1

, x8) with respect to which g takes the following local canonical form

g = (gi

0 /4

/4 B

(2)

where /4 is the unit 4 x 4 matrix and B is a 4 x 4 symmetric matrix whose entries are functions of the coordinates (x1,..., x8). Note that g is of neutral signature (+ + + +

----), and that the parallel null 4-plane D is spanned locally by {d1; d2, d4},

d

where d® = —-, i = 1,..., 8. dx®

In this paper, we consider specific Walker metrics on M8 with B of the form

B=

/ a 0

0 0

0 0

0 0

(3)

where a, b are smooth functions of the coordinates (x1

2.2. Almost Norden — Walker manifolds. We can construct various g-orthogonal almost complex structures y on a Walker 8-manifold Mg with metrics g as in (2), (3) so that (Mg, y, g) is a (neutral) almost Norden manifold. The structure y defined

by

ydi = #3, yd2 = d4, yd3 = -di, y^4 = -d2, y^5 = ^(a + b)^3 - dr, yds = -dg,

ydr = - 2(a + b)di + d5, ydg = ^6.

is one of the simplest examples of such an almost complex structure.

y

y

^ =H )

0

/0 0 -1 0

0 0 0 -1 0 0

1 0 0 0 (a + b)/2 0

0 1 0 0 0 0

0 0 0 0 0 0

0 0 0 0 0 0

0 0 0 0 -1 0

0 0 0 0 0 -1

0 -(a + b)/2 0\

0 0 1

0 0

(4)

with respect to the natural frame (djj , i = 1,..., 8.

Remark 2. From (4) we see that, in the case a = -6, y is integrable.

2.3. Integrability of <p. We consider the general case.

The almost complex structure f of an almost Norden - Walker manifold is integrable if and only if

(Ngjfc = - - & + ^j = 0.

(5)

Since Njk = — Nj, we need only consider Njk (j < k). By explicit calculations, the nonzero components of the Nijonhuis tensor are as follows:

N1 = N7 = N?7 = —N|5 = i(ai + bi), 3 1

N57 = 4(0 + b)(ai + bi),

N5 = N47 = N7 = -Nf5 = -(a2 + 62),

N

1 7

47

27

2

1

-N35 = -N35 = N37 = — (as + 63)

2

1

N^7 = -7 (a + 6)(as + 63),

(6)

N^7 = N5 = N235 = N37 = 2(a4 + 64), N¿6 = -NTs = N|8 = N37 = - !(ae + 65), Ns = -N6V = N¿6 = N78 = - 1(as + 6s).

From (6), we have

Theorem 2. Tfte almost complex structure y 0/ an almost Norden - Walker manifold is integrable if and only if the following PDEs hold:

ai + 61 = 0, a2 + 62 = 0, a3 + 63 = 0, a4 + 64 = 0, a6 + 66 = 0, as + 6s = 0.

(7)

Corollary 1. The almost complex structure f of an almost Norden - Walker manifolds is integrable if and only if

a = -b + (8)

where £ is a function of x5 and x7 only.

(2)

B =

0

0 0

is always Norden Walker.

0 6(x1,...,x8 00

0 0

0 0

3. Norden — Walker — Einstein metrics

We now turn our attention to the Einstein conditions for a Walker metric (2), (3) with a Mid 6 given by (8). For a Mid b in (8), we put f = ^(a — b) = a — ^£ = — b + ^

1

Since a = f +2£ Mid b = — f +2it follows that B in (2) is as follows:

B =

f (x1,...,x8) + 1 £(x5,x7) 0 0

0 0 0 0 0 0 —f (x1,...,x8) + 1 £(x5,x7) 0

(9)

\ 0 0 0 0/

Let Rij and S denote,respectively, the Ricci tensor and the scalar curvature of

metric (2) with B given as in (9). The Einstein tensor is defined by Gij = Rij- — -Sg

8

(10)

and has the following nonzero components:

G25 = 1 fl2, G17 = — G35 = — 1 fi3, G45 = 1 f 14, G56 = 1 fl6, G58 = 1 f18, G27 = — 1 f23, G47 = — 1 f34, G67 = — 1 f36, G78 = — 1 f38, G15 = 1(3f11 + f33)j G26 = G48 = — 1 (f11 — f33), G37 = — ^ (f 11 + 3f33), G57 = 1(f 17 + f1f3 — f35),

3 1 1

G55 = —f26 — f37 — f48 + ^f (f11 — f33) + g£(3f11 + 5f33) — ^f3 ,

G77 = f 15 + f26 + f48 — 3 f (f11 — f33) + 1 £(5f11 + 3f33) — 1 f2.

A metric g with B as in (9) is Norden - Walker - Einstein if all the above components Gij =0.

Theorem 3. A Norden - Walker metric g is a Norden - Walker - Einstein one if the following PDEs hold:

a1 — b1 = 0, a2 — b2 = 0, a3 — b3 = 0, a4 — b4 = 0.

Proof. The assertion follows from (10) and the relation f = 1(a — b). □

From Theorem 2 and Theorem 3, we have

g

the following PDEs hold:

a1 = a2 = a3 = a4 = b1 = b2 = b3 = b4 = 0, a6 + b6 = 0, a8 + b8 = 0.

4. Holomorphic Norden — Walker (Kahler — Norden — Walker) metrics

Let (M8, y, g) be an almost Norden - Walker manifold. If

($Vg)fcij = yrdmgij — yrdfcgmj + gmj (diy™ — dfcy™) + gimdjy™ = 0, (11)

then, by virtue of Theorem 1, y is integrable and the triple (M8, y, g) is called a holomorphic Norden Walker or a Kahler Norden Walker manifold. Taking into account Remark 1, we see that an almost Kahler Norden Walker manifold with conditions = 0 Mid = 0 does not exist.

Substitute (2) and (3) into (11). Since (Ф^g)ijk = (Фvg)ikj, we need only consider (&vg)ijk (j < k). By explicit calculations, the nonzero components of the tensor Ф^д are as follows:

(Ф^д)155 = аз, (Ф^д)157 = 2(bi - ai), (Ф^д)т = Ьз,

(ф^5)255 = «4, (Ф^#)257 = ^- 02), (Ф^#)277 = &4,

(Ф^5)з55 = —«1, (Ф^#)з57 = 2(Ьз - аз), (Фу^)з77 = —bl,

(Фу£)455 = — «2, (Ф^^)457 = ^(&4 — 04), (Ф^^)477 = — &2,

(Ф^5)517 = —(Ф^5)715 = ^(ai + bi), (Ф^^)527 = — (Ф^)725 = 2(«2 + 62),

(Фу£)537 = — (Ф^£)735 = ^(аз + 63), (Ф^)547 = — (Ф^£)745 = ^(«4 + 64),

(Фу£)555 = ^Я + 6)«з — «7, (Ф^^)557 = —65,

(Ф^£)567 = — (Ф^£)756 = 2(а6 + 6б), (Ф^^)577 = + 6)6з + «7, (Ф^&)578 = —(Ф^5)758 = 2 (<38 + bg), (Ф^#)б55 = — «8,

(Ф^^)б57 = ^(6б — «б), (Ф^#)б77 = — 68, (Ф^^)755 = — ^Я + 6)«1 — 65,

1

"2(

1

2(

(Ф^^)757 = — «7, (Фу£)777 = — ^ (« + 6)6i + 65,

(Ф^^)855 = «6, (Ф^^)857 = ^(68 — «8), (Ф^#)877 = 6б.

From the above equations, we have

Theorem 4. triple (M8, f, g) ¿s a Kahler - Norden - Walker manifold if and only if the following PDEs hold:

ai = a2 = a3 = a4 = a6 = a7 = as = 0,

bi = b2 = b3 = b4 = b5 = b6 = bg = 0.

Corollary 4. manifold (Mg,f, g) ¿s Kahler - Norden - Walker if and only if the matrix B in (2) is as follows:

B

(«(x5) 0 0 0

0 0 0 0

0 0 6(x7) 0

V 0 0 0 0

(13)

5. Kahler — Norden — Walker — Einstein metrics

We now turn our attention to the Einstein conditions for a Walker metric (2), (3) with a Mid 6 given by (12). In this case, as a = a(i5^d b = b(x7), B in (3) is of the form (13).

Let Rij and S denote, respectively, the Ricci tensor and the scalar curvature of metric (2) with B given as in (13). We see that all the components of the Einstein

tensor defined by Gij = Rij - - Sgj are zero.

8

Thus, we have

Theorem 5. A metric g with B as in (13) is always Kahler Norden Walker Einstein.

6. On a relation between the Goldberg conjecture of almost Norden — Walker and Kahler — Norden — Walker manifolds

Let (M2n, y, g) be an almost Norden manifold, and choose a y-compatible 2-form typ on M2n, where typ(X,Y) = h(yX,Y), h(X,Y) = g(X,Y) + g(yX,yY). Then we can propose an almost Norden version of the Goldberg conjecture as follows [19]: if (Gi) M2n is compact, (G2) g is Einstein, and (G3) the y-compatible 2-form is closed, then y must be integrable.

We now define two subfamilies in the set of all compact Norden Walker 8-manifolds:

KNW = {(M8,y,g): Ф^ = 0},

GNW = {(M8,y,g): M8 with conditions (G2), (G3)} .

Theorem 6. Let M8 G KNW. Then M8 is of type GNW, i е., M8 G GNW.

Proof. Suppose that M8 G KNW Then, from Theorem 5, we see that g is Einstein. By virtue of Theorem 1 (Vy = 0), for we have

(VQp)(Z; X, Y) = (Vz g)(yX, Y) - (Vz g)(X, yY)+

+ g((Vz y)X, Y) - g(X, (Vz y)Y) = 0,

Vg

p. 149],

dtyp = A(Vtyp),

where is the covariant difierential of and A is the alternation, we have

dtyp = 0,

i.e., is closed. Thus, the proof is completed. □

This paper was supported by The Scientific and Technological Research Council of Turkey, with number ТВ AG (108T590).

Резюме B

B

Уокера. Для указанных структур исследуются вопросы интегрируемости, условия келе-ровости и эйпштейповости.

Ключевые слова: 8-мериое многообразие Уокера, метрика Нордепа, голоморфная метрика, метрика Эйнштейна.

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Поступила в редакцию 12.08.09

Салимов Ариф Агаджан оглы доктор физико-математических паук, профессор, кафедра алгебры и геометрии Бакинского государственного университета, г. Баку, Азербайджан и отделение математики факультета паук Университета Ататюрка, г. Эрзурум, Турция.

Е-шаП: asalimov0atauni.edu.tr

Исчан Мурат доктор паук, отделение математики факультета паук Университета Ататюрка, г. Эрзурум, Турция.

Е-шаП: jn.ismjiMatauni.edu.tr