Differential geometry of Walker manifolds Текст научной статьи по специальности «Математика»

Том 153, кн. 3

УЧЕНЫЕ ЗАПИСКИ КАЗАНСКОГО УНИВЕРСИТЕТА

Физико-математические пауки

2011

UDK 530.12

DIFFERENTIAL GEOMETRY OF WALKER MANIFOLDS

A.A. Salimov, M. Iscan, S. Turanli

Abstract

In t.lie present, paper, we focus our attention ou t.lie iiit.egrabilit.y and liolomorpliic conditions of a Norden-Walker structure (M,gN+ ,ip). We also give a characterization of a KahlerNorden - Walker metric gN+ .

Key words: Norden Walker structure. Walker manifolds, pure tensor field, Käliler Norden Walker metrics, liolomorpliic tensor field, twin metrics.

Introduction

Let M be a CX -manifold of finite dimension 4. We denote by 3rs(M) the module over F(M) of all CX-tensor fields of type (r, s) on M, i.e., of contravariant degree r and covariant degree s, where F(M) is the algebra of CX-functions on M.

A neutral metric wg on a 4-manifold M is said to be Walker metric if there exists a 2-dimensional null distribution D on M, which is parallel with respect to wg. From Walker's theorem [1], there is a system of coordinates with respect to which wg takes the local canonical form

wg = (w9ij I

0 0 1 0

0 0 0 1

1 0 a c

0 1 c b

(i)

where a, b, c are smooth functions of the coordinates (x,y,z,t). The parallel null 2-plane D is spanned locally by {dx, dywhere dx = d/dx, dy = d/dy.

In [2, Fact 1], a proper almost complex structure with respect to wg is defined as a wg- orthogonal almost complex structure ^ so that y is a standard generator of a positive n/2 rotation on D,ï.e., ydx = dy and ydy = — dx. Then for the Walker metric wg, such a proper almost complex structure y is determined uniquely as

/0 -1 -c

1 0 \{a-b)

0 0 0

0 0 1

\{a-b)\ c -1 0

(2)

/

In [3], for such a proper almost complex structure ^ on Walker 4-manifold M, an almost Norden structure (gN +, is constructed, where gN + ^s a metric on M, with properties gN ) = — gN +(X,Y). In fact, as one of these examples, such

a metric takes the form (see Proposition 6 in [3]):

+ =

( 0 -2 0

-2 0 -a

0 -a 0

\-b -2c i(l -ab)

-b \ -2c i(l - ab) -2bc

(3)

Wo may call this an almost Norden Walker metric. The construction of such a structure in [3] is to find a Norden metric for a given almost complex structure, which is different form the Walker metric.

In [3], for a given proper almost complex structure y, an another Norden - Walker metric GN + is also constructed:

GN+

/-2 0 -a -2c \

0 2 0 b

—a 0 ^(1—a2) —ac

y—2c b ~ -ac \{b2 - Ac2 - I))

(4)

The purpose of the present paper is to study Kahlor and qnasi-Kahlor conditions of Norden - Walker metrics gN + and GN + .

1. Kahler — Norden — Walker metrics

Let y be an affinor field on M, i.e., y G 9}(M). A tensor field t of type (r, s) is called pure tensor field with respect to y if

' 1 r

= i(xi,...,xs; y£,...,£)

= t(xi,...,xs; £,..., y£)

12 r ,

for any X1,X2,...,Xs G 9q(M) and £,£,...,£ G 90(M), where y is the adjoint operator of y defined by

( y£)(x)= £(yX),

X G3fi(M),£ G91(M).

*

We denote by 3rs (M) the module of all pure tensor fields of type (r, s) on M with respect to the affinor field y. We now fix a positive integer A. If K and L are pure tensor fields of types (p1,q1^d (p2, q2) respectively, then the tensor product of K L

K < L = (LSl'."rp

)

31■ ■ ■ 3q1

is also a pure tensor field.

**

We shall now make the direct sum 9(M) = ®r (M) into an algebra over the

c *

real number R by defining the pure product (denoted by <g) or " o ") of K G (M)

*

and L G (M) as follows:

C C

< : (K, L) ^ (K<L)

K

ii...m\...ipi Lrl---rp2

si...m\...sq

for A < pi,

(A is a fixed positive integer),

j .. .. X. . ji LSi.. . £. . . rp2 for p < p2,qi

(p is a fixed positive integer), for pi =0, p2 = 0, for qi =0, q2 = 0.

In particular, let K = X e 9q(M^^d L e Aq(M) be a q-form. Then the pure

C

product X ® L coincides with the interior product iX L.

Definition 1 [4]. Lety e =o 9r(M) be a tensor algebra

*

over R. A map : 9(M) ^ 9(M) is called a ^-operator on M if

a) is linear with respect to constant coefficients;

b) ^ : 9*r(M) ^ »r+1(M) for all r, s;

c) ^(K ® L) = (^K) ® L + K ® ^Lfor all K, L e 9(M);

d) Y = — (LYy)X X, Y e 90(M), where LY is the Lie derivation with respect Y

e) (iyw) = (d(«Yw))(yX) — (d(«y (w o y)))(X) = (yX)(iyw) — X(ivyw) for all w e 9}(M) mid X, Y e 90(M), where iyw = w(Y) = w ® Y.

Let (M, wg) be a Walker 4-manifold with a Norden - Walker metric gN + and proper almost complex structure y. If the Nijenhuis tensor field Nv e 92(M) vanishes, then y is a complex structure and moreover M is a C-holomorphic manifold X2(C) whose transition functions are C-holomorphic map pings. Nv = 0 is equivalent to the condition Vy = 0, where V is a torsion-free affine connection. A metric gN + is a Norden - Walker metric [3. 5 8] if

gN + (yX,Y )= gN + (X,yY) (5)

for any X, Y e 90(M), i.e., gN + is pure with respect to the proper almost complex structure y. If (M, y) is an almost complex manifold with Norden- Walker metric gN +, we say that (M, y, gN +) is an almost Norden- Walker manifold. If y is integrable, we

say that (M, y, gN +) is a Norden - Walker manifold. *

Let t e 9r(X2(C)) be a complex tensor field on X2(C). The real model of such a tensor field is a pure tensor field t e 9S(M) with respect to y, which in general is

not C-holomorphic. When y is a proper complex structure on M and the tensor field

*

vanishes, the complex tensor field t on X2(C) is said to be holomorphic [9]. Thus

*

t X2 ( C) M

t

1 2 r

(^t)(X,Yi,Y2,...,Yr,£,£,...,O = 0

1 2 r

for any X, Yq, ..., Yr e 90(M) mid e 90(M), where

(^ t) (X,Yi,...,Yr,£\...,£r) = (yX) t (Yi,...,Yr,£\...,£r) —

S

— Xt (yYi,..., Yr, e1,..., r) + t (Yi,..., (Lya y) x, ..., Yr, e1,..., er) —

A=1

r

— ]Tt(Yi,...,Yr,e},...,LvxeM — Lx(eM◦ y),...,er). (6)

M=1

In a Norden Walker (almost Norden Walker) manifold a Norden Walker metric gN + is called holomorphic (almost holomorphic) if

(^gN+)(X, Y, Z) = (yX) (gN+ (Y, Z)) — X (gN+ (yY, Z)) +

+ gN+ ((Lyy) X, Z) + gN + (Y, (Lzy) X) = 0

for any X, Y, Z G ^(M) .If (M, y, gN +) is a Norden - Walker manifold with a holo-morphic Norden - Walker metric gN +, we say that (M, y, gN +) is a holomorphic Norden Walker manifold.

In some aspects, holomorphic Norden Walker manifolds are similar to Kahler Norden Walker 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. An almost Norden Walker manifold is a holomorphic Norden

y

respect to the Levi - Civita connection of gN + .

Proof. By virtue of (5) and with Vg = 0 we have

gN+(Z, (Vyy)X) = gN+((VYy)Z, X). (7)

Using (7). we can transform (6) as follows:

($vgN+)(X,Z1,Z2) = -gN+((Vx y)Z1,Z2) +

+ gN+((Vziy)X,Z2)+ gN+(Z1, (Vz2y)X). (8)

From this we have

($vgN+)(Z2,Z1,X) = -gN+((Vz2 y)Z1,X) +

+ gN+((Vziy)Z2,X)+ gN+(Z1, (Vxy)Z2). (9)

If we add (8) and (9). we find

gN +)(X, Z1, Z2) + ($vgN +)(Z2, Z1, X) = 2gN +(X, (Vziy)Z2). (10)

By substituing $vgN + = 0 in (10), we find Vy = 0. Conversely, if Vy = 0, then the condition $vgN + = 0 follows from (8). Thus the proof is complete.

□

Remark. Recall that a Kahler Norden Walker manifold can be defined as a triple (M, y, gN +) which consists of a manifold M endowed with a proper almost complex structure y and a pseudo-Riemannian Norden - Walker metric gN + such th at Vy = 0, where V is the Levi - Civita connection of gN +. Therefore, there exist a one-to-one correspondence between Kahler Norden Walker manifolds and complex manifolds with a holomorphic Norden Walker metric as they were defined in [9].

Let (M, y, gN +) be an almost Norden - Walker manifold. If

(^gN+)kj = ymdmg.N+ - yrdkgm+ + g^ym - dkyD + gN+djy^ = 0, (n)

then by virtue of Theorem 1 the triple (M, y, gN +) is called a holomorphic NordenWalker or a Kahler Norden Walker manifold. By substituting (2) and (3) in (11), we obtain

(*,gN+U = (^gN+U =(*,gN+Lt = (^gN+L* = -by + 2^

(*,gN+) xyz = (*,gN+) xzy = -«y, ($^gN+)= (*,gN+) ** = -b* - 2cy, = = = "5(«b)y + (ac)x,

xtt =4ccx - bbx - 2(&c)tf, ($vgN+) yxz = (^£N+)yzx = ay> (*,gN+)yxt = = bx + 2cy, (*,gN+)yyt = = -by + 2cx,

(*<e9N+)yzz=aay, {^gN+)yzt = {^gN+)ytz = k(ob)x + (ac)y,

{^9N+)ytt=4ccy - 66y + 2(6c)*> = (%gN+)zz:x=h - Ha ~ b)*>

= 2(bx - ax), = =2cx - ay + by,

($^+)sa!t= {$v9N+).tx = 2cbx -Ha- b)by + 2c, - cax - at + bcx,

{%gN+) zyy = 4cy, ) zyz = {%gN+) zzy = cax - aay + l(ab)y-at + 2cz,

= = 2bcy + 2ccx - bz - (ac)y + cby, (^gN+)zzz = abz,

= -^(a+5)ai-±(a-5)(a&)!, + (5c)z+acz, (12)

($vgN+)ztt = 2c(bc)x +4ccz - (a - b)(bc)y - 2cat - bbz, ($v£N+)txx = -4cx,

gN+)txy = (^£W+)tyx = bx-2cy - ax, = = at - acx-2cz,

($^Ar+)ixi= (*>p9N+)ttx = bbx ~ Hab)x - 2ccx - cby + bz, (%9N+)tyy = 2(by - ay), (%gN+)tyz = (%gN+)tzy=bz - i(a - b)ax - (ac)y, {%gN+)tyt= {%gN+)tty = (b - a)cx - 4ccy + 2cz - at - ±5(a - b)y,

+)tzz = aat - 2acz, ($vgN +)ttt = (b - a)(bc)x - 2c(bc)y + 2bcz - bat, (^>f9N+)tzt = (^>f9N+)ttz = — jia ~ b)(ab)x — ^c(ab)y-\-cat — 2ccz + ^(a + b)b~. From those equations we have

Theorem 2. The triple(M, y, gN+) is Kahler-Norden- Walker if and, only if the following PDEs hold:

ax = ay = cx = cy = bx = by = bz = 0, at - 2cz = 0. (13)

Example. Let c = 0 (for Walker metrics wg with c = 0, see [10]). Then the triple (M, y, gN +) with metric

gN+

( 0 -2 0 —b(t) \

-2 0 —a(z) 0

0 —o{z) 0 ±(1 - a(z)b(t.))

\-b(t) 0 ±(1 -a(z)b(t.)) " 0

is always Kahlor Nordon Walker.

Let (M, y, g) be an almost Hermitian manifold. The Goldberg conjecture [11, 12] states that an almost Hermitian manifold (M, y, g) must be Kahler (or y must be integrable) if the following three conditions are imposed: (Gi) if M is compact and (G2) g is Einstein, and (G3) if the fundamental 2-form is closed. Despite many papers by various authors concerning the Goldberg conjecture, there are only two papers by Sekigawa [13, 14] which obtained substantial results to the original Goldberg conjecture.

(M, y, g)

(G1^(G2^d (G3). If the scalar curvature of M is nonnegative, then y must be integrable.

Let now (M, y, wg) be an Hermitian - Walker manifold with the proper almost complex structure y and the metric wg (see (1)). From Theorem 1, we have

Theorem 3. Let (M, y, wg) be an Hermitian - Walker manifold, with the proper almost complex structure y. The proper almost complex structure y on a Walker manifold (M, wg) is integrable if + = 0, where gN + is a Norden - Walker metric defined by (3).

2. Twin Norden — Walker metrics

Let (M, y, gN +) be an almost Norden - Walker manifold. The associated NordenWalker metric of almost Norden Walker manifold is defined by

G(X,Y) = (gN + o y)(X,Y) (14)

for all vector fields X and Y on M. One can easily prove that G is a Norden - Walker metric GN + (see (4)), which is called the twin metric of gN + and it plays a role similar to the Kahler form in Hermitian Geometry. We shall now apply the -operator to the pure metric GN + :

(^GN +)(X, Y, Z) = (yX) (Gn + (Y, Z)) - X (Gn + (yY, Z)) + + GN + ((LYy) X, Z)+ GN + (Y, (LZy) X) = = (LvxGN + - Lx(GN + o y))(Y, Z) + + GN + (Y,yLxZ) - GN +(yY,LxZ) =

= (<^GN+)(X,yY,Z)+ Gn+(Nv(X,Y ),Z). (15)

Tims (15) implies the following

Theorem 4. In an almost Norden - Walker manifold (M, y, gN +), we have

= (<^gN+) o y + gN+ o (Nv).

Corollary 1. In a Norden - Walker manifold (M, y, gN + ) the following conditions are equivalent:

a) + =0,

b) + =0.

We denote by VgN+ the covariant differentiation of Levi-Civita connection of Norden metric gN +. Then we have

VgN + GN+ = (VgN + gN+) o y + O (VgN + y) = g^ O (VgN + y),

which implies VgN + GN + = 0 % virtue of Theorem 1 (VgN + y = 0). Therefore, we have

Theorem 5. Let (M, y, gN+) be a Kahler-Norden-Walker manifold. Then the Levi - Civita connection of Norden - Walker metric gN + coincides with the Levi - Civita connection of twin Norden - Walker metric GN + .

3. Quasi-Kähler — Norden — Walker manifolds

The basis class of non-integrable almost complex manifolds with Norden metric is the class of the quasi-Kähler manifolds. An almost Norden manifold (M, y, gN +) is called qnasi-Kähler [15] if

a gN +((Vxy)Y, Z) = 0,

X , Y

where a is the cyclic sum by three arguments.

By setting (Lyy)X = Ly(yX)-y(LyX) = Vy(yX)-VvxY-^(VyX)+y(VxY)

and using (8), we see that +)(X, Y, Z) may be expressed as

($vgN+)(X, Y, Z) = -gN+((Vx y)Y Z) + gN+((Vy y)Z, X) + gN+((Vz y)X, Y).

If we add +)(X,Y,Z^d +)(Z,Y,X), then by virtue of

+(Z, (Vyy)X) = +((Vyy)Z,X), we find

+)(X,Y,Z) + +)(Z,Y,X) = 2gN +((Vy y)Z,X).

Since + )(X, Y,Z) = + )(X, Z, Y), from last equation we have

($vgN+)(X, Y, Z) + ($vgw+)(Y Z, X) + ($vgN+)(Z, X, Y) = a aN+((Vxy)Y Z).

X, Y , Z

Tims we have

Theorem 6. Let (M, y, +) &e an almost Norden - Walker manifold. Then the Norden - Walker metric + is quasi-Kähler Norden Walker if and only if

+)(X, Y, Z) + +)(Y Z, X) + +)(Z, X, Y) = 0 (16)

for any X, Y, Z e 90(M2„).

From (1.) and (16) we have

Theorem 7. A triple (M, y, +) is a quasi-Kähler-Norden-Walker manifold if and only if the following PDEs hold:

bx = by = bz = 0, ay - 2cx = 0, Ox + 2cy = 0, (6 - a)cK - 2ccy + 2cz - at =0.

We thank Professor Yasuo Matsushita for valuable comments. This paper is supported by The Scientific and Technological Research Council of Turkey (TBAG-108T590).

Резюме

А.А. Салимое, M. Исчаи, С. Тура,или, Дифферепциальпая геометрия многообразий Уокера.

В статье рассматриваются интегрируемость и голоморфность структуры Нордепа Уокера (M,gN+,tp), а также дается характеризация метрики Кэлера-Нордена-Уокера

gN+■

Ключевые слова: структура Нордепа Уокера, многообразие Уокера, чистое тензорное поле, метрика Кэлера Нордепа Уокера, голоморфное тензорное поле, двойная метрика.

References

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16. Salimov A.A., Iscan M. Some properties of Norden-Walker metrics // Kodai Math. J. -2010. - V. 33. - P. 283-293.

Поступила в редакцию 27.10.10

Salimov, Arif A. Doctor of Science. Professor. Department of Mathematics, Faculty of Sciences, Atat.iirk University, Erzurum, Turkey.

Салимов, Ариф Агаджан оглы доктор физико-математических паук, профессор отделения математики факультета естественных паук Университета Ататюрка, г. Эрзурум, Турция.

E-mail: asalimoveatauni.edu.tr

Iscan, Murât PhD, Assistant Professor, Department of Mathematics, Faculty of Sciences, Atat.iirk University, Erzurum, Turkey.

Исчан, Мурат доктор паук, и.о. доцента отделения математики факультета естественных паук Университета Ататюрка, г. Эрзурум, Турция.

E-mail: jn.iscanMatauni.ejlu.tr

Turanli, Sibel Research Assistant, Department of Mathematics, Faculty of Sciences, Atat.iirk University, Erzurum, Turkey.

Туранли, Сибел аспирант отделения математики факультета естественных паук Университета Ататюрка, г. Эрзурум, Турция.

E-mail: sibelturanliOhotmail.com