J. geom. 59 (1997) 67 - 76 0047-2468/97/020067-10 $1.50+0.20/0 �9 Birkh/iuser Verlag, Basel, 1997 ] Journal of Geometry

O N D U A L H O L O M O R P H I C A L L Y P R O J E C T I V E L Y F L A T A F F I N E C O N N E C T I O N S 1

Stefan Ivanov

Given a complex Riemannian metric h and a torsion-free complex affine connection V on a complex manifold, a dual holomorphically-planar curve of V is defined as a curve whose tangent complex plane, generated by its tangent 1-form, is parallel along the curve. The corresponding dual holomorphicaUy projective group is defined as a group of transforma- tions of connections preserving dual holomorphically- planar curves. The class of connec- tions complex semi-compatible with the metric h and pairs of complex semi-conjugate con- nections are defined using the relations between their holomorphically-planea" curves and their dual hotomorphically- planar curves. The dual holomorphically-projective curvature tensor for a connection complex semi-compatible with h is determined as an invariant of the dual holomorphicalIy-projective group. Dual holomorphically- projectively flat connec- tions complex semi-compatible with h are characterized as connections with vanishing dual holomorphically-projective curvature tensor.

1 I N T R O D U C T I O N

We define a dual holomorphically-planar ( briefly dual H-planar) curve as a curve whose tan-

gent complex space generated by its tangent 1-form is parallel along the curve and determine

dual holomorphically-projective (briefly dual H-projective) transformations as the transfor-

mations preserving unparameterized dual H-planar curves. We introduce complex semi-

conjugate connections and the class of complex semi-compatible connections as invariants of

the dual H-projective transformations. We study the corresponding dual holomorphically-

projective (briefly dual H-projective) curvature tensor and prove that for complex semi-

compatible connections this tensor is an invariant of the class of the dual H-projective trans-

formation. We characterize the dual H-projectively flat complex semi-compatible connection

1Research partialy supported by Contract MM 423/1994 with the Ministry of Science and Education of Bulgaria and by Contract 219/1994 with the University of Sofia "St. K1. Ohridski'.

68 Ivanov

as those with vanishing dual H-projective curvature tensor.

In [4] we apply this considerations to complex affine hypersurface theory: we determine condi-

tions of the fundamental theorem for non-degenerate complex affine hypersurface immersions,

first in terms of the induced affine onnection V, which has to be dual H-projectively flat,

and secondly, in terms of its semi-conjugate connection V*, which has to be H-projectively

fiat.

2 D U A L H - P R O J E C T I V E T R A N S F O R M A T I O N S

An n-dimensional complex Riemannian manifold (M, H) is a complex manifold M endowed

with a non-degenerate symmetric complex bilinear form H. If the local components of H

with respect to a local holomorphic coordinates z 1, ..., z ~ are holomorphic functions then

(M, H) is said to be a complex analytic Riemannian manifold.

Every complex Riemannian manifold can be considered as a real 2n-dimensional manifold

(M, h, J ) with the induced complex structure d and the induced pseudo Riemannian metric

h which has to be of signature (n, n). Moreover, these two structures are related by the

following equality h(JX, Y) = h(X, JY) = h(JY, X), where X , Y are vector fields on M.

The triple (M, h, or) is called also a complex Riemannian manifold.

In this paper we use the real approach in order to make clear the proof of our main result

in section 4.

Let (M, h, J ) be a complex-Riemannian manifold and V be a complex affine connection i.e.

V J = 0 and let c : le + M,I~ = ( -e ,e ) , e > 0, be a smooth curve on M with tangent

vector field ~. Let co~ be the 1-form corresponding to the tangent vector ~ (with respect to

the metric h), defined by w~(X) := h(4,X), for every tangent vector X.

D E F I N I T I O N . We call a smooth curve c a dual II-planar curve of the connection V if the

space span{wa,a0ja} is parallel along the curve i.e. the following holds

(2.1) V~c0~ = fw~ + qwj~,

where f and q are smooth functions on c.

Let Y be an arbitrary vector field on M. From (2.1) we obtain:

(2.2) (V~h)(+, Y) 4- h(Vafi, Y) = fh(O, Y) 4, qh(a~, Y).

A parameter s of a dual H-planar curve c(s) we call an affine parameter if the function f in

(2.1) is identically zero i.e.

(2.3) (Vah)(&, Y) 4- h(V~&, Y) = qh(J6, Y).

Ivanov 69

P R O P O S I T I O N 2.1 For every dual H-planar curve there exists an affine parameter. Such

an atone parameter is unique up to affine transformations of R .

Proof. It is analogous to that for the standard geodesics. Q .E .D.

Further by a dual H-planar curve we shall mean a dual H-planar curve together with an

ai~ne parameter.

Let S be an (1,2)-tensor on M satisfying the condition:

(2.4) h(S(X, Z), Z) = (Vxh)(Y, Z).

Then (2.3) is equivalent to the equation:

(2.5) + s(a, = qJ .

The equation (2.5) is an ordinary differential equation describing dual H-planar curves. By

the fundamental theorem of ordinary differential equations we obtain

P R O P O S I T I O N 2.2 Let (M, h, J) be a complex-Riemannian manifold and q is a smooth

(fixed) function on M. Let p be a point on M and X be an arbitrary vector at the point p.

There locally exists a dual H-planar curve trough the point p with tangent vector at p equal

to the given vector X , satisfying (2.5).

D E F I N I T I O N . Let (M, h, J) be a complex Riemannian manifold and V, V' be complex

affine connections. We call the connections V and V' dual H-projectively equivalent if they

have common (unparameterized) dual H-planar curves.

P R O P O S I T I O N 2.3 Let (M, h, J) be a complex Riemannian manifold of dimension 2n >

4. Two torsion-free complex affine connections V and ~7, are dual H-projectively equivalent

iff there exists a vector field T on M, such that:

(2.6) Vb~Y = V x Y - h(X, Y ) T + h(X, JY )T .

Proof: Let Q be the difference (1,2) tensor between ~7 and V' i.e. Q(X, Y) = WxY- ~7xY. Since V and V' are torsion-free complex connections the tensor Q has the following

properties:

(2.7) Q(X, Y ) = Q(Y, x ) ; Q(X, JY ) = JQ(X, Y ) = Q(JX, Y) .

Let p be a point on M, X be an arbitrary vector at p and q be a smooth :function on M.

Let c(t) be the dual H-planar curve of V, parameterized by an afl=ine parameter t, through

the point p such that dp = X. By the condition of the theorem c(t) is also a dual H-planar

curve of ~7' (t may not be an affine parameter). From (2.3) and (2.2) we get

(2.8) V'coe - Vecoe = c~(5)~oe + 5(JS)wae,

70 Ivanov

where oe and fl are smooth functions on the tangent bundle T M of M.

Let Z be an arbi trary vector at p. Restricting (2.8) at the point p we get:

(2.9) h(Q(X, Z),X) = o~(X)h(X, Z) + fl(JX)h(JX, Z).

Since the point p and the vectors X, Z are arbitrary we conclude that (2.9) is valid at every

point p on M and for every two vectors at p. From (2.9) and (2.7) we get c~(X) = - f l ( Z ) ,

since 2n > 4. Then the equality (2.9) takes the form

(2.10) h( Q( X, Z),X) = ~( X)h( X, Z) - c~( J X )h( J X, Z ).

Since 2n > 4 it follows from (2.10) that a has to be a 1-form on M and the following equality

holds true

(2.11) h(Q(X, Z), Y) + h(Q(Y, Z), X) = cx(X)h(Y; Z) + v~(Y)h(X, Z ) -

o~( J X)h( JY, Z) - c~( JY)h( J X, Z),

for arbi trary vectors X, Y, Z at any point p on M.

Subtracting from this equation the one obtained by interchanging Y and Z and using (2.7)

we obtain:

(2.12) h(Q(X, Z), Y) - h(Q(X, Y), Z) = a(Y)h(X, Z) - a(Z)h(X, V ) -

c~( JY)h( JX, Z) + ~( J Z)h( JX, Y).

Interchanging X and Z in (2.12) and adding the result to (2.11) we get h(Q(X, Z),Y) = a(Y)h(X, Z) - a(JY)h(JX, Z) which is equivalent to (2.6) since h is non-degenerate.

Conversely, if (2.6) holds it is easy to check that XT' has the same dual H-planar curves as

V which completes the proof of the theorem. Q . E . D .

For torsion-free complex affine connections V and V ~ we will speak of a dual H-projective

change from V to V ' or of dual H-projective equivalence of V and XT' if (2.6) holds.

Dual H-projective transformations appeared in a natural way in complex atone hypersurface

theory (see [1], [2], [4]).

3 C O M P L E X S E M I - C O M P A T I B L E A N D C O M P L E X

S E M I - C O N J U G A T E C O N N E C T I O N S

Let (M, h, J, V) be a complex-Riemannian manifold and V be a torsion-free complex affine

connection on M. Let V* be an affine connection such that every dual H-planar curve of V

is a H-planar curve of V*. (We recall that a smooth curve c is said to be a H-planar curve

Ivanov 71

if its complex tangent space span{h, J8} is parallel along the curve.)

Suppose V* satisfies the Coda.zzi equation:

(3.13) (V~h)(Y, Z) = (V~,h)(X, Z).

The Codazzi equation implies that the tensor V*h is totMly pure tensor, i.e.

(V*jxh)(Y, Z) = (V*xh)(JY, Z) = (V~h)(Y, JZ).

Using Proposition 2.2, from (2.5) and (2.4) it follows that the connection V* is torsion-free complex affine connection iff the tensor Vh satisfies the following identity:

(3.14) (Vxh)(Z, Z) - (Vyh)(X, Z) = -T(X)h(Z, Z) + ~(Y)h(X, Z)+

T( J X)h( JY, Z) - "r( JY)h( JX, Z),

where r is an 1-forln on M.

D E F I N I T I O N . We shall say that a torsion-free complex afflne connection g 7 is complex semi-compatible with the metric h by 1-form r if the equation (3.14) holds. If T = 0 then V

is compatible with h.

A complex Riemannian manifold (M, J, h) is a complex analytic Riemannian manifoM if the

Levi-Civita connection V h of the metric h is a complex connection (see [3]). If a complex

Riemannian manifold (M, J, h) admits an affine connection complex semi-compatible with

the metric h then h has to be complex analytic Riemanniaa metric. Indeed, by the well

known theorem (see [5], p.132) we have:

2h(V~Y, Z) = 2h(VxY, Z) + (Vxh)(Y, Z) + (Vyh)(X, Z) - (Vzh)(Y,X).

Using (3.14) we get

(3.15) 2h(V~Y, Z) = 2h(VxY, Z) + (Vxh)(Y, Z) - T(Y)h(X, Z) + T(Z)h(X, Y)+

r(JY)h(X, JZ) - r(JZ)h(X, JY).

From (3.15) we get ~ h j = 0. Hence, h is a complex analytic Riemannian metric (see [3]).

If V is an affine connection complex semi-compatible with the complex analytic metric h

by a 1-form 7 then there exists a unique affine connection V* compatible with h* such that

every dual-geodesic of V is a geodesic of V*. The connection V* is defined by:

(3.16) h(V*xY, Z) = h(VxY, Z) + (Vxh)(Y, Z) - T(Y)h(X, Z) + r(JY)h(JX, Z).

From (3.16) we easily obtain:

(3.17) Xh(Y, Z) = h(V~]r, Z) + h(Y, "fizZ) + r(Y)h(X, Z) - T(JY)h(JX, Z).

72 Ivanov

D E F I N I T I O N . An affine connection V* on M is said to be semi- conjugate to V by t-form

T relative to the metric h if the equation (3.17) holds, tf r = 0 then V* is conjugate to V.

From(3.17) it follows that the connection V* is aJways a complex connection.

P R O P O S I T I O N 3.1 Let V* be a complex semi-conjugate to 2 7 by r. Then 27* is torsion-

free iff 27 is complex semi-compatible with the metric h by -r.

Proof: Let K be the difference (1.2) tensor between 27 and 27*, i.e. K = 27* - V. Since both

connections are complex connections and 27 is torsion-free the tensor K has the following

properties:

(3.18) K(X, JY ) = J K ( X , Y ) ; K ( X , Y ) - K ( Y , X ) = T*(X,Y) ,

where T* is the torsion tensor of 27*. From (3.17) we obtain:

(3.19) (27xh)(Y,Z) = h ( K ( X , Y ) , Z ) + T(Y)h(X,Z) - ~'(JY)h(JX, Z).

From (3.19) using (3.18) we get that T* = 0 iff (3.14) holds, since h is non-degenerate.

Q.E.D.

Complex semi-compatible connections and complex semi-conjugate connections arise in a

natural way from complex aftlne hypersurface theory (see [1], [21, [4]).

From (2.6) and (3.14) it follows that the class of complex semi-compatible connections is

an invariant of the class of dual H-projective transformations. Moreover, (3.16) implies

that for every fixed class of complex compatible connections there exists exactly one linear

connection, which is complex semi-conjugate with all the connections of this class.

P R O P O S I T I O N 3.2 Let (M, h, J) be a complex analytic Riemannian manifold. Let 27 be

an afflne connection complex semi-compatible with the metric h by r and 27* be the corre-

sponding aJfine connection Complex semi-conjugate to 27 by T. Then the curvature tensors

R* and R for 27* and 27, respectively, are related by the following equality:

(3.20) h(R*(X, Z)Z, U) + h(R(X, Y)Z, U) =

h(X, U) [(V{,T)Z -- r (Y) r (Z) + T(JY)'r(JZ)I --

h(Y, u ) - , - ( x ) T ( z ) + T ( J X ) T ( J Z ) I -

h( X, JU) [(V{,r)JZ - r( JY)r ( Z ) -- r( Y)r( J Z)] +

h(Y, J g ) [(27*xr)JZ - T(JX)r(Z) -- r (X)v(JZ)] .

Ivanov 73

Proof." From (3.17) we obtain:

XYh(Z, U) = h(V~V~Z, U) + h(V~Z, VxU) + h(VxVyU, Z) + h(V*xZ, V~U)+

T(V~Z)h(X, U) - ~'(JV~Z)h(JX, U) + ~(Z)h(X, Vy, U)-

T( J Z)h( JX, Vy, U) + X(T( Z) )h(Y, U) - X('r( J Z)h( JY, U)+

-r(Z) [h(V~Y, U) + h(Y, Vx, U) + T(Y)h(X, U) - ~'(JY)h(JX, U)] -

T(JZ) [h(V*xJY, U) + h(dr, Vx, U) + -c(JV)h(X, U) + v(Y)h(JX, U)].

Subtracting fi'om this equation the one obtained by interchanging X and Y and the equation

[X, Y]h(Z, U) = h(V[*x,v]Z, U) + h(Z, V[x,v]U) + T(Z)h([X, Y], U) - T(JZ)h(J[X, Y], U)

gives the desired result. Q.E.D.

4 D U A L H O L O M O R P H I C A L L Y - P R O J E C T I V E L L Y

F L A T C O N N E C T I O N S

An atone connection %7 is sMd to be dual H-projectivety fiat if around each point there is a

dual It-projective change of V to a flat atone connection.

Let (M, h, V) be a 2n-dimensional complex analytic Riemamfian manifold and V be an affine

connection complex semi-compatible with h. We define the dual H-projective curvature

tensor DH W of V by the equality:

(4.21) DHW(X, Y)Z = R(X, Y)Z - h(Y, Z)M(X) + h(X, Z)M(Y)+

h(Y, JZ)JM(X) - h(X, JZ)JM(Y) ,

where X, Y are vector fields on M and the tensor M is given by:

(4.22) M(X) = -Ric#(X) + trh(Ric~) x trh(J o Ric) j x . 2(n - 1) 2(n - 1)

In (4.22) we denote by Ric#(X) the Ricci operator, corresponding to the Ricci tensor Ric, defined by h(Ric#(X), Y) := Ric(X, Y) and by trh we denote the trace with respect to h.

We have

T H E O R E M 4.1 The dual H-projective tensor for an a.~ne connection complex semi - compatible with a complex Riemannian metric is an invariant of the class of the dual H- projective transformations.

74 Ivanov

Proof: Let V be an affine connection complex semi-compatible with h by T and V' be dual

H-projectively equivalent to V. From (2.6) using (3.14) for the curvature tensors R and R'

of V and V t respectively, we calculate:

(4.23) R'(X, Y ) Z = R(X, Y )Z - h(Y, Z)L(X) + h(X, Z)L(Y)+

where

(4.24)

+h(Y, JZ )JL (X) - h(X, JZ )JL(Y) ,

L(X) = V x T - [a(X) + ~(X)] T + [ a ( JX) + T(JX)] JT,

where ~ is the 1-form corresponding to the vector field T (with respect to the metric h)

defined by ~ (X) := h(X, T).

Contracting twice in (4.23) we get:

tr R%x trig o Ri jX] 2 L ( X ) = - R i c # ( X ) + 2 ( n _ l ) 2 ( n - ~ j -

trhRid x trhJ o R i d . . ] - (Ric ' )*(x) + 2(n - 1) G

Substituting the last equality into (4.23) we get D H W = DHW'. Q . E . D .

Further we need the following result:

L E M M A 4.2 Let V be an affine connection complex semi-compatible with h by "r and with

zero dual H-projective curvature tensor~ and let V* be the affine connection complex semi-

conjugate to V by T. If 2n >_ 6 then:

i) The curvature tensor R* of %7, is totally pure.

ii) The connection 27* is Ricci-symmetric and the following formulae are valid:

(4.25) tr(R) = -2d-c; tr(J o R) = --2(T o J); tr(R*) = tr(J o R*) = O.

Proof: Contracting in (3.20) we obtain:

(4.26) tr(R*) + tr(R) = -2&-; tr(J o R*) + tr(J o R) = - 2 d ( v o J) ;

Since D H W = 0 from (4.21) and (3.20) we get:

(4.27) R*(X, Y ) Z = P(Y, Z )X - P(X, Z )Y - P(Y, J Z ) J X + P(X, JZ)JY ,

where

(4.28) P(X, Y) = M(X, Y) + ( V x r ) Y - T(X)T(Y) + r ( J X ) r ( J Y ) .

In (4.28) M(X, Y) = h(M(X), Y) and M(X) is given by (4.22). The first Bianchi identity

implies that the tensor P has to be symmetric and totally pure, since 2n > 6. From (4.27) we

get that R* is a totally pure tensor and from (4.26) we get (4.25) which proves the assertion.

Q . E . D .

Now, we shall formulate and prove our main result:

Ivanov 75

T H E O R E M 4.3 Let (M, J, h) be a 2n-dimensional complex analytic Riemannian manifold

and • be an a]fine connection complex semi-compatible with h by ~'. Then:

i) If dim M > 6, then V is dual H-projeetively fiat iff D H W = O.

ii) For dim M = 4, let the curvature tensor R of V satisfy the conditions

(4.29) t rR(X, Y) = -2d~'(X, Y), tr(J o R)(X, Y) = -2d(T o J)(X, Y),

Then V is dual H: projectively fiat iff D H W = 0 and the following condition holds:

(4.30) (~ x M ) Y - ( V y M ) X = r ( X ) M ( Y ) - r ( Y ) M ( X ) -

- T ( J X ) J M ( Y ) + r ( J Y ) J M ( X ) .

Proof: Let D H W = O. To prove that V is dual H-projectively flat it is sufficient to prove

the existence of a solution of the following system of partial differential equations:

(4.31) V x T = M ( X ) + In(X) + ~(X)] T - [a(JX) + ~'(JX)] JT,

where M is given by (4.22), c~ is the 1-form corresponding to the vector field T with respect

to the metric h and X is a vector field on M.

Using D H W = 0 and (3.14) the integrability conditions of the system (4.31) take the form:

(4.32) ( V x M ) Y - ( V y ~ l ) X - T(X)M(Y) + T(Y)M(X) + T ( J X ) J M ( Y ) -

--T( J Y ) J M ( X ) : -- Idol(X, Y) + d'r(X, Y)] T + [d(a o J)(X, Y) + d(T o J) (X, Y)] JT.

From (4.31) using (3.14) and Lemma4.2 we get de~+dr = d(aoJ)+d(~'oJ) = 0. Substituting

this equality into (4.32) we have that the integrability conditions of the system (4.31) are :

(4.33) (V x M ) Y - (Vy M ) X = T(X)M(Y) - T ( Y ) M ( X ) -

- T ( J X ) J M ( Y ) + T(JY)JM(X) .

We have to consider two cases.

CASE 1. 2n > 6. From the second Bianchi identity~ using D H W = 0 and (3.14) we

obtain:

(4.34) crv, x,y(h(Y, Z ) ( ( V v M ) X - ( V x M ) V - "c(V)M(X) + T(X)M(V)+

"r(JV)JM(X) - "r(JX)J3I(V)) - h(JY, Z ) ( J ( V v M ) X - J ( V x M ) V -

"r( V)JJt:I( X ) + ~'( X )J M ( V) - r( J V ) M ( X) + 7-( J X ) J M ( V) ) = 0,

where X, Y, Z, V are vector fields on M and ~rv, x,y denotes the cyclic sum of ~ X, Y. Since

2n > 6 from (4.34) we get (4.33) which proves the sufficient part of i).

76 Ivanov

CASE 2. 2n = 4. By the conditions of the theorem DHW(R) = 0 and the integrability

conditions of the system (4.31) are exactly the conditions (4.30) and the sufficient part of ii)

follows.

Further, the necessary part of i) follows immediately from Theorem 4.1. If V is dual H-

projectively flat we have that (4.33) is valid and the necessary part of ii) follows. This

completes the proof of Theorem 4.3. Q.E.D.

I would like to thank the referee for indicating omissions in the text and for the helpful

advices during the preparation of the final form of the paper.

R E F E R E N C E S

[1] DILLEN, F., The aj]ine differential geometry of komplex hypersurfaces, Med. Konink. Acad. Wetensch. Belg. 52(1990)1,91- 112.

[2] DILLEN, F., VRANCKEN, L., V E R S T R A E L E N , L., Complex affine differential geometry, Atti. Acad. Peloritana Pericolanti CL Sci. Fis. Mat. Nat. vol. LXVI (1988), 231 - :260.

[3] GANC HEV,G. , IVANOV,S.,Characteristic curvatures on complex Riemannian ma- nifolds, Riv. Mat. Univ. Parma 51(1992), 1 5 5 - 162.

[4] IVANOV, S., On the fundamental theorem for non-degenerate complex affine hyper- surface immersions, to appear.

[5] S C H O U T E N , J.A., Ricci calculus 2nd ed., Springer 1954.

University of Sofia, Faculty of Mathematics and Informatics, Department of Geometry, bul. James Bouchier 5, 1126 Sofia, BULGARIA. E-mail: ivanovsp @fmi.uni-sofia.bg

Eingegangen am 7. Februar 1995; in r e v i d i e r t e r Form am 16. Juni 1995