Journal o f Geomet ry

Vol . 53 (1995)

0047 -2468 /95 /020089 -1151 .50 + 0 .20 /0

(c) 1995 Birkh~iuser Ver lag, Basel

O N D U A L - P 1 L 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 t

S t e f a n I v a n o v

Given a pseudo Riemanniax~ metric h and a torsion-free affine eom~ection V on a. smooth n-manifold M , a d u n geodesic curve of V is defined as a curve whose tangent 1-form is parNlel along the curve. The corresponding duN-projective group is defined as a group of t ransformat ions of connections preserving dual-geodesic curves. The class of connections semi compat ible with the metric h and pairs of semi-conjugate connections are defined using the relations between their geodesics and duN-geodesics. The duN-project ive curvature tensor for a connection semi-compatible with h is determined as an invariant of the d u n projective group~ DuN-projectively flat connections semi-compatible with h are characterized as connections with vanishing duN-projective curvature tensor. As an applicat ion we recover the fundamenta l theorem for non-degenerate hypersurface immersions.

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

We define a. dual-geodesic curve as a curve whose tangent 1-form is parallel along the curve and

determine dual-project ive t ransformations as the transformatio.~.s preserving unparameter ized dual-

geodesic curves. We introduce semi-conjugate connections and th.e ciass of semi-compatible con-

nections as invariants of the dual-projective transformations. We study the corresponding duN-

projective curvature tensor and prove tha t for semi-compatibie connections this tensor is an in-

variant of the class of duN-projective t ransformation, and characterize the duN-projeetively flat

semi compat ible connection as those with vanishing duN-projective curvature tensor.

Following the ideas of [3] and [8], we determine conditions for the fundamenta l theorem for non-

degenerate affine hypersurface imersions, first in terms of the induced affine connect ion V, which

has to be dual-projectively flat~ and on secondly~ in terms of its semi-conjugate connection V*,

which has to be projectively flat o

Research partiMy supported by Contract MM 18/1991 witi~ the Ministry of Science and Education of Bulgaria ~nd by Contract with the University af Sofia.

90 Ivanov

2, DUAL PROJECTIVE TRANSFORMATIONS,

Let ( M , h ) be a pseudo-Riemannian manifold and V 'be an a~ne connection and let c : ]~ --*

M,I~ = ( - e , Q , e > 0, be a smooth curve on M with tangent vector field &Let ~ be the 1-form

corresponding to the tangent vector d (with respect to the metric h), defined by ~d(X) := h(d, X).

D E F I N I T I O N . We call a curve c a dual-geodesic curve of the connection V if the space span{~a}

is parallel along the curve i.e. the following holds

where ,~ is an 1-form.

A parameter s of a dual geodesic curve c(s) we call an afflne parameter if the 1-form ~d is parallel

along the curve i.e.

(2.2) V~wd = 0.

P R O P O S I T I O N 2o1~ For every dual-geodesic curve there exists an aJfine parameter, Such an

ajfine parameter is unique up to aJfine transformations of R .

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

Further by a dun1 geodesic we shall mean a dual geodesic ~ogether with an affine parameter.

Let Y be an arbitrary vector field on M, From (2.2) we obtain:

(2.3) (V~,~)(e, Y) + h(Vee, Y) = O.

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

(2.4) h(S(X , X), Y) = (Vx h)(X, Y) .

Then (2.3) is equivalent to the equation:

(2.5) V~d + S(~, ~) = O.

The equation (25) is an ordinary differential equation describing dual geodesic 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 p be a point on M and X be an arbitrary vecwr at the point p. The~'e

locally exists a unique dual geodesic curve trough the point p with tangent vector at p equal to the

given vector X.

D E F I N I T I O N . Let (M, h) be a pseudo Riemannian manifold and V, V / be affine connections.

We call the connections V and V t dnal-projectiveIy equivalent if they have corm'non dual geodesic

curves.

Ivanov 91

P R O P O S I T I O N 2.3. Let (M~h) be a pseudo Riemannian manifold. Two torsion-free a]flne

conn~tions V and V' are dual-projectively equivalent iff there exists an I-form (~ on M, such that:

(2.6) V ~ Y = V~:Y - hCX, Y > # ,

where c~# is" the vector field corresponding to a(with respect to the metric h) defined by h(a #, X) :=

~(x).

Proof: Let Q be the difference (1,2) tensor between V and X7 ' i.e. Q ( X , Y ) = V ~ Y - V x Y . Since

V and W are torsion-free the tensor Q is a symmetric tensor i.e. Q ( X , Y ) = Q(Y ,X) .

Let p be a point on M and X be an arbitrary vector at p.

Let c(t) be the dual-geodesic of V, parameterized by an affi.ne parameter t, through the point p

such that ~p = X. By the condition of the theorem c(t) is also a dual-geodesic of V ' (t may not be

an affine parameter). From (2.1) and (2.2) we get

(2.7) V~w~ - V c ~ = a(~)w~,

where a is the 1-form on M.

Let Z be an arbitrary vector at p. Restricting (2.7) at the point p we get:

(2~8) h(O(X~ Z), X) = ~(X)h(X , Z)~

Since the point p and the vectors X, Z are arbitrary we conclude that (2.8) is valid at every point

p on M and for every two vectors at p. From. (2.8) we get

(2.9) h(O(X, Z ) , Y ) + h(O(Y, Z), X ) = a(.Z)h(Y, Z) + a (Y)h (X , Z),

for arbitrary vectors X, I7, Z at any point p on M.

Subtracting from this equation ;he one obtained by interchanging Y and Z and using the symmetries

of Q we obtain:

(2.10) h(Q(X, Z), Y) - h(Q(X, Y), Z) = ce(Y)h(X, Z) - a (Z)h(X , Y).

Interchanging X and Z in (2A0) and adding the resnlt to (2.9) we get h(Q(X, Z), Y) = a (Y)h(X , Z)

which is equivalent to (2.6) since h is non- degenerate.

Conversely, if (2.6) holds it is easy to check that V ' has the same unparameterized dual-geodesics

as V which completes the proof of the theorem. Q.EoDo

From Proposition 2.3 we get:

T H E O R E M 2.4, Two ~orsion-fi'ee aj~ne connections V and W on a pseudo Riemannian manifold

(M, h) have common dual geodesic curves with respect to the common affine parameter iff V and

V' coincide.

For torsion-free affine connections V and V ~ we will speak of a dual- projective change from V to

92 Ivanov

V ' or of dual-projective equivalence of V and V' if (2.6) holds. If c~ = dr162 is a positive smooth

function) is a closed !-form the transformations (Z6) are known as 1- conformal equivalence of

statistical manifolds, provided h' = c)h (see [1],[9])o Such transformations are considered also in

[14], [4].

3. S E M I - C O M P A T I B L E A N D 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, V) be a pseudo-Riemannian manifold and V be a torsion-free affine connection on M.

Let V* be an affine connection such that every dual-geodesic curve of V is a geodesic curve of V*

with respect to the same affine parameter. From (2.3) we have

(3.1) V3~Y = V x Y + S(X ,Y ) ,

where the (1,2) tensor S satisfies (2A). From (3.1) and (2.4) it follows that the connection V* is

~:orsion-free iff the tensor Vh is totally symmetric i.e.

(3.2) (Vxh)(Y, Z) = (Vrh)(X, Z)o

A finear connection V is said to be compatible with the metric h if the tensor Vh is ~ Godazzi

tensor i. e. the equation (3.2) holds~ Compatible connections are considered in [3],[12].

Thus, if V is a torsion-free afline connection compatible with the metric h then there exists a unique

torsion-free u~ne connection V* such that every dual-geodesic of V is a geodesic of V* with respect

to the same ~fi%e parameteL The connection V* is defined by:

(3.3) ~(VkY, Z) = h ( V ~ > Z) + l V x ~ ) ( r , Z)

Dora (3.2) and (3.3) we easily obtain:

(3.4) (v~vh)(Y, z) = -(Vxh)(Y, Z)

(3,5) xh(Y, z) = h(v~Y, z) + h(Y, v x z )

Thus, the connection V* is also compatible with the metric h. From (3.4) and (3.1) it follows also

that every dual-geodesic of V* is geodesic of V with respect to the same affine parameter.

~wo ~ffine connections (V, V*) are said to be conjugate with respect to the metric h if the equation

(3~ holds. If both courtections V and V* are torsiomfree then the manifold (M, h, V, V*) is called

a statistical manifol& For the geometry of conjugate connections see [3], [12], [15]~ [7], [8]. For the

connection bet~ween conjugate connections and information geometry see [1].

Let vj be a torsion-free af[ine connecdo, compatible with h and V* be its conjugate connection. If

V ~ ~s dual projective eq-aivalent to V from (2.6), (3.2) and (3.3) we easily obtain:

~3,6~ ~vxhi(Y zi - (v~h)(x, z) = -~(x)h(Y, z) + ~(Y)h(x, z)

Ivanov 93

(3.7) Xh(Y, z) = h(V)Y, Z)+ h(Y, V)Z) + ~(Y)h(X, Z),

It is not dificult to see that (3.6) and (3.7) are equivalent conditions iff V ~ and V* are torsion-free

connections

D E F I N I T I O N . We shall say that a torsion-free affine connection V is semi-compatible with the

metric h by 1-form a if the equation (3.6) holds,

It is clear that the class of semi-compatible anne connections is closed by a dual-projective trans-

formation.

D E F I N I T I O N ~ A torsion-free anne connectioll V* on M !.s said to be semi- conjugate to V rda-

tire ~o the metric h if the equation (3.7) holds~ We shaJi say that V* is semi-conjugate to V by a.

P R O P O S I T I O N 3.1. Let a torsion-free affine connection V* be semi-conjugate to a torsion-free

a]fine connection V by a 1-form "r~ Then the curvature tensors R* and R for V* and V respectively,

are related by the equality:

(3.8) h(R*(x, Y)Z, ~) + h(R(X, Y)U, Z) =

h(X~ U)[(V~,T)Z - T(Y)T(Z)] -- h(Y~ U)[(V~-)Z - ~(X)T(Z)]~

Proof: From (3.7)we obtain:

xYh(z , u) : h(V;:V~ Z~ U) + h(V~,Z~ VxU) + ~-(V~,Z)h(X, U)+

+h(VxVyV~ Z) + h(V~Z, VyU) + ~(Z)h(X~ VvU)+

+X~(Z)h(Y, U) + ~(Z)[h(V~Y, U) + h(Y, VxU) + ~(Y)h(X, U)].

Subtracting from this equation the one obtained by interchanging X and. Y and the equation

IX, Ylh(Z, U) = h(Vi~x,ylZ~ U) + h(Z, V i x y l U ) + ~(Z)h([X~Y], U)

gives the desired result~ Q.E.Do

4. D U A L - P R O J E C T I V E F L A T S E M I - C O M P A T I B L E A F F I N E C O N N E C T I O N S

An affine connection V is said to be dual-projectively fiat if around each point there is a dual-

projective change of V to a fiat anne co~anectiom

Let ( M , h , V) be an n-dimensional pseudo-Riemannian manifold and V be an affine connection

semi-compatible with h. Let r = trh(Ric) denote the trace of the Ricci tensor of V with respect to

ho The dual-projective curvature tensor D W of V is defined by [4], [8],[9]:

(4o!) DW(X,y)Z = ~(X,Y)Z - h(Y, Z)M(X) + h(X, Z)M(Y),

94 Ivanov

where

(4.2) M ( X ) = - R i c e ( x ) + r / ( n - 1)X~

X, Y are vector ~.elds on M aad Rice(X) denote the Ricci operator, defined by

h ( R i c e ( x ) , Y) := Rio(X, Y )

The following results are a generalizations of the theorems which were proved for Riemannian maw

ifoids in [4] and for statistical manifolds in [9].

T H E O R E M 4.1o a) The dual-projective curvature tensor for an aJfine connection V semi-

compatible with a metric h is an invariant of the class of dual- projective transformations~

b) The dual-projective curvature tensor is an invariant of the conformal transformations of the

metric h.

T H E O R E M 4,2o Let (M,h , V) be an n-dimensional pseudo-Riemannian manifold and V be an

a Jfine connection semi-compatible with h b9 i-form r, Then:

i) If n >_ 3, then V is dual-projectively flat iff D W = O;

ii) I f n = 2, then D W is identically zero.

Let the curvature tensor R of V satisfies the condition:

t rR = -dr;

Then V is dual-projectzveiy fiat iff the ]bllowmg condition holds:

(4.3) ~r[X ~ (VxR ic# ) (Y ) I = r (Rie#(Y) ) .

Proof of Theorem 4.i. Let V be an a n n e connection semi-compatible with h by r a~.d V" be

dual-projectively equivalent to V. Prom (2~ using (3,a) for the curvature tensors R and R ~ of V

and V' respectively, we calculate:

(4.4)

whe.l'e

(4.5)

R' (X , g ) Z : R ( X , Y ) Z - h(Y, Z ) L ( X ) + h(X, Z ) L ( Y ) ,

L ( X ) = V x ~ ~ - (~ (X) + T (X) )~#

Contrac'~ing twice in (4.4) we get:

.r(X) = l - r a c e ( X ) + r / ( n - 1)X] - [ - R i c ' # ( X ) + r~/(n - i)X].

Substituting the last equality into (4.4) we get D W = DW' which proves a) of Theorem 4.1.

The condition b) follows immediately from (4ol) and (4.2)~

Q.E.D.

Ivanov 95

Further we need the following

L E M M A 4.3. Let n >_ 3 and V be an aJfine connection semi-compatible with h by ~- and with

zero dual-projective curvature tensor, and let V* be the aJ~ne connection semi-conjugate to V by

r. Then V* is Ricci-symmetric and the following formulae are valid:

(4.6) tr(R) = - d r ; tr(R*) = 0o

Proof of the Lemma 4.3. Contracting in (3.8) we obtain:

(4.7) tr(R*) + tr(R) = -d~-.

Since D W = 0 from (4.1) and (3.8) we get:

R*(X~ Y )Z = [M(Y, Z) + ( V ~ r ) Z - r(Y)r( Z)]X - [M(X, Z) + (V*xv)Z - r (X)r (Z) IY

where M(X~ Y) = h(M(X) Y) and M(X) is given by (4.2). Then the first Bianchi identity implies

that the ter~sor

[M(X,Z) + ( V ~ r ) Z - r( X)r( Z)]

must be symmetric, since n >_ 3. Using the expression (4.2) we get:

(4.8) tr(R)(Y~ Z) = - Ric(g, Z) + Ric(Z, Y) = -dr(Y , Z)~

Substituting (408) into (4.7) we get the proof of the Lemma 4.3. QoE~Do

Proof of Theorem ~.20 Let DW = 00 To prove that V is dual-projectively fiat it is sufficient to

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

(4.9) V x ~ # = ~ ( X ) + [~(X) + ~-(X)]~ #,

where M is given by (4.2) and X is a vector field on M.

Using D W = 0 and (3.6) the integrability conditions of the system (409) take the form:

(4A0) ( V x M ) Y - ( V y M ) X - r ( X ) M ( Y ) + r ( Y ) M ( X ) = - (da(X , Y) + dr(X, Y))c~ #.

From (4.9) using (3.6) and Lemma 4.3 we get da + dr = 0. Substituting this equality into (4.10)

we have that the integrability conditions of the system (4.9) are :

(4.11) ( V x M ) Y - ( V y M ) X = r ( X ) M ( Y ) - T(Y)M(X) .

We have to consider two cases.

CASE 1.n >_ 30 From the second Bianehi identity, using DW = 0 and (3.6) we obtain:

(4.12) av, x,y[h(Y, Z ) [ (VvM)X - ( V x M ) V - r (V)M(X) + r(X)M(V)]] = O,

where X, Y, Z, V are vector fields on M and rrv, x,y denotes the cyclic sum of V, X, Y Since n >_ 3

from (4.12) we get (4.11) which proves the sufficient part of i).

96 [vanov

C A S E 2.n = 2. The sufficient part of ii) follows from the following

L E M M A 4.4. I f n = 2 the conditions (4.11) and (4.3) are equivalent.

Proof o f the L e m m a $.~o Taking the trace in (4.11) and using (4.2) we get (4.3)~ For t.he inverse, if

(4.3) holds and n = 2 it is easy to verify (case by case) that [4.1i) also holds which proves Lemma

4~

Further,the necessary part of i) follows immediately from Theorem 4.1.If V is duM-projectively fiat

we have that (4.11) is valid. Then the necessary part ofii) follows from Lemma 4.4~ This completes

the proof of Theorem 4.2. Q .E .D .

5. A P P L I C A T I O N S T O A F F I N E H Y P E R S U R F A C E T H E O R Y . ,

In this section, we recall several definitions at~d preliminary facts on affine hypersurface theory. For

more details, see [10], [11], [15]o

Let M ~ be a smooth manifold of dimension n >_ 2o A pair (f , () is called an affine immersion of

M ~ into the (n+l)-dimensional airline space (R ~+1, D) equipped by the usual flat connection D,

if f is an immersion of M ~ into R ~+1 and ~ is a transversal vector field along f . For a given

affine immersion (f~ ~) of M ~ the induced torsion-free affine connection V and the induced second

fundame~tM form h are determined by:

(5.~) D x f . ( Y ) = f . ( V x Y ) + h ( X , Y ) { ,

where X and Y are vector fields on M K

If we change ~ to ~ = (~ + f . ( Q ) ) / ~ , where Q is a ;angent vector field on M ~ and X a nowhere..

vanishing function, then the corresponding objects change as follows:

(5.2) h = Ah; ~ ? x Y = V x Y - h ( X , Y ) Q ; r = r + ~ - d( lnA) ,

where ~ is ~he 1-form defined by ~?(X) = h ( X , Q ) .

If h is nondegenerate the immersiov., is said to be non-degenerate or regular. It is easy '~,o see from

(5.2) that this condition is independent of the choice of ~. In this case ~7 and V are dual-projective

equivalent~

For an affine immersion (f , ~) we also write

(5.3) Dx~ = - / , ( S X ) + ~ ( x ) ~

where S is the affi_ne shape operator and r is the transversal connection form for the immersion

(f ,~) . The affine immersion ( f , ( ) is said to be equiaffine [101 (or ~ a relative normal [15]) if the

transversal connection form r is zero. If moreover the so called apoiarity condition ( t r (Vh) = 0) is

satisfied the immersion is the classical Blaschke immersion [2].

Ivanov 97

Let R denote the curvature tensor of V. We have the following fundamenta l equations:

~ ( x , z ) z = h(u z ) s x - h (X , Z ) S Y , aar

(Vxh)(Y, Z) - (Vyh)(X, Z) = -r(X)h(Y, Z) + r(Y)h(X, Z), CodazziI.

( v x s ) Y - ( v y s ) x = ~ ( x ) s z - ~ ( Y ) s x ~ Co~azz~Ii.

h(X, SY) - h(Y, SX) = dr(X,Y), Ricci.

Conversely, these four equations are the sufficient integrabili ty conditions for the existence of a

local affine immersion of M n into R n+l ( see [13])

If the immersion ( f , ~) is non-degenerate an immediate consequence of the Gauss and Ricci equa-

tions is the following formula

t r R ( X , Z ) = - d w ( X , Z ) o

The Codazzi I equat ion impfies tha t the connection V induced from a non-degenerate affine im-

mersion ( f , ( ) is semi-compatible with the induced second fundamenta l form h by the t ransversal

connection 1-form v.

Fur ther we need the notions of the conormal map of a given immersion ( f , ( ) . Let Rn+l denote the

dual vector space to the vector space R ~+1 associated to the affine space R~+I . For each p E M s,

let v~ be a covektor (vp 6 R~+I) , uniquely determined by the conditions

(5.4) vp(~p) = 1, vp(f ,Y) = 0,

for every vector Y at p E M n. The map v : p E M ~ ~ vp C I~+1 is called the conormal map of

( f ,~ ) , and v is called the affine conormal. Using (5.1) and (5.3) from (5.4) we obtain:

(5.5) v.(Y)(~) = --T(Y); v.(Y)( f .Z) = -h(Y, Z),

From (5,5) it follows tha t the conormal map is an immersion M ~ ~ Rr~+l - {0}, if f is nonde-

generate. Then (5.4) and. (5.5) show tha t v itself is a t ransversal field to v (Mn) . Let we consider

v ( M ~) as a centro-affine hypersurface (by taking - v as a t ransversal vector at each point) . We can

then write the s t ructure equat ion :

(5.6) Dx(v,(Y)) = v,(V~cY) - h~(X,Y)v,

where V ~ is another torsion free affine connection on M ~ and h* is another symmetr ic (0.2) tensor

on M~. We have

P R O P O S I T I O N 5.1. For non-degenerate a,!fine immersion (f, ~) the following formula is valid:

(5 7) Xh(Y,Z) = h(VxY, Z) + h(Y V~:Z) + r(Z)h(X,Y).

Proof: Differentiating in (5.5), applying (5.1), (5.3), (.5.6) and (.5.5) we get the proof by a straight-

forward calculations. Q.EoDo

98 Ivanov

We note that (5.7) is proved in [15] in a more genera], situation.

The formula (5~ implies that the affine connection V* induced by the conormai map of a non-.

degenerate a n n e immersion (f , ~) is seml-conjugate to the induced connection V by the transversal

connection 1-form r o

Comparing the fundamentaI equations of Gauss, Codazzi and Ricci for non-degenerate affine im-

mersion (f , ~) with the conditions of Theorem 4.2 we get that the connection V is dual projective

flat. The converse is also true and it is called the fundamentM theorem for non-degenerate a n n e

immersions:

T H E O R E M 5.2. Let (M~h~V) be an n-dimensional connected and simply connected pseudo-

Riemannian manifold with metric h and torsion-free aJfine connection ~7. Let R be the curvature

tensor of V and r be a 1-form on M. Then the following conditions are equivalent:

i) There exists a non-degenerate a~ne immersion f : M n ~ t t ~+1 such that the induced funda-

mental form, the induced connection and the induced transversal connection form of the immersion

are h~ V, v, respectively. Such an immersion is unique up to a]fine transformation on R n+l.

ii) The affine connection V is semi-compatible with h by r and the unique aJfine connection V*

semi-conjugate to V by r is projectively flat and with symmetric Ricci tensor.

iii) The connection V is semi-compatible with h by v.

For n >_ 3 V is a dual-projective fiat affine connection.

For n = 2, V satisfies the condition: tr R = - d r and V is a dual-projective flat.

We note thet if v = 0 the equivalence of i) and ii) is the main result of [3] and equivalence of i) and

iii) is the main result of [8].

Proof: We reduce the general case (r r 0) to the case r = 0.

First of all we observe that from (5.2) and (5.6) if we change the transversal vector field ( of an

affine immersion (f, ~) by the rule $ = ~ + f~(Q), where Q is a "tangent vector field then the a n n e

connection X7 changes by a dual-projective transformation and the a n n e connection V* induced

by the conormal map does not change. If we take ~ = - r in (5.2) we have ? = 0. Using Theorem

4.1 and Theorem 4.2 it is easy to see that the general case r # O is reduced to the case T = 0 and

the results follow from the main resuits of [3] and [8]0 Q.EoD..

REMARK L if f : M n ~ R ~+1 is an arlene immersion then (5.2) and Theorem 4A imply that

DW does not depends on the normalization of the immersion. Thus, DW depends oIfiy on the

hypersurface f itself. In particular, for non-degenerate anne immersion, we have DW=0.

Conversely, (for dim _> 3 ) Theorem 5.2 implies that for given conformal class {h} of non-degenerate

metrics and given class {V} of semi-compatible with metrics of (h} anne connections~ there exists

nov.-degenerate hypersurf~ce affine immersion f~ such that {h} and {V} are realized as induced

structures from the immersion f if and on!y if DW=0.

Geometric formulations of the fundamentM theorem for hoiomorphic affine immersions is consid-

Ivanov 99

ered in [5] and for atone Kaehler immersic~ns, in I6].

I would like to thank Prof.Dr. Uo Simon for pointing out the eonformal invariaitee of the dual-

projective curvature tensor~ and also to the referee for the helpful advices during the preparation

of the finM form of the paper.

R E F E R E N C E S

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(1990), 221 - 231.

[4 i IVANOV, S., Symmetric quarter metric connections. Math. and Education in Math.,Proe. Spring Conf. UBM Sunny Beach (1987), 220 - 227.

[5] IVANOV, S., Complex semi-conjugate connections and the fundamental theorem in complex aJfine differential geometry.~ to appear.

[6] IVANOV~ S., On Radon's theorem for aJfine Kaehler immersions., to appear in Differential Geometry and its Applications, Proe. 5th Into Conf~ on Diff. Geom. and Appl. ~ Opera 1992.

[7] KUROSE, T., Dual connections and affine geometry, Math. Z. 203 (1990) 115 - 121.

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University of Sofia, Faculty of Mathematics and lnformatics, Department of Geometry~ bul. James Bouehier 5, i126 Sofia, BULGARIA.

Eingegangen am 22. November 1993; in r e v i d i e r t e r Form am 25. J u l i 1994