173

APPENDICES

A p p e n d i x A. A digest of the theory of connections.

Let us recall briefly some well-known eoneepts and formulas from

the theory of connections. (Cf.[KN I].) "Differentiable" always means

"of class C ~ ".

Let M be a differentiable manifold, G a Lie group and P(M,G)

a principal fibre bundle over M. Thus G acts on P freely to the

right and this action is simply transitive on each fibre of P.

For u aP, let Gu denote the subspace of Tu(P ) consisting of

vectors which are tangent to the fibre through u. A connection

in the bundle P(M,O) is an assignement of a subspace Qu of T (P)

to each u ~ P such that

(a) Tu(P) = Gu + Qu (direct sum of vector spaces)

( b ) Qua = ( R a ) * Q u for e v e r y u e P a n d a ~ G w h e r e R a i s t h e

transformation of P induced by a ~ G, R u = ua, a (c) Qu depends differentiably on u.

For each u e P, G is called the vertical subspace of Tu(P), u and Qu is called the horizontal subspace of Tu(P ) with respect to

the connection V Each tangent vector X 6Tu(P ) can be uniquely

written as X = Y + Z, where Y ~ G u and Z ~ Qu"

If co is a vector-valued differential k-form on P(M,G), then

the exterior covariant differential D~ (with respect to P ) is de-

fined by the formula

(D~) (X1, .... Xk+l) = (d~) (I'LXI, .... hXk+l)

where hX. denote the horizontal components of X.. i 1 Given a connection ~ in P~ we define a 1-form ~ on P with

values in the Lie algebra ~ of G as follows: for each X ~Tu(P )

take the vertical component Y of X and define ~ (X) as the unique

element A of the Lie algebra ~ such that (d(u.exp tA)/dt)t=o = Y.

Now, ~ (X) = 0 if and only if X is horizontal. The form ~ is cal-

led the connection form of the given connection V .

In particular, the set of all tangent frames to a manifold M is

a principal fibre bundle L(M,GL(n,R)), called the principal frame

bundle of M.

An affine connection on M is a bilinear map ~: ~(M)~ ~(M)--~

' ) ~(M) written in the form (X,Y) m~xY , satisfying the follo-

wing axioms~

174

(i) VfxY = f(VxY) x, Y e ~ ( M ) , f ef(M).

(ii) VxfY = (xf)Y + f(VxY)

It is well-known that, for a point p 6 M, the value (~xY)p depends

only on the value X and on the germ of Y at p. Thus, for any P

vector u &Tp(M) and any local vector field Y in a neighborhood

of p we can define the vector Qu Ye Tp(M), The vector ~u ¥ is cal-

led the eovariant derivative of Y with respect to u.

There is a bijeetive correspondence between the connections in

the principal frame bundle L(M) = L(M,GL(n,R)) and the affine con-

nections on M. Therefore, we often identify the corresponding ob-

jects.

Let now G C GL(n,R) be a Lie subgroup and P(M,G) ~ L(M,GL(n,R))

a principal subbundle, i.e., a G-structure on M. Each connection in

P(M,G) can be extended in a unique way to a connection in L(M).

Let P be a connection in P, ~ its connection form and ~ = D~

the exterior eovariant differential of ~ . 3~ is called the curvatu-

re form of the connection r We have the first structural equation

1 ( A1 ) d~(X,Y) =-~[~(X),~(Y)] + _O.(x,Y) X, Y e T u ( P ) , u 6 P ,

(with values in the Lie algebra ~ c g l ( n , R ) ) .

Further, let us recall that the canonical form ~ of P (inde-

pendently of P ) is the Rn-valued 1-form on P defined by 0(X) =

= u-l.~(X) for u eP, X ~Tu(P ). Here ~: P---~ M denotes the bundle

projection, and eaeh u ~ P is considered as a map of R n onto

%(u)(M). The torsion form ~ of ~ on P is defined as the exte-

rior covariant differential DO of 0 with respect to r . It is

uniquely determined by the second struetural equation

dS(X,Y) = -~ [~ (X) 'O(Y) - ~ ( Y ) - O ( X ) ] + @ (X,Y), ( A2 )

for X, YeTu(P), u~P (with values in Rn).

Consider the torsion and curvature tensor fields of the corres-

ponding affine corkneetion on M:

( A 3 ) T(X,Y) = V x Y - ~ ' y X - ~ X , Y ] x ,Y,Z ~ ( M )

( A4 ) R ( x , Y ) z = [Vx ,Vy~z - V~x,y~z

Then T and R are also defined by the following formulas by means

of ~ and -(~ respectively:

175

( ~5 ) T ( x , Y ) = u ( 2 e ( ~ , ? ) ) f o r x , Y ~ T x ( M )

( A6 ) R(X,Y)Z = u ( 2 n ( ~ , F ) ( u - l z ) ) f o r X, Y, Z e T x ( M ) ,

where u is any element of P such that X(u) : X, and ~, ~&%(P)

are arbitrary lifts of X, Y respee%ively.

Remark. Considering u.(2/l(~,V)) as a " s i n g u l a r frame" at

can also write instead of (A6)~

( A7 ) R ( x , Y ) Z = u . ( 2 ~ ( ~ , T ) ) ( u - l z ) .

X~ we

For the torsion tensor field T and the curvature tensor field R

of an affine connection ~ on M we have the well-known Bianehi

identities:

( A8 ) ~ ( R ( X , Y ) Z = ~ { T ( T ( X , Y ) , Z ) + ( V x T ) ( Y , Z ) ~

( A9 ) ~ { ( V z R ) ( x , Y ) + R ( T ( X , ~ ) , z ) ~ : o .

(Here ~ denotes the cyclic sum with respect to X~ Y, Z.)

The following result is known from the theory of connections:

Let P be a connection in L(M), ~ the corresponding affine eon-

neetion on the manifold M, and f: M--+M a diffeomorphism. Then

the connection ~ is invarian% with respect %o the induced automor-

phism ~: L(M) ~ L(M) of the frame bundle if and only if V is in-

variant with respeet 9o f in the sense that

( AI0 ) f~(~X Y) = ~f~X f~Y for every X, Y~ ~(M).

f is then called an affine transformation of the affine mani-

fold (M,V). More ~enerally, if (M,V)~ (M',V') are two manifolds with the

affine connections, then a diffeomorphism f: M---~M" is called an

affine map if (AI0) holds with the symbol ?" on the right-hand side.

Let ~ be a fixed connection in L(M). Let I ~ R be an arbit-

rary interval. A differentiable curve 6: I--->L(M) is said to be

horizontal if all the tangent ~eetors (d~/dt) are horizontal with

respect to ~ To any differentiable eurve ~: I---->M and any pair

(to,Uo) where to ~ I and Uo~ ~-l(~(to) ) there is a unique hori-

zontal e~r~e ~ I--~ L(M) such that Z= = ~ and #(%0 ) = u ° If

I = <a,b ) is a closed finite interval, then we can assign in this

way a unique frame Ul$ ~-l(~(b)) to any frame Uo6 =-l(~(a)). Hen-

176

ee we get an isomorphism h ~ T ( )M----> )M which is independent l ~ ~ a T~(b of the choice of u e ~- (~(a)) and it is called the parallel trans-

o port along ~.

A differentiable map v(t): I > T(M) is called a vector field a-

long ~: I---> M if v(t) 6T (t)M for each t ~ Z . Such a vector field

v(t) is said to be parallel if~ for any subinterval < tl~t2>C I~ the

vector v(t2) is the parallel translate of v(tl) along the are

7 1 < t l , t 2 ) "

A c u r v e ~ : I - - - ~ M i s c a l l e d a g e o d e s i c i f t h e t a n g e n t v e c t o r

field v(t) = d_~ along ~ is parallel It is well-known that, for dt

each point p~M and each tangent vector X~Tp(M), there is a unique

maximal geodesic ~X: X----~ M (Z ~ O) w i t h the properties ~x(O) = p,

(d~x(0)/dt) = X. The affine connection ~ corresponding to ~ is

said to be complete if all maximal geodesies ~X ~ X ~T(M), are defi-

ned on the whole real line R.

Let ~ : I---> M be a regular differentiable curve. Then the paral-

lelism along ~ can be expressed in terms of the affine connection

as follows: a veetor field v(t) along ~ is parallel if and on-

ly if Vd_x(v(t)) = 0 for all t el. Here Vd_ v(t ) is uniquely defi-

dt dt

ned~ for each t~ I, as the eovariant derivative ~d~W, where W is

dt

an arbitrary (differentiable) local vector field in the neighborhood

of ~(t) such that v(~) = W (~) for all ~ near t .

For every differentiable veetor field X on M, we have defined

the operator ~X: Y---~X Y on vector fields from ~(M). There is a

unique extension of this operator to the algebra ~(M) of all dif-

ferentiable tensor fields on M such that the following holds=

(i) V x is a derivation of ~(M) with respect to the tensor

product~

(ii) V X preserves the type of any tensor field~

(iii) ~X commutes with the contractions,

(iv) VX Y has the usual meaning for any vector field Y, and

Vxf = Xf for any differentiable function.

Let (M,V) be a manifold with an affine connection. To any point

p ~M there is a neighborhood No of the null vector in Tp(M) such

that the geodesics ~X are defined at the value t = l~ for all

Xe N O . Define the exponential map EXpp: N o > M by the formula

EXpp(X) = ~X(1), XG N o . The image EXpp(No) = Np is called a normal

177

neighborhood of p if the map EXpp is a diffeomorphism of N o on

Np. A normal neighborhood always exists to each point p~ M.

The normal coordinates in a normal neighborhood are defined by

the map (xl, .... Xn)~ > EXpp(xlel+...+Xnen) where 4 el, .... en~ is a

basis in Tp(M).

Let ~: < a,b > ~M be a closed piece-wise differentiable curve~

i.e. such that ~(a) = ~(b) = p. We can define the parallel transport

along the whole of ~ using the parallel transports along the diffe-

rentiable pieces of ~ . All transformations h~: Tp(M)~Tp(M) ari-

sing in this way form a Lie group ~(p), called the holonomy group of

M with reference point p. If M is connected, then all holonomy

groups ~(x), x~M, are mutually isomorphic. Choosing a frame u o at p~M, we can define the holonomy group @ (u o) ~ GL(n,~) with

reference frame U o ; here ~ (Uo) ~ ~(p). Any two holonomy groups

(Uo) , ~ (Ul) , (Uo,U 1 g L(M)) are conjugate to each other in GL(n,R).

A p p e n d i x B. Some theorems from differential geometry.

Xffine maps, isometries, holonomy. (Cf. [KN I] for more details.)

B_~I). Let M, M" be manifolds with affine connections (or Riemannian

manifolds) and let f,g, be affine maps (or isometries, respectively)

of M into M'. If M is connected and (f~)x : (g~)x for some

x gM, then f and g coincide on M.

B2). Let M be a connected~ simply connected analytic manifold with

an analytic affine connection. Let M" be an analytic manifold with

a complete analytic affine connection. Then every affine map fu of

a connected open subset U of M into M" can be uniquely extended

to an affine map f of M into M'.

B3). Let M and M" be analytic Riemannian manifolds. If M is

connected and simply connected and if M" is complete, then every

isometric immersion fu of a connected open subset U of M into

M" can be uniquely extended to an isometric immersion f of M in-

to M'.

B_~). Let M and M" be connected and simply connected~ complete a-

nalytic Riemannian manifolds. Then every isometry between connected

open subsets of M and M" can be uniquely extended to an isometry

between M and M'.

BS). Let M be a connected differentiable manifold with an affine

178

connection, and let ~(x) be the holonomy group of M with refe-

rence point x. Then the Lie algebra ~(x) of ~(x) is equal to

the subspace of End(Tx(M)) spanned by all endomorphisms of the form

(~R)(X,Y) = ~-~R(~X,zY)~, where X, Y~ Tx(M ) and ~ is the paral-

lel transport along an arbitrary piecewise differentiable curve star-

ting from xo

Affine manifolds with parallel curvature and parallel torsion.

B_~6). Let M be a differentiable manifold with an affine connection

such that ~T = VR = 0. With respect to any atlas consisting of nor-

mal coordinate systems, M is an analytic manifold and the connec-

tion V is also analytic.

B7). Let M and M" be differentiable manifolds with the affine

connections V and V" respectively. Assume VT = ~R = 0, V'T" =

= ~'R" = 0. If F is a linear isomorphism of T (M) onto T (M') Xo Yo

and maps the tensors T and R at x into the tensors T ° x x o Yo

O O

and R" at Yo respectively, then there is an affine isomorphism f Yo

of a normal neighborhood U onto a normal neighborhood V such x Yo o -i

= = F. Explicitly, we have f = Exp o FuEXPx that f(Xo) Yo and f~x O Yo 0

BS). Let M and M" in BT) be connected, simply connected and com-

plete. Then there exists a unique affine isomorphism f of M onto

M" such that f(Xo) = Yo and the differential of f at x ° coinci-

des with F.

Bg). Let M be a differentiable manifold with an affine connection

such that VT = VR = O. Then the Lie algebra g(x) of the holonomy

group ~(x) is spanned by all endomorphisms of the form R(X,Y),

x, Y 6 Tx(M).

Remark. The proof of B9) follows immediately from BS) and from the

property "~R = R.

179

R E F E R E N C E S

[B]

[Boll

[Be2]

[Bo3]

[Ca]

[Ch]

[CPI]

[cP2]

[cw]

IF1]

[E21

[E3]

[GLZ]

[GL2]

[Gr]

[H]

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M.Cahen, M.Parker: Sur des classes d'espaces pseudo-rieman- niens sym6triques. Bull. See.Math.Belg. 22 (1970), No 4, 339-354.

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M.Oahen~ N.Wallaeh: Lorentzian symmetric spaces. Bull.Am.Math. See. 76 (1970), 585-591.

A.S.Fedenko: Homogeneous ~ -spaces and spaces with symmetries. Vestnik BGU (Minsk), Serie I, 1972, No 2, 25-30 (Russian).

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A.Gray: Riemannian manifolds with geodesic symmetries of order 3. J.Differential Geometry 7 (1972)~ No 3-4, 343-369.

S.Helgason: Differential Geometry and Symmetric Spaces. Pure and Appl.Math. No 127 1962~ Academic Press, New York and London.

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[H~ S.Helgason: Differential Geometry, Lie Groups, and Symmetric spaces, Pure and Appl.Math., 1978, Academic Press, New York - San Francisco - London.

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0.Kowalski: Riemannian manifolds with genersl symmetries. Math. Z. 136 (1974), No 2, 137-150.

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182

S U B J E C T I N D E X

admissible s-structure

affine connection

complete

affine

locally symmetric space

map

reduetive space

symmetric space

s-structure

transformation

amal~amation (of s-manifolds)

amalgamating deeomposition

automorphism of a regular s-manifold

autoparallel submanifold

Bianchi identities

canonical connection

of a local regular s-structure

of a reductive homogeneous space

of a regular s-manifold

canonical form on the principal frame bundle

Cartan (-) connection

eenterless infinitesimal s-manifold

characteristic variety (in the theory of ei~envalues)

closed subset of eigenYalues

complexification

of a regular s-manifold

of a ~eneralized symmetric Riemannian space

connection form

eovariant derivative

curvature form

curvature tensor field

derivation (of a manifold with multiplication)

direct product of regular s-manifolds

direct sum of infinitesimal s-manifolds

iii

173

176

45 3-75

41

45 168 175

99 99 147 89

33, 175

25, 70 29

47, 52 174

41 1 0 0

116

108

103 105 173 174 174 174

48 93 914

183

elementary transveetion

exterior covariant differential

Fitting Lemma

Fitting 0-component, l-component

foliations (in regular s-manifolds, in generalized

symmetric spaces)

free point (of a Riemannian manifold)

generalized affine symmetric space

of semi-simple type

of solvable type

primitive

unitary

generalized pointwise symmetric R. space

generalized symmetric Hermitian space

generalized symmetric Riemannian space

proper

without infinitesimal rotations

group of automorphisms of a regular s-manifold

group of transveetions

of an affine manifold

of a regular s-manifold

Hermitian regular s-structure (s-manifold)

holonomy ~roup

induced G-invariant metric

infinitesimal

isomorphism of regular s-manifolds

model of a (locally) regular s-manifold

s-manifold

integrable tensor field S

invariant

almost complex strueture

almost Hermitian structure

complex structure

foliation

infinitesimal submanifold

Kahlerian structure

submanifold

tensor field

57 173

53 53

91, 98

128

111

114

114

115

113

164

102

8

134 lO4 47

36 58

102

177

ii

74

74, 82

73, 82

88

84 86 86 91 90 87 90 84

184

irreducible

generalized affine symmetric space

infinitesimal s-manifold

regular s-manifold

Riemannian space

set of eigenvalues

isomorphism

of infinitesimal s-manifolds

of regular s-manifolds

k-symmetrie Riemannian manifold

Killing form

lattice point (in the theory of eigenvalues)

Levi decomposition

local

automorphism

geodesic symmetry

isomorphism

regular s-structure

symmetry

local regular s-triplet

effective

prime

locally regular s-manifold

Riemannian

Lorentzian symmetric space

metrizable regular s-manifold

nearly Kahler structure

Nijenhuis tensor

order

of a free point

of a generalized affine symmetric space

of a generalized symmetric Riemannian space

of a periodic tensor structure

of an s-structure

para]lel transport

periodic tensor structure

m61 94

99 20

109

74, 82

74, 81

8 ii

121

97

69 l, 45

69 23, 68, 81

24, 69 76 77 77 69 81

172

65

171 86, 88

128 113

8 128

4

176 128

185

pointed

affine manifold

regular s-manifold

prime

reductive homogeneous space

regular homogeneous s-manifold

primitive generalized affine symmetric space

principal frame bundle

proper generalized symmetric R. space

pseudoduality

pseudo-Riemannian symmetric space

41

6o

41

53 115

173

134

106, 107

172

reducible

infinitesimal s-manifold

regular s-manifold

set of eigenvalues

Reduction Theorem (in the theory of connections)

reductive homogeneous space

regular s-manifold

of finite order

of semi-simple type

of solvable type

regular homogeneous s-manifold

regular s-structure

parallel, non-parallel

Riemannian space

generalized symmetric

homogeneous

k-symmetric

locally symmetric

symmetric

without infinitesimal rotations

94

93

lO9

36

i0, 27

47

62

96

96

53 74, 81

22, 23, 168

8

9 8

1

1

65, 104

semi-simple symmetries

set of eigenvalues of a regular s-manifold

similarity (of locally regular s-manifolds)

s-structure

of finite order

of infinite order

of order k

regular

Riemannian

112

108

69

4

14, 113

113

4

7, 74, 81 4

186

structural equations (of a eornuection)

symmetry

affine

geodesic

generalized

in a distributive groupoid

Symmetric space (by O.Loos)

system of eigenvalues (of a regular s-structure)

O-variety (in the theory of eigenvalues)

torsion form

torsion tensor field

weakly invaria~t

subspaee

submanifold

tensor field

174

45 l, 45

2, 47

46 , 47

45

ll8

116 174 1?4

92

92

88

187

N O T A T I O N INDEX

(See also the list of standard denotations. )

s i P

S i P

(M,~) 1

t s x : x ~ M~ 2

CI(~. sx~ ) 2

Cl(Sp) 4

T(M,p) 4

I(Mp) 4

0(%) 5

Cl(Sp) 5

s 6

cl(~ ~x~ ,p) 7

G ~, (G~) * 9 , 53

K£ 4l

~ ( M , v ) 36

'rr (M) 36, 58

Tr'*(M) 3 9

(M,~) 47

Ant (M) 47

Der(M)

-i x °y

L(~)

(M, ~s~ )

(G,~, ~ )

gOA ' glA

~(M,-I% ~ )

c la (-~ s ~ )

~ n

.,A. n

n

0

@

121,

49

49

50

52

53

53

58

65

117

119

119

12o

121

i73

174

174

174