MARTIN LINDEN AND HELMUT RECKZIEGEL

O N A F F I N E M A P S B E T W E E N A F F I N E L Y

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

ABSTRACT. Every affine map between two affinely connected manifolds is the composition of an affine submersion and an affine immersion; and the inverse image of an autoparallel submanifold by such maps is the union of autoparallel submanifolds. Furthermore, the Lie group of all affine transformations of an affinely connected manifold carries the compact-open topology.

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

In this article we present three theorems from the theory of an n e manifolds, which are generalizations of results of linear algebra. In connection with manifolds the attribute 'aNne' shall express that they are equipped with linear connections. Of course, the affine spaces of linear algebra are an n e manifolds, if we choose the canonical differentiation of vector fields as covariant differentiation. The results of linear algebra which shall be generalized are the following:

1. The image f(N) of an anne map f : N ~ M is an an n e subspace of M. 2. The inverse image f - I ( L ) of an aNne subspace L c N by an affine map

f : M ~ N is an aNne subspace of M. 3. The Lie group of all affine maps of an affine space carries the

compact-open topology. In the context of affine manifolds we use the notion of affine maps in the sense of Eells and Sampson (see [3]); in affine spaces of linear algebra this notion coincides with the classical one. Instead of 'aNne subspaces' we speak of 'autoparallel submanifolds' in the theory of affine manifolds (see [6, II, p. 53]); exact definitions are given in Section 1.

The simplest example of an affine map into an arbitrary a n n e manifold M is the parametrization 7:1 ~ M of a geodesic; as 7(I) may have self-intersections, we cannot expect submanifolds as images of affine maps in general. The appropriate modification of the first problem, therefore, is to ask whether every affine map f can be represented as the composition i o s of an aNne submersion s and an affine immersion i. In Theorem 1 we give a positive answer. In the special ease of Riemannian manifolds Vilms already solved the problem under additional completeness conditions (see [-9, Th. 2.2]). However, the method of his proof cannot be used in the general setting. Instead we construct 'universal' affine immersions in Section 2. They are also used to treat

Geometriae Dedicata 33: 91-98, 1990, © 1990 Kluwer Academic Publishers. Printed in the Netherlands.

92 M A R T I N L I N D E N A N D H E L M U T R E C K Z I E G E L

the second problem. Again the example of geodesics shows that the inverse image of an autoparallel submanifold by an affine map is not an autoparallel submanifold in general; but in fact it is the union of autoparallel submanifolds of different dimensions (Theorem 2).

In contrast to the first two questions, the third one is true without any modification: For every connected affine manifold M the Lie transformation group 9.I(M) of affine diffeomorphisms M ~ M carries the compact-open topology (Theorem 3). In the case that M is geodesically complete, this result was already obtained by Nomizu in 1953 (see [7]). But in general investigations the authors use topologies of 92(M), which seem to be slightly finer as the compact-open topology (see [4] and I-6, I, p. 229]). Our contribution to this question is rather small, as we have only combined Lemma 2 of Hano and Morimoto [4] with Kobayashi's description [5, p. 41] of 9.I(M). Since the proof of Lemma 2 of [4] is of geometric importance, we shall repeat it here in modern terminology; as profit, we can offer an improved version of their result. As application of Theorem 3 we state that for every pseudo-Riemannian manifold M the Lie group ~(M) of isometries also carries the compact-open topology; this result is widely known in the positive-definite case.

1. P R E L I M I N A R I E S A N D N O T A T I O N S

Let M and N be connected affine manifolds. 1 The covariant derivative, torsion tensor and exponential map of M will be denoted by V, T and exp: ~ ~ M, respectively; furthermore we put ~p .'= ~ c~ TpM and expp := exp I ~p for all p ~ M (see [6]). The corresponding data of N are denoted by the same symbols; but sometimes - to avoid confusion - we shall also use indices M and N. A differentiable map f : N ~ M is said to be affine iff it 'commutes with covariant differentiation':

(1) V ~ f , Y = f ,V~Y

for all vector fields X and Y of N; in more geometric terms: f is affine iff for every parallel vector field Y along a curve of N the image f , Y is also parallel (see [3], [9]). f is said to be totally 9eodesic iff the image f o 7 of each geodesic 7 of N is a geodesic of M, i.e. iff

(2) f o exp s = exp M o f , ] ~N.

1 In this article all manifolds, maps . . . are supposed to be of class C °°.

AFFINE MAPS 93

P R O P O S I T I O N 1. (a) f is affine if and only if f is totally geodesic and

f , rN(u,v) = r M ( f , u , f , v ) for all p e N and u, ve TpN.

(b) I f f is totally geodesic, then f has constant rank; if, furthermore, f is bijective, it is therefore a diffeomorphism.

Proof. The proof of (a) can be found in [9]; (b) follows from (2) by use of

normal neighbourhoods and Sard's theorem.

A further important notion in the theory of affine manifolds is that of autoparallel submanifolds: An (immersed) submanifold N of M is said to be autoparallel iff for all vector fields X, Y of N the covariant derivative V~ Y is tangent to N again; in more geometric terms, N is autoparallel if the tangent

bundle TN of N is invariant with respect to parallel displacement (in M) along

curves of N.

2. UNIVERSAL AFFINE IMMERSIONS

Let M be an affine manifold, and let r be an integer with 0 < r < dim M. We suppose the existence of an affine map into M of rank r.

T H E O R E M 1. There exists an r-dimensional affine manifold S and an affine immersion i: S ~ M, which has the following universal property: For each affine map f : N ~ M of rank r from any affine manifold N into M there exists exactly one affine submersion f : N ~ S, such that

(3) f = io f .

REMARK. As i: S ~ M is the solution of a universal problem, it is unique up to affine diffeomorphisms. Moreover, it has the following property:

(4) Vp, q~S: ( i , TpS = i, T q S ~ p = q).

Of course, in general S is not connected.

Proof. Let ~ denote the class of all affine maps of rank r into M. On account of our assumptions, ~ is not empty. The basis of the construction of S is the following local information, which follows immediately from the rank theorem:

ASSERTION 1. For every (f : N ~ M) ~ ~ and every point p ~ N there exists a neighbourhood U ofp in N such that A := f (U) is an r-dimensional regular 2 and autoparallel submanifold of M and f [ U is an affine submersion onto A.

2 That is, the topology of A is induced by the ambient space M.

94 M A R T I N L I N D E N A N D H E L M U T R E C K Z I E G E L

As consequence, the set 6 of all r-dimensional regular, autoparallel submani-

folds A c M is not empty. For each A ~ 6 and p ~ A let Ap denote the germ of

A at p, i.e. Ap = {B ~ 61 there exists a neighbourhood U of p in M such that

A n U = B n U}. In virtue of the autoparallelity of A, for each p E A there exists a neighbourhood V of 0 e TpM such that

expp(TpA n V) ~ Ap.

Therefore, we obtain: If A, B ~ 6 have a common point p, then

(5) Ap = Bp o TpA = TpB.

Now, we denote the set of all these germs Ap(A ~ 6, p ~ A) by S and the map

S ~ M, Ap ~ p by i. This set S can canonically be equipped with the structure of an r-dimensional C oo manifold such that for each A E 6

A'.={ApIp~A}

is an open set of S and the restriction i[ A: A ~ A a C °o diffeomorphism; see [1,

5.8.12]. In particular, i is an immersion. Furthermore, on S we can canonically

define a covariant derivative: If X, Yare vector fields of S, then for every A ~ 6

the vector field (V~i, Y)[A is tangential to A, since A is autoparallel; hence

(V~i, Y)p ~ i, TpS for all p ~ S; thus there exists one and only one vector field of S which we denote by V s Y such that i ,Vx s Y -- Vx~i, Y; obviously the operator

V s satisfies the Koszul axioms of liner connections. With respect to this

connection i is an affine map because of the last formula. In contrast to the

general construction in [1], in our situation we get

ASSERTION 2. S is a paracompact manifold.

In fact, from (5) we deduce immediately that S is a Hausdorff space. Since i is an immersion, we can therefore deduce the paracompactness of S modifying an

argumentation of Chevalley [2], as we learned from Professor P. Dombrowski, Cologne: Let C be a connected component of S, d a countable atlas of Coo

charts of M and I the finite set of multi-indices v = (vl . . . . . v,) with

1 ~< vl < -.. < vr ~< m ,= dim M. For each chart c ~ d with coordinate func-

tions xl . . . . . Xm: U ~ ~ and for each v ~ I let ~(c, v) denote the set of points p ~ i - ~ ( U ) n C such that the map ~p.'=(x . . . . . . . xvr)oi has rank r at p. By

means of [2, p. 97, Lemma 3] each connected component of ~(c, v) has countable topology. As the collection of all sets ~(c ,v ) (c~d , v~I) is a

countable open covering of C (remember that i is an immersion!) [-2, p. 96, Lemma 2] therefore shows that C, too, has countable topology. Thus

Assertion 2 is proved. To finish the proof we have to establish the universal property for i: S ~ N.

A F F I N E MAPS 95

For that let ( f : N ~ M)eq~ and p e N be given. Then we may assume the situation of Assertion 1, by which we can define f(p)'.= Am. By means of formula (5) this definition is independent of the choice of U; thus we have f l g = (ilA)-1 o (fl U). As ilA is an affine diffeomorphism onto A and f l g an affine submersion onto A, f is an affine submersion into S with i o f = f . Furthermore, it follows immediately that f is determined uniquely by f . Finally, the construction of S and statement (5) imply formula (4).

3. T H E I N V E R S E I M A G E OF AN A U T O P A R A L L E L S U B M A N I F O L D BY

AN A F F I N E MAP

In this section we show that the inverse image of an autoparallel submanifold by an affine map is the disjoint union of a finite number of autoparallel submanifolds; more exactly:

THEOREM 2. Let f : M ~ N be an affine map, L an autoparallel submanifold of N,

¢p ..= {v e TpMI f , v e TI(~)L } for every p e f - I(L),

and

K,,= {pe f - l ( L ) ] d i m ~p= r} for every reN .

Then each of the subsets K,, which is not empty, can canonically be equipped with the structure of an r-dimensional autoparallel submanifold such that TpKr = ~p for every p e Kr.

REMARK. In contrast to the general theory of submanifolds, in our situation we need not demand the transversality of f and L; here the transversality would even imply f - I ( L ) = K, with r = dim M - dim N + dim L.

Proof. First we will illuminate the local situation.

ASSERTION. In the situation of Theorem 2for every point Po e Kr there exists an open neighbourhood U of Po in M and an open neighbourhood B of qo := f(Po) in L such that A:=( f IU) - I (B) is an r-dimensional autoparallel submanifold of M with Poe A c Kr; moreover, Vp e A: TpA = ~p.

To prove this statement we choose normal neighbourhoods of Po (resp. qo); more exactly, let V c 9p~ (resp. 17" c ~qSo) be an open neighbourhood of the zero vector in TpoM (resp. TqoN ) such that the exponential map o fM (resp. N) is a diffeomorphism from V (resp. 17) onto an open neighbourhood U of Po (resp. U of qo); additionally we may assume that B ,= exP~o(TqoL ~ 17) is an open neighbourhood of qo in L and that f . (V) is contained in 17". Then A := exppo(~po ~ V) is an r-dimensional regular submanifold of U; because of

96 MARTIN LINDEN AND HELMUT RECKZIEGEL

formula (2) it equals ( f [U) - I(B) and Vp e A: TpA = {v ~ TpM [ f , v ~ Tytv)L} = iv- As f is affine and L autoparalM, the latter formula immediately implies the autoparallelity of A.

In the cases r = 0 and r = dim M, Theorem 2 follows trivially from the Assertion. In the other cases we use the universal r-dimensional affine immersion i: S --* M of Theorem 1 to finish the proof. Because of the preceding Assertion and statement (4), the set G := (q e S [i(q) e Kr and i, TqS = ~i(q)} is an open part of S and the affine immersion i[ G is an bijection onto K,. Carrying the differentiable structure of G to K, by i[ G the set K, obviously becomes an r-dimensional autoparalM submanifold of M with TpK, = ¢p for every p s K,.

4. THE SPACES (5(N,M) AND 91(N,M)

For two connected affine manifolds M and N let if(N, M) denote the space of all continuous maps N ~ M equipped with the compact-open topology. The sets ffi(N, M) and 9/(N, M) of all totally geodesic resp. affine maps N -~ M shall be considered as topological subspaces of ~(N, M). In the following we repeat the fundamental investigation of Hano and Morimoto [4] making use of the exponential map in order to prove

T H E O R E M (Hano and Morimoto [4, Lemma 2]). (a) 9.1(N, M) and ffJ(N, M) are closed subspaces of ~(N, M).

(b) I f ( f , ) is a sequence in {b( N, M) converging to f ~ ffi( N, M ), then for every v ~ TN the sequence (f , ,v) converges to f , v in TM.

Proof. Let (f,) be a sequence a in ffi(N, M) converging in ~(N, M) to a map f ~ if(N, M). Simultaneously we shall prove f ~ 15(N, M) - hence ~(N, M) is closed in if(N, M) - and statement (b). For that let p be an arbitrary fixed point of N, put q .'= f(p) and denote the canonical projection T M -~ M by re. As the map E: ~ ~ M × M, v ~ (rr(v), exp(v)) has maximal rank at the zero vector of

TqM, there exists a neighbourhood U of q and a neighbourhood W c ~ c~ re- I(U) of the zero section of TMI U, such that E[ Wmaps Wdiffeomorphically onto U x U. This configuration has the following crucial property:

(6) If c: [0, 1] ~ M is a geodesic with c([0, 1]) c U, then ~(0) e W.

Furthermore, let us choose a starshaped neighbourhood V= ~ of the zero vector of TpN with f o exp,(V) = U. If, now, v ~ V is an arbitrary vector and ? the geodesic t .-, exp(tv), t ~ [0, 1] in N, then f(?([0, 1]) is contained in U. As (fn) converges to f with respect to the compact-open topology, nearly all

a As E(N, M) satisfies the second axiom of countability, we may argue by sequences.

AFFINE MAPS 97

geodesic arcs f . o ? run in U, lim(f, o ?(0)) = f o ?(0) and lim(f, o y(1)) = f o ?(1). Applying (6) to these geodesics we find that nearly all vectors f . , v = (jr, o ?)(0) are contained in W and that these vectors satisfy the equation E(f , ,v) =

(f,(7(0)), f,(7(1))). Consequently, the sequence (f,,,v) converges in T M to a vector w • W satisfying E(w) = ( f o 7(0), f o ?(1)), i.e. w • TqM and expq(w) = f o expp(V).

As v was arbitrarily chosen in V and the maps f , , : TpN ~ TI.(p)M are linear, we even have proved the existence of a linear map A: TpN ~ TI(p)M

such that

(7) Vv • ¥ : f o exp~(v) = exp~(Av).

Thus, f is differentiable on a neighbourhood of (the arbitrary point) p and A is the tangent map f , at p. Formula (7), therefore, shows f • f f~ (N ,M) . Furthermore, statement (b) is obviously true for all v • TN. Finally, if in the preceding situation the maps f . are affine, then we can combine the last results with part (a) of Proposition 1 and deduce that f is affine, too. Hence also 9.I(N, M) is closed in ~(N, M). []

Now we use the theorem of Hano and Morimoto and Kobayashi's investiga- tion [5, Chap. II.1] to prove:

THEOREM 3. For every connected affine manifold M the Lie transformation

9roup 9.I(M) of affine diffeomorphisms M ~ M carries the compact-open topology.

Proof. Let ZL denote the topology of the Lie group 9A(M) and Zco the compact-open topology of 9~(M). From the general theory of transformation groups it is well known that ZL is finer than Zco. To prove the converse we denote the linear frame bundle of M by LM, choose an arbitrary frame (Vl,. . . , Vm) • L M and define the map ~P: 9~(M)~ LM, f ~ ( f , vl , . . . , f , Vm). As each map f • 9.1(M) is uniquely determined by ~( f ) , this map is injective. From [5, p. 41] we know that the 'orbit' N ,= ~(9.1(M)) is a closed submanifold of L M and that the topological and differentiable structure of the Lie group ~(M) can be obtained by pulling back these structures from N via 4). If now (f,) is a sequence in 9.I(M) converging to a map f • ~(M) with respect to Zco, then the theorem of Hano and Morimoto shows that the sequence (¢P(fn))

converges to qb(f) in N. Hence, (f,) converges to f also with respect to ZL, i.e. the topology Zco is finer than ZL. []

Let us finish the article by applying Theorem 3 to a connected pseudo-Riemann- ian manifold M in order to determine the topology of its isometry group

98 MARTIN LINDEN AND HELMUT RECKZIEGEL

3 ( M ) (compare [8, p. 255]). If we s imul taneous ly consider M as affine

manifo ld with respect to the Lev i -C iv i t a connect ion, then 3 ( M ) is a closed

subgroup of 9.I(M); therefore we obtain:

C O R O L L A R Y . For every connected pseudo-Riemannian manifold M the Lie

9roup Z~(M) carries the compact-open topology.

REFERENCES

1. Bourbaki, N., Vari~t~s diff~rentielles et analytiques. Fascicule de r~sultats. Paragraphes 1 ~ 7, Hermann, Paris, 1967.

2. Chevalley, C., Theory of Lie Groups, Vol. 1, Princeton Univ. Press, 1946. 3. Eells, J. and Sampson, J. H., 'Harmonic mappings of Riemannian manifolds', Amer. J. Math. 86

(1964), 109-160. 4. Hano, J. and Morimoto, A., 'Note on the group of affine transformations of an affinely

connected manifold', Naooya Math. J. $ (1955), 71-81. 5. Kobayashi, S., Transformation Groups in Differential Geometry, Springer, Berlin, Heidelberg,

New York, 1972. 6. Kobayashi, S. and Nomizu, K., Foundations of Differential Geometry, Vols I and II, Interscience

Publishers, New York, London, 1963/69. 7. Nomizu, K., 'On the group of attine transformations of an ailinely connected manifold', Proe.

Amer. Math. Soc. 4 (1953), 816-823. 8. O'Neill, B., Semi-Riemannian Geometry, Academic Press, New York, 1983. 9. Vilms, J., 'Totally geodesic mappings', J. Diff. Geom. 4 (1970), 73-79.

Authors' address:

Mar t in Linden and He lmut

Reckziegel,

Mathemat i sches Ins t i tu t der

Universi t / i t K61n,

Weyer t a l 86-90,

D-5000 K61n 41,

F.R.G.

(Received, March 3l, 1989)