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On geometric objects and lie groups of transformations
MATHEMATICS
ON GEOMETRIC OBJECTS AND LIE GROUPS OF TRANSFORMATIONS BY
NICOLAAS H. KUIPER AND KENTARO YANO (Communicated by Prof. J. A. ScHOUTEN at the meeting of June 25, 1955)
Introduction EHRESMANN defined prolongations (prolongements [1, 2]) of a 0'-manifold. A prolongation is a principal fibre bundle which is for a great deal determined by the base space a o•-manifold and a natural number, the object class t < s. In differential geometry one often is led to fibre bundles which are not principal, but of which the associated principal fibre bundle is of the kind defined by EHRESMANN. Following HAANTJES and LAMAN [3] these fibre bundles will be called geometric object bundles. Geometric objects or geometric object bundles have been defined by GoLAB, HAANTJES, LAMAN, NIJENHUIS, ScHOUTEN, WAGNER and others. In § 1 we give a definition of a geometric object bundle. In § 2 we consider transformations in the base space of a geometric object bundle, and their prolongations in the bundle. In § 3 we obtain a main theorem on geometric objects on a Lie-group space with applications. § 4 deals with Lie. groups of transformations of a manifold and the existence of invariant geometric objects. In § 5 we define Lie-derivatives of geometric object fields and give some applications. More applications will be given in a second paper. 1.
Geometric object bundles In this paper A will denote a fixed pseudo-group of homeomorphic mappings of differentiability class 0', s> 0, in n-dimensional number space R"', which contains the group of translations: § l.
TCA.
Two homeomorphic mappings CfJk : Uk-+ Vk k= i, j of neighbourhoods Uk of an n-manifold X into R .. are called A-compatible 1 ) if
1.1,
=
cp1 cp,- 1 j cpi ( U, r'l U1) EA.
(The mapping A;, is only defined in the neighbourhood mentioned on the right hand side of the vertical bar.) A manifold with a local A-structure or a A-manifold is a manifold covered by a complete set of mappings CfJk: uk-+ vk of the above kind, 1)
Compare VEBLEN and WHITEHEAD [4].
412
any two of which are A-compatible. The homeomorphic mappings are called A-coordinate systems, A-reference systems or just reference systems. In the sequel X will be a A-manifold. Points of X will be indicated by x; points of R" will be indicated by z; in particular the point (0, 0, ... , 0) by 0. Two reference-systems q;, and f{J; both covering x C X are called jetequivalent at x if their restrictions to some neighbourhood of x are identical. The jet-equivalence class of {x, q;.} is called a jet of kind A or A-jet and it will be denoted by j(q;.(x), x; q;i) =j(q;;(x), x; f{J;); x is called the source of this jet, z=q;,(x) is the butt of this jet 2). Iff C A, z E R" is covered by /, then the jet determined by f with source z is denoted by j(z 1 , z; f)=j(z 1 , z), where
Z 1 =f(z).
Jets of this kind will sometimes be called auto-jets. If the butt of a first jet coincides with the source of a second jet, then the product can be formed: j(Za, zl; M1) =:'j(zs, z2; f2) · j(z2, ~; f1) j(z2, x; fq;)=j(z2, ~; f)·j(zv x; q;).
The jet with source ~ E R" and butt ~ E R" obtained from a (unique) translation t(~, z1) is itself denoted by t(z2, z1). Proposition 1. The jets of the kind j(O, 0; f) form a group Ll. We introduce a non-Hausdorf topology in Ll with respect to a O•-A-manifold by the definition: a neighbourhood in Ll consists of all jets that can be represented by functions whose systems of derivatives up to the sth, at .the source of the jet, form a neighbourhood in the suitable number space. Proposition 2. Any jet of the kind j(z 1 , z; f) admits a unique factorization as follows short:
t(z 1 , O)·j[O, 0; t(O,
0)]-t(O, z)
Z 1 )·f·t(z,
j(z', z)=t(z 1 , O)·j(O, O)·t(O, z).
The mapping 01 : j(z 1 , z; f)-+ j[O, 0; t(O, Z 1 ) · f·t(z, 0)] is a homomorphism of the pseudo-group of auto-jets onto the group of jets with source = =butt= 0. Proposition 3. Two jets j(z, x; q;1 ) and j(z 1 , x; q;2 ) with the same source x EX determine a unique auto-jet j(z z) by division: 1
J'( Z I ,
X; (/)2 ) =
J'( Z I ,
Z; (/)2 (/)1-1) •
,
J'(Z,
X; (/JI ) •
Proposition 4. If (/Jv q;2 , q;3 determine three jets with the same source x and with butts z, = q;,(x), then the quotient-auto-jets obey j(Za, Z1; (/Js f{J1 1) = j(Za, Z2; f{Js f{J2 1 ) · j(z2, ~; f{J2 f{J1 1 ) 8)
The notion and word jet was introduced by
EHRESMANN
[1, 2].
413
and this product rule also holds for the images of these jets under 0;, which we denote as follows: or jii
j31 = j32. j21
E
LJ.
Theorem l. The entities X, Y, G, h defined below determine a unique fibre bundle B with base space X, fibre of the kind Y, group G, and homomorphism h : Ll -+ G, which is called the geometric object bundle over X of the kind (Y, G, h). X is a C•-.A-manifold of dimension n.
Y is an analytic manifold. G is a Lie group of analytic transformations of Y.
h is a continuous homomorphism of Ll onto G. The bundle is defined as follows. Let cpi : Ui-+ Vi(Ui C X, ViC Rn) be A-reference-systems covering X 3). In the set of triples (i, x E Ui, y E Y) we introduce the equivalence, called identification: (i,x,y)"-'(j,x,giiy) for xEUJ"lU;
gii = hj;i; jii = t(O,z;) · j(z;,zi; cp;cpi- 1 ) • t(z,,O) EA zi
=
cpi (x)
z;
=
cp; (x).
An equivalence class is by definition a point of the fibre bundle. The equivalence classes with a fixed x form the fibre of the bundle at x. The bundle projection n is the mapping of the fibre at x, onto x. The fibre n-1 (x)= Y., is homeomorphic with Y. The mapping: class of (i, x, y) -+ (x, y)
(l.l)
is a homeomorphic mapping of n-1( Ui) onto ui by i=(nXcp{). We then have forb En-1 (Ui) n- 1 (Ui)
b If b
E
i
-~
~ Ui
nb X cpib
X
X
y and will be denoted
Y IPi~ Vi X Y rp·
~
cpinb X cpib.
n-1 ( ui n U;), then cptb = gii cpib, and if b
E
n-1 ( ui n U; n Uk),
then
j (zk> zi; cpk cpi- 1 ) = j (zk, z;; cpk cp;- 1 ) • j (z;, zi, cp; cpi- 1 ) jki
=
jki . jii
hence, because h is a continuous homomorphism,
gki = gki. gii• Also gii is a continuous function of x. 3) If we require moreover that the set of A-reference systems is complete, that is not contained in a (A-compatible) bigger set, then the definitions are independent of the particular set of subsets {Ui} of X. Compare STEENROD [8].
414
The mappings ( l.l) which fulfill all the conditions just mentioned define the structure of fibre bundle in the point set n-1 (X). STEENROD [8]. The fibre bundle B so obtained is the object-bundle required in theorem l. If Y' C Y is invariant under G, then X, Y', G, h determine a unique object-bundle B', which can be considered as imbedded in B. We call B' a subbundle of B. A cross-section of B is called a geometric object field or geometric object of the kind (Y, G, h). One point of B: b E Y., is called a geometric object at x. Example: Let A be the pseudo-group of all 0' reversible homeomorphisms in R"'. r be the invariant subgroup of Ll consisting of those jets that can be obtained from homeomorphisms in A, that are expressed by functions Z~=Z~ (zt, ... , z.,)
which have the same k-th derivatives for k= 0, 1, ... , l < t ~ s at the point 0, as the functions that express the identity homeomorphism:
his the homorphism which maps a jet in Ll onto its image in G=LifF. The object-bundle B so defined is a manifold of differentiability class s-t. A O•-cross-section in B (r < s-t) is called object (field) of object class t and (of course) of differentiability class r. HAANTJES and LAMAN [3] determined all transitive geometric objectbundles of object-class t= I and dimension n+ l (dimension Y = 1). Tensor-bundles are bundles of class t= I. Affine connections (parallel displacement) can be defined in the bundle of tangent vectors of a 0 2-manifold. The affine connections are themselves cross-sections in a bundle of object-class t= 2. Every affine connection belongs to a class of projectively equivalent affine connections, which class determines a unique normal projective connection. E. CARTAN [5]. Such a normal projective connection is a cross-section in a bundle of objectclass t= 2. It is not easy to determine whether a given connection which is defined in a general way in a fibre bundle can be considered as a geometric object field. As an example we mention a projective (conformal) connection in a fibre-bundle with fibre the projective n-space (n-sphere), with group the projective (Moebius) group and without a fixed oblique cross-section. EHRESMANN [6], KuiPER [7]. Every product bundle is a geometric object bundle however (G=LI/LI = 1). Another example of a geometric object bundle is obtained from the tensor-bundle of covariant tensors of kind t,.;. under the identification: t,.A""""'"'(!t,.A
(!
> 0.
If t,.A is symmetric and positive definite then the geometric object is called a conformal metric. It is of class t= I. The normal conformal con-
415
nection determined by a conformal metric is a geometric object of class t=2. Other examples of geometric object-bundles are obtained by taking for A a subgroup of all reversible C•(s= 1, 2, ... =or w) homeomorphisms, for example consisting of all homeomorphisms that leave invariant a fibred structure or a complex structure m Rn. We conclude this paragraph with: Theorem 2. If X is a C•-product spa,ce of k circles and a euclidean n-k-space, then any geometric object bundle over X is a product bundle. The same is true for any open sub-space of such a space X. Proof : The universal covering X of X admits the space ~ of rows of n numbers (Zt, ... , Zn) as one coordinate system u that covers X and such that the fundamental group of X is generated by the k transformations for j =I, ... and k respectively (a{ is the unit matrix). We chose coordinate systems in X that are the product of the natural induced homeomorphism of a sufficiently small neighbourhood in X onto one in X and this coordinate-system x: X --+ Rn. The only element in L1 that is obtained from pairs of such coordinate systems for X, is the identity. Then the only element in a that occurs in the description of the object bundle is also the identity, and therefore the bundle is a product bundle. As the circle is base-space of non-trivial fibre bundles, this implies Theorem 2 ' . its base space.
Not every fibre bundle is a geometric object bundle over
An interesting problem is the characterisation of all object-bundles among the fibre bundles.
Prolongations of A-transformations in X Let b be a cross-section or a geometric object-field in the object-bundle (B, X, n, Y, a, h, A)· b(x)= Y., n b. A homeomorphism rJ : U'--+ U in X is called a A-point-transformation if in case ({J : U--+ V is a A-referencesystem, the same is true for ((JrJ : U'--+ V. § 2.
o~~Q_!E~~~
in*
in
t I v*
t Iv
o~~~LbJ
D
:n:F'.x"I
-+I
U"
416 The A-point transformation 'YJ has a unique prolongation {EHRESMANN [1]; cp* and (cp'YJ)* are defined in (1.1)} 'YJ* : n- 1 (U') ~ n- 1 (U)
defined by, if b'
E
n-1 (U'),:
(4.1) Prolongation commutes with projection: n7J* = 7Jn. (4.1) can be understood as follows: If we use the reference systems cp and cp'YJ for U and U' then the prolongation 'YJ* of 'YJ is expressed by the pair {'YJ : U'---'>- U, and identity in Y}. An expression in terms of two arbitrary reference systems cp: U---'>- V =cpU and "P: U'---'>- V'='lfJU', instead of cp and cp'YJ respectively, is as follows (x' =nb'). (4.2)
'Yj•b'
=
n-1 7Jb'n ("' (cp*)-1 h [t(O, CfJ'YJX') • j(cp'YJX 1 , 'ljJX'; ffJ'YJ'IjJ- 1) • • t('1jJX 0) J'lfJ*b'. 1
,
Substituting 'lfJ=cp'YJ in (4.2) we obtain (4.1) again. (4.2) is independent of the particular reference system cp for U. This can be seen from straightforward computation. From (4.1) we have: ('YJ-1)* = ('YJ*)-1. Therefore (4.2) is also independent of the reference system "P for U' hence independent of reference systems used. Now suppose we have two A-point transformations. 'YJ : U'---'>- U and , : U"---'>- U', hence the product 'YJ'V : U"---'>- U. Using the reference systems cp : U ---'>- V, cp'YJ : U' ---'>- V and cp'YJP : U" ---'>- V, we observe that the prolongations 'YJ*, ,*, ('YJP)* are respectively expressed by:
Hence ('YJP)* =
( 'YJ, identity) ( "; identity) ('YJP, identity). 'YJ*P*, and we have the
Theorem 3. Every A-transformation in X has a unique prolongation in B. The mapping which assigns to every A-transformation its prolongation is a group-isomorphism. From the definitions we also have: Any sub bundle B' C B is invariant (not point wise) under the prolongation of any A-transformation in X.
§ 3. Geometric objects on Lie groups Theorem 4. Let .7t' be a Lie group of transformations operating on the left on the group space H of .7t'. Any object bundle B with base space Hand fibre space Y is an analytic product bundle H x Y with left invariant analytic cross-sections H x y (y E Y).
417
Proof: Choose a fibre Y, C B, and a point b E Y,. Let 'Yj(t) E :Y? be a transformation of H, 'YJ*(t) E :Y?* its prolongation in B, and t a point of the abstract analytic group of :Y?. The set :Y?*b consists of one point in each fibre and is an analytic cross-section because 'YJ(t) is an analytic transformation, which depends also analytically on t. A point b' C :Y?*b is characterised by bEY, and n(b')=x'. The analytic correspondence b' --+ x' x b of B = n-1(H) onto H x Y ,, so obtained, is l- l, and 'YJ* E :Y?* the prolongation of 'YJ E :Y? is represented under this representation by
'YJ*(x' X y)=1JX X y. Application: Theorem 4': Ann-dimensional Lie group has the following left invariant geometric object fields: an absolute parallelism; many affine connections among which symmetric connections; Riemannian metrics of any signature; for n even many almost-complex structures and almost-hermitian metrics ; Finsler metrics. 1
In all these cases we define suitably the geometric object at one point of the group and the required object-field consists of the images of this geometric object under the prolongations in the fibre-bundle of the transformations of the group.
Geometric objects and transitive groups of transformations In this § H is a transitive group of C00 -A-point transformations of a C 00 -A-manifold X, base space of an object-bundle B. I, is the subgroup of all transformations in H that leave x E X fixed. The prolongations in B of I, and H are I; and H*. I., and I; are called group of isotropy of H(H*) at x. § 4.
Theorem 5. The object bundle B over X admits a cross-section b invariant under all prolongations in H*, if and only if the isotropy group I; at x has a fixed point in Y, (for some x EX, and then for any x EX). Proof: The necessity is obvious. To prove the sufficiency we consider a point b.,,:; Y, invariant under I!. For any two transformations / 1 and f2 in H, which map x onto the same point x', the prolongations fj_, 1; obey: {f;)- 1 fi E I; {f;)-1 t~ b, = b,
fib,=
t: b,.
Therefore the point set {/* b,}, f* E H*, contains exactly one point in the fibre Y.,, for any x' EX. The group properties imply that this crosssection {f* b,} is invariant under H*. In the applications it often occurs that the homomorphism-onto I;--+ I; I Y,, defined by restriction of the transformations of I, to the fibre Y ,, is an isomorphism. This is the case when the restriction of an element 'YJ* of I; to Y., uniquely determines 'YJ*. In the proof of many
418
theorems on groups of transformations leaving invariant some geometric object we therefore may restrict ourselves to considerations concerning one fibre Y.,. For example: Theorem 6. {I} Let xn be a space with a) a Riemannian metric, b) an affine connection, c) a Kiihlerian metric, d) an affine connection with an invariant almost complex structure, with a N -dimensional group of structure preserving transformations. {II} Let N° be the dimension of the group of a) motions in a space of constant curvature, b) affinities in the affine space, c) motions in a Fubini-space, d) complex-analytic affinities in complex affine space. Then N
=
N° implies that
N~=n(n+
xn is of the kind mentioned under {II}.
l)/2, Ng=n(n+ I) and putting n=2m
N~=m 2 + 2m,
N~=2m 2 +2m.
Proof: In a space with an affine connection, of which cases abed are examples, an affine point transformation with fixed point x is determined by its prolongation restricted to the tangent space at x. This ensures the faithfulness of the representation of I; in the tangentspace. The dimension D of the isotropy-group obeys N -n < D < N°-n, hence N < N°. Next suppose N =N°. In all cases abed, there is an affine c~mnection. Let .Q be the curvature tensor of this connection andS the (anti-symmetric) torsion-tensor (S = 0 for the cases a and c), the vanishing of which characterises the cases mentioned under {II}. In cases b and d we find, among the prolongations of the point transformations in X with invariant point x, restricted to the tangent space Y.,, those which are geometrical multiplications of the tangent space Y.,. The curvature-tensor and the torsion tensor must be invariant under the representation of these multiplications in the related tensor-spaces. These representations are also non-trivial geometrical multiplications. Hence .Q= 0, S = 0. This proves b and d. In cases a( c) the (holomorphic) sectional curvature is invariant under all orthogonal (unitary) transformations in the tangent space ~,tt x. AsH is transitive the (holomorphic) sectional curvature. is the same for all (holomorphic) sections at all points and the space is " of constant (holomorphic) curvature ". This proves cases a and c. Lie-derivatives Let b be a geometric object field in the bundle B over X. Let N be a neighbourhood of the identity of a Lie group H, which operates as a group of A-transformations § 5.
17(t) : U--+ U(t)
in X. tEN is a point in the group-space. n(t) is the corresponding trans-
419
formation in X. Suppose x n-l(x)= Y., is defined by: (5.1)
E
U(t) for all tEN. Then a mapping of N into
L: t--+ Y.,
n
[1J*(t)·b].
In case N = H, H acts as a Lie group of transformations on the image point set L(H) in Y .,. Under the mapping (5.1) the tangent space at the unit-element of H is mapped into the tangent space of the point b(x)=b n Y., with respect to the fibre. This mapping is called the Lie-differential of the geometric object b, at x, with respect to the given Lie group. For any parametrised differentiable curve t(s) in H, with t(O) is the identity, the image of the vector dtfds under the Lie differential is called the Lie-derivative of b, at x, with respect to the parameter s. It is a tangent to a parametrised curve in Y.,. For x variable we get a field of such tangents also called the Lie-derivative. Theorem 7. The Lie-derivative of a geometric object (field) of differen· liability-class > 1 is a geometric object (field). Proof: If X, Y, G, h are the entities which determine the fibre-bundle B, in which b is a 0'-cross-section r > 1, then the Lie-derivative !£'b is a C-'- 1 cross-section in the fibre-bundle determined in a unique way by: X, Y 1 , G, h1 where Y1 is the space of all tangent vectors at all points of Y and h1 is obtained from h by replacing any analytic transformation in Y by its prolongation in Y 1 • This replacement is an isomorphism according to theorem 3. Theorem 8 . A C•-geometric object field b in the bundle B over X, r > I, is invariant under the prolongations of a connected Lie-group H of A -transformations of X, if and only if the Lie-differential of b at any point x EX with respect to H vanishes. The necessity is obvious. The sufficiency is not equally obvious however (!). Suppose the Lie-differential of b at every point x EX with respect to H vanishes. Suppose, for a fixed x, that the set of points
Yx n 17*(t) · b
t EH
is not one point. Then a curve t(s) with a point t(I)=t1 exists in H such that the tangent vector d
ds { Y.,
n 17* (t ( s)) · b}
does not vanish for s = I. Let X=1](t 1 ) ·x'. The prolongation YJ*(t1 ) maps Yx' onto Y., and this mapping is under reference systems represented by an element of G operating in Y. Hence it carries non vanishing tangent vectors of Yx, onto such vectors of Y., and vice versa. Therefore the curve with parameter s:
420
has a non-vanishing derivative for s= 1, that is at the point Y.,, n b. The Lie-differential of b at x' with respect to H is then not zero in contradiction with the assumptions. REFERENCES
I. EHRESMANN, CHARLES, Les prolongements d'une variete differentiable. Attidel IV Congresso dell' Unione Matematico Italiana Taormina, 9 pages, 25-31 Ott. (1951). 2. , Introduction a la theorie des structures infinitesimales et des pseudogroupes de Lie. Colloque intemational du C.N.R.S. Geometrie differentielle Strasbourg, 97-117 (1953). 3. HAANTJES, J. and G. LAMAN, On the definition of geometric objects. Proc. Akad. Amsterdam, 56, Series A = Indagationes Math. 15, 208-222 (1953). 4. VEBLEN, 0. and J. H. C. WHITEHEAD, The foundations of differential geometry. Cambr. Tract., Ch. II § 8 and Ch. III, 29 (1932). 5. CARTAN, E., Sur les varietes a connection projective. Bull. Soc. Math. France, 52, 205-241 (1924). 6. EHRESMANN, CHARLES, Les connections infinitesimales dans un espace fibre differentiable. Colloque de topologie 1950 du C.B.R.M., 29-55. 7. KuiPER, N. H., Einstein spaces and connections, Proc. Akad. Amsterdam 53, = Indagationes Math. 12, 504-521 (1950). 8. STEENROD, N., The topology of fibre-bundles (Princeton University Press, 1951).
ON GEOMETRIC OBJECTS AND LIE GROUPS OF TRANSFORMATIONS BY
NICOLAAS H. KUIPER AND KENTARO YANO (Communicated by Prof. J. A. ScHOUTEN at the meeting of June 25, 1955)
Introduction EHRESMANN defined prolongations (prolongements [1, 2]) of a 0'-manifold. A prolongation is a principal fibre bundle which is for a great deal determined by the base space a o•-manifold and a natural number, the object class t < s. In differential geometry one often is led to fibre bundles which are not principal, but of which the associated principal fibre bundle is of the kind defined by EHRESMANN. Following HAANTJES and LAMAN [3] these fibre bundles will be called geometric object bundles. Geometric objects or geometric object bundles have been defined by GoLAB, HAANTJES, LAMAN, NIJENHUIS, ScHOUTEN, WAGNER and others. In § 1 we give a definition of a geometric object bundle. In § 2 we consider transformations in the base space of a geometric object bundle, and their prolongations in the bundle. In § 3 we obtain a main theorem on geometric objects on a Lie-group space with applications. § 4 deals with Lie. groups of transformations of a manifold and the existence of invariant geometric objects. In § 5 we define Lie-derivatives of geometric object fields and give some applications. More applications will be given in a second paper. 1.
Geometric object bundles In this paper A will denote a fixed pseudo-group of homeomorphic mappings of differentiability class 0', s> 0, in n-dimensional number space R"', which contains the group of translations: § l.
TCA.
Two homeomorphic mappings CfJk : Uk-+ Vk k= i, j of neighbourhoods Uk of an n-manifold X into R .. are called A-compatible 1 ) if
1.1,
=
cp1 cp,- 1 j cpi ( U, r'l U1) EA.
(The mapping A;, is only defined in the neighbourhood mentioned on the right hand side of the vertical bar.) A manifold with a local A-structure or a A-manifold is a manifold covered by a complete set of mappings CfJk: uk-+ vk of the above kind, 1)
Compare VEBLEN and WHITEHEAD [4].
412
any two of which are A-compatible. The homeomorphic mappings are called A-coordinate systems, A-reference systems or just reference systems. In the sequel X will be a A-manifold. Points of X will be indicated by x; points of R" will be indicated by z; in particular the point (0, 0, ... , 0) by 0. Two reference-systems q;, and f{J; both covering x C X are called jetequivalent at x if their restrictions to some neighbourhood of x are identical. The jet-equivalence class of {x, q;.} is called a jet of kind A or A-jet and it will be denoted by j(q;.(x), x; q;i) =j(q;;(x), x; f{J;); x is called the source of this jet, z=q;,(x) is the butt of this jet 2). Iff C A, z E R" is covered by /, then the jet determined by f with source z is denoted by j(z 1 , z; f)=j(z 1 , z), where
Z 1 =f(z).
Jets of this kind will sometimes be called auto-jets. If the butt of a first jet coincides with the source of a second jet, then the product can be formed: j(Za, zl; M1) =:'j(zs, z2; f2) · j(z2, ~; f1) j(z2, x; fq;)=j(z2, ~; f)·j(zv x; q;).
The jet with source ~ E R" and butt ~ E R" obtained from a (unique) translation t(~, z1) is itself denoted by t(z2, z1). Proposition 1. The jets of the kind j(O, 0; f) form a group Ll. We introduce a non-Hausdorf topology in Ll with respect to a O•-A-manifold by the definition: a neighbourhood in Ll consists of all jets that can be represented by functions whose systems of derivatives up to the sth, at .the source of the jet, form a neighbourhood in the suitable number space. Proposition 2. Any jet of the kind j(z 1 , z; f) admits a unique factorization as follows short:
t(z 1 , O)·j[O, 0; t(O,
0)]-t(O, z)
Z 1 )·f·t(z,
j(z', z)=t(z 1 , O)·j(O, O)·t(O, z).
The mapping 01 : j(z 1 , z; f)-+ j[O, 0; t(O, Z 1 ) · f·t(z, 0)] is a homomorphism of the pseudo-group of auto-jets onto the group of jets with source = =butt= 0. Proposition 3. Two jets j(z, x; q;1 ) and j(z 1 , x; q;2 ) with the same source x EX determine a unique auto-jet j(z z) by division: 1
J'( Z I ,
X; (/)2 ) =
J'( Z I ,
Z; (/)2 (/)1-1) •
,
J'(Z,
X; (/JI ) •
Proposition 4. If (/Jv q;2 , q;3 determine three jets with the same source x and with butts z, = q;,(x), then the quotient-auto-jets obey j(Za, Z1; (/Js f{J1 1) = j(Za, Z2; f{Js f{J2 1 ) · j(z2, ~; f{J2 f{J1 1 ) 8)
The notion and word jet was introduced by
EHRESMANN
[1, 2].
413
and this product rule also holds for the images of these jets under 0;, which we denote as follows: or jii
j31 = j32. j21
E
LJ.
Theorem l. The entities X, Y, G, h defined below determine a unique fibre bundle B with base space X, fibre of the kind Y, group G, and homomorphism h : Ll -+ G, which is called the geometric object bundle over X of the kind (Y, G, h). X is a C•-.A-manifold of dimension n.
Y is an analytic manifold. G is a Lie group of analytic transformations of Y.
h is a continuous homomorphism of Ll onto G. The bundle is defined as follows. Let cpi : Ui-+ Vi(Ui C X, ViC Rn) be A-reference-systems covering X 3). In the set of triples (i, x E Ui, y E Y) we introduce the equivalence, called identification: (i,x,y)"-'(j,x,giiy) for xEUJ"lU;
gii = hj;i; jii = t(O,z;) · j(z;,zi; cp;cpi- 1 ) • t(z,,O) EA zi
=
cpi (x)
z;
=
cp; (x).
An equivalence class is by definition a point of the fibre bundle. The equivalence classes with a fixed x form the fibre of the bundle at x. The bundle projection n is the mapping of the fibre at x, onto x. The fibre n-1 (x)= Y., is homeomorphic with Y. The mapping: class of (i, x, y) -+ (x, y)
(l.l)
is a homeomorphic mapping of n-1( Ui) onto ui by i=(nXcp{). We then have forb En-1 (Ui) n- 1 (Ui)
b If b
E
i
-~
~ Ui
nb X cpib
X
X
y and will be denoted
Y IPi~ Vi X Y rp·
~
cpinb X cpib.
n-1 ( ui n U;), then cptb = gii cpib, and if b
E
n-1 ( ui n U; n Uk),
then
j (zk> zi; cpk cpi- 1 ) = j (zk, z;; cpk cp;- 1 ) • j (z;, zi, cp; cpi- 1 ) jki
=
jki . jii
hence, because h is a continuous homomorphism,
gki = gki. gii• Also gii is a continuous function of x. 3) If we require moreover that the set of A-reference systems is complete, that is not contained in a (A-compatible) bigger set, then the definitions are independent of the particular set of subsets {Ui} of X. Compare STEENROD [8].
414
The mappings ( l.l) which fulfill all the conditions just mentioned define the structure of fibre bundle in the point set n-1 (X). STEENROD [8]. The fibre bundle B so obtained is the object-bundle required in theorem l. If Y' C Y is invariant under G, then X, Y', G, h determine a unique object-bundle B', which can be considered as imbedded in B. We call B' a subbundle of B. A cross-section of B is called a geometric object field or geometric object of the kind (Y, G, h). One point of B: b E Y., is called a geometric object at x. Example: Let A be the pseudo-group of all 0' reversible homeomorphisms in R"'. r be the invariant subgroup of Ll consisting of those jets that can be obtained from homeomorphisms in A, that are expressed by functions Z~=Z~ (zt, ... , z.,)
which have the same k-th derivatives for k= 0, 1, ... , l < t ~ s at the point 0, as the functions that express the identity homeomorphism:
his the homorphism which maps a jet in Ll onto its image in G=LifF. The object-bundle B so defined is a manifold of differentiability class s-t. A O•-cross-section in B (r < s-t) is called object (field) of object class t and (of course) of differentiability class r. HAANTJES and LAMAN [3] determined all transitive geometric objectbundles of object-class t= I and dimension n+ l (dimension Y = 1). Tensor-bundles are bundles of class t= I. Affine connections (parallel displacement) can be defined in the bundle of tangent vectors of a 0 2-manifold. The affine connections are themselves cross-sections in a bundle of object-class t= 2. Every affine connection belongs to a class of projectively equivalent affine connections, which class determines a unique normal projective connection. E. CARTAN [5]. Such a normal projective connection is a cross-section in a bundle of objectclass t= 2. It is not easy to determine whether a given connection which is defined in a general way in a fibre bundle can be considered as a geometric object field. As an example we mention a projective (conformal) connection in a fibre-bundle with fibre the projective n-space (n-sphere), with group the projective (Moebius) group and without a fixed oblique cross-section. EHRESMANN [6], KuiPER [7]. Every product bundle is a geometric object bundle however (G=LI/LI = 1). Another example of a geometric object bundle is obtained from the tensor-bundle of covariant tensors of kind t,.;. under the identification: t,.A""""'"'(!t,.A
(!
> 0.
If t,.A is symmetric and positive definite then the geometric object is called a conformal metric. It is of class t= I. The normal conformal con-
415
nection determined by a conformal metric is a geometric object of class t=2. Other examples of geometric object-bundles are obtained by taking for A a subgroup of all reversible C•(s= 1, 2, ... =or w) homeomorphisms, for example consisting of all homeomorphisms that leave invariant a fibred structure or a complex structure m Rn. We conclude this paragraph with: Theorem 2. If X is a C•-product spa,ce of k circles and a euclidean n-k-space, then any geometric object bundle over X is a product bundle. The same is true for any open sub-space of such a space X. Proof : The universal covering X of X admits the space ~ of rows of n numbers (Zt, ... , Zn) as one coordinate system u that covers X and such that the fundamental group of X is generated by the k transformations for j =I, ... and k respectively (a{ is the unit matrix). We chose coordinate systems in X that are the product of the natural induced homeomorphism of a sufficiently small neighbourhood in X onto one in X and this coordinate-system x: X --+ Rn. The only element in L1 that is obtained from pairs of such coordinate systems for X, is the identity. Then the only element in a that occurs in the description of the object bundle is also the identity, and therefore the bundle is a product bundle. As the circle is base-space of non-trivial fibre bundles, this implies Theorem 2 ' . its base space.
Not every fibre bundle is a geometric object bundle over
An interesting problem is the characterisation of all object-bundles among the fibre bundles.
Prolongations of A-transformations in X Let b be a cross-section or a geometric object-field in the object-bundle (B, X, n, Y, a, h, A)· b(x)= Y., n b. A homeomorphism rJ : U'--+ U in X is called a A-point-transformation if in case ({J : U--+ V is a A-referencesystem, the same is true for ((JrJ : U'--+ V. § 2.
o~~Q_!E~~~
in*
in
t I v*
t Iv
o~~~LbJ
D
:n:F'.x"I
-+I
U"
416 The A-point transformation 'YJ has a unique prolongation {EHRESMANN [1]; cp* and (cp'YJ)* are defined in (1.1)} 'YJ* : n- 1 (U') ~ n- 1 (U)
defined by, if b'
E
n-1 (U'),:
(4.1) Prolongation commutes with projection: n7J* = 7Jn. (4.1) can be understood as follows: If we use the reference systems cp and cp'YJ for U and U' then the prolongation 'YJ* of 'YJ is expressed by the pair {'YJ : U'---'>- U, and identity in Y}. An expression in terms of two arbitrary reference systems cp: U---'>- V =cpU and "P: U'---'>- V'='lfJU', instead of cp and cp'YJ respectively, is as follows (x' =nb'). (4.2)
'Yj•b'
=
n-1 7Jb'n ("' (cp*)-1 h [t(O, CfJ'YJX') • j(cp'YJX 1 , 'ljJX'; ffJ'YJ'IjJ- 1) • • t('1jJX 0) J'lfJ*b'. 1
,
Substituting 'lfJ=cp'YJ in (4.2) we obtain (4.1) again. (4.2) is independent of the particular reference system cp for U. This can be seen from straightforward computation. From (4.1) we have: ('YJ-1)* = ('YJ*)-1. Therefore (4.2) is also independent of the reference system "P for U' hence independent of reference systems used. Now suppose we have two A-point transformations. 'YJ : U'---'>- U and , : U"---'>- U', hence the product 'YJ'V : U"---'>- U. Using the reference systems cp : U ---'>- V, cp'YJ : U' ---'>- V and cp'YJP : U" ---'>- V, we observe that the prolongations 'YJ*, ,*, ('YJP)* are respectively expressed by:
Hence ('YJP)* =
( 'YJ, identity) ( "; identity) ('YJP, identity). 'YJ*P*, and we have the
Theorem 3. Every A-transformation in X has a unique prolongation in B. The mapping which assigns to every A-transformation its prolongation is a group-isomorphism. From the definitions we also have: Any sub bundle B' C B is invariant (not point wise) under the prolongation of any A-transformation in X.
§ 3. Geometric objects on Lie groups Theorem 4. Let .7t' be a Lie group of transformations operating on the left on the group space H of .7t'. Any object bundle B with base space Hand fibre space Y is an analytic product bundle H x Y with left invariant analytic cross-sections H x y (y E Y).
417
Proof: Choose a fibre Y, C B, and a point b E Y,. Let 'Yj(t) E :Y? be a transformation of H, 'YJ*(t) E :Y?* its prolongation in B, and t a point of the abstract analytic group of :Y?. The set :Y?*b consists of one point in each fibre and is an analytic cross-section because 'YJ(t) is an analytic transformation, which depends also analytically on t. A point b' C :Y?*b is characterised by bEY, and n(b')=x'. The analytic correspondence b' --+ x' x b of B = n-1(H) onto H x Y ,, so obtained, is l- l, and 'YJ* E :Y?* the prolongation of 'YJ E :Y? is represented under this representation by
'YJ*(x' X y)=1JX X y. Application: Theorem 4': Ann-dimensional Lie group has the following left invariant geometric object fields: an absolute parallelism; many affine connections among which symmetric connections; Riemannian metrics of any signature; for n even many almost-complex structures and almost-hermitian metrics ; Finsler metrics. 1
In all these cases we define suitably the geometric object at one point of the group and the required object-field consists of the images of this geometric object under the prolongations in the fibre-bundle of the transformations of the group.
Geometric objects and transitive groups of transformations In this § H is a transitive group of C00 -A-point transformations of a C 00 -A-manifold X, base space of an object-bundle B. I, is the subgroup of all transformations in H that leave x E X fixed. The prolongations in B of I, and H are I; and H*. I., and I; are called group of isotropy of H(H*) at x. § 4.
Theorem 5. The object bundle B over X admits a cross-section b invariant under all prolongations in H*, if and only if the isotropy group I; at x has a fixed point in Y, (for some x EX, and then for any x EX). Proof: The necessity is obvious. To prove the sufficiency we consider a point b.,,:; Y, invariant under I!. For any two transformations / 1 and f2 in H, which map x onto the same point x', the prolongations fj_, 1; obey: {f;)- 1 fi E I; {f;)-1 t~ b, = b,
fib,=
t: b,.
Therefore the point set {/* b,}, f* E H*, contains exactly one point in the fibre Y.,, for any x' EX. The group properties imply that this crosssection {f* b,} is invariant under H*. In the applications it often occurs that the homomorphism-onto I;--+ I; I Y,, defined by restriction of the transformations of I, to the fibre Y ,, is an isomorphism. This is the case when the restriction of an element 'YJ* of I; to Y., uniquely determines 'YJ*. In the proof of many
418
theorems on groups of transformations leaving invariant some geometric object we therefore may restrict ourselves to considerations concerning one fibre Y.,. For example: Theorem 6. {I} Let xn be a space with a) a Riemannian metric, b) an affine connection, c) a Kiihlerian metric, d) an affine connection with an invariant almost complex structure, with a N -dimensional group of structure preserving transformations. {II} Let N° be the dimension of the group of a) motions in a space of constant curvature, b) affinities in the affine space, c) motions in a Fubini-space, d) complex-analytic affinities in complex affine space. Then N
=
N° implies that
N~=n(n+
xn is of the kind mentioned under {II}.
l)/2, Ng=n(n+ I) and putting n=2m
N~=m 2 + 2m,
N~=2m 2 +2m.
Proof: In a space with an affine connection, of which cases abed are examples, an affine point transformation with fixed point x is determined by its prolongation restricted to the tangent space at x. This ensures the faithfulness of the representation of I; in the tangentspace. The dimension D of the isotropy-group obeys N -n < D < N°-n, hence N < N°. Next suppose N =N°. In all cases abed, there is an affine c~mnection. Let .Q be the curvature tensor of this connection andS the (anti-symmetric) torsion-tensor (S = 0 for the cases a and c), the vanishing of which characterises the cases mentioned under {II}. In cases b and d we find, among the prolongations of the point transformations in X with invariant point x, restricted to the tangent space Y.,, those which are geometrical multiplications of the tangent space Y.,. The curvature-tensor and the torsion tensor must be invariant under the representation of these multiplications in the related tensor-spaces. These representations are also non-trivial geometrical multiplications. Hence .Q= 0, S = 0. This proves b and d. In cases a( c) the (holomorphic) sectional curvature is invariant under all orthogonal (unitary) transformations in the tangent space ~,tt x. AsH is transitive the (holomorphic) sectional curvature. is the same for all (holomorphic) sections at all points and the space is " of constant (holomorphic) curvature ". This proves cases a and c. Lie-derivatives Let b be a geometric object field in the bundle B over X. Let N be a neighbourhood of the identity of a Lie group H, which operates as a group of A-transformations § 5.
17(t) : U--+ U(t)
in X. tEN is a point in the group-space. n(t) is the corresponding trans-
419
formation in X. Suppose x n-l(x)= Y., is defined by: (5.1)
E
U(t) for all tEN. Then a mapping of N into
L: t--+ Y.,
n
[1J*(t)·b].
In case N = H, H acts as a Lie group of transformations on the image point set L(H) in Y .,. Under the mapping (5.1) the tangent space at the unit-element of H is mapped into the tangent space of the point b(x)=b n Y., with respect to the fibre. This mapping is called the Lie-differential of the geometric object b, at x, with respect to the given Lie group. For any parametrised differentiable curve t(s) in H, with t(O) is the identity, the image of the vector dtfds under the Lie differential is called the Lie-derivative of b, at x, with respect to the parameter s. It is a tangent to a parametrised curve in Y.,. For x variable we get a field of such tangents also called the Lie-derivative. Theorem 7. The Lie-derivative of a geometric object (field) of differen· liability-class > 1 is a geometric object (field). Proof: If X, Y, G, h are the entities which determine the fibre-bundle B, in which b is a 0'-cross-section r > 1, then the Lie-derivative !£'b is a C-'- 1 cross-section in the fibre-bundle determined in a unique way by: X, Y 1 , G, h1 where Y1 is the space of all tangent vectors at all points of Y and h1 is obtained from h by replacing any analytic transformation in Y by its prolongation in Y 1 • This replacement is an isomorphism according to theorem 3. Theorem 8 . A C•-geometric object field b in the bundle B over X, r > I, is invariant under the prolongations of a connected Lie-group H of A -transformations of X, if and only if the Lie-differential of b at any point x EX with respect to H vanishes. The necessity is obvious. The sufficiency is not equally obvious however (!). Suppose the Lie-differential of b at every point x EX with respect to H vanishes. Suppose, for a fixed x, that the set of points
Yx n 17*(t) · b
t EH
is not one point. Then a curve t(s) with a point t(I)=t1 exists in H such that the tangent vector d
ds { Y.,
n 17* (t ( s)) · b}
does not vanish for s = I. Let X=1](t 1 ) ·x'. The prolongation YJ*(t1 ) maps Yx' onto Y., and this mapping is under reference systems represented by an element of G operating in Y. Hence it carries non vanishing tangent vectors of Yx, onto such vectors of Y., and vice versa. Therefore the curve with parameter s:
420
has a non-vanishing derivative for s= 1, that is at the point Y.,, n b. The Lie-differential of b at x' with respect to H is then not zero in contradiction with the assumptions. REFERENCES
I. EHRESMANN, CHARLES, Les prolongements d'une variete differentiable. Attidel IV Congresso dell' Unione Matematico Italiana Taormina, 9 pages, 25-31 Ott. (1951). 2. , Introduction a la theorie des structures infinitesimales et des pseudogroupes de Lie. Colloque intemational du C.N.R.S. Geometrie differentielle Strasbourg, 97-117 (1953). 3. HAANTJES, J. and G. LAMAN, On the definition of geometric objects. Proc. Akad. Amsterdam, 56, Series A = Indagationes Math. 15, 208-222 (1953). 4. VEBLEN, 0. and J. H. C. WHITEHEAD, The foundations of differential geometry. Cambr. Tract., Ch. II § 8 and Ch. III, 29 (1932). 5. CARTAN, E., Sur les varietes a connection projective. Bull. Soc. Math. France, 52, 205-241 (1924). 6. EHRESMANN, CHARLES, Les connections infinitesimales dans un espace fibre differentiable. Colloque de topologie 1950 du C.B.R.M., 29-55. 7. KuiPER, N. H., Einstein spaces and connections, Proc. Akad. Amsterdam 53, = Indagationes Math. 12, 504-521 (1950). 8. STEENROD, N., The topology of fibre-bundles (Princeton University Press, 1951).