Differential Geometry, in honor of K. Yano
Kinokuniya, Tokyo, 1972, 317-334.

NATURAL

BUNDLES AND THEIRPROPERTIES

GENERAL—Geometric objects revisited—ALBERT

1.

NIJENHUISHistorical BackgroundIn the early development of the “Ricci calculus”, now known as classical differential geometry and tensor analysis, it was realized that objects
more general than tensors occurred naturally. The most common examples
were affine connections (Christoffel symbols) and what are now called jetsof tensor fields and connections. All these occurred in almost every significant formula: the Christoffel symbols are built up from jets of thefundamental tensor field; the curvature tensor is built up from jets of theconnection parameters; every covariant derivative is built up from jets ofsome tensor field.
Tensors were defined by the manner of transformation of their components under a change of coordinates in the underlying manifold, or
under a “non-holonomic” change of basis in the tangent space. This invariance-under-transformation type of definition was completely in tune

with Felix Klein's Erlanger Program of 1872 which defines the geometricproperties of a space as the set of invariants of the group of transformations under which the space is invariant. See [1872.1] and [1921.1, p. 460].
The desirability of the formalization of objects more general than tensors
was expressed by G. Ricci [1887.1 and 1889.1]. The underlying idea wasthat these objects would have components which, under a change of coordinates, change in a manner which (unlike tensors) is not necessarily linear
and does not necessarily depend only on the first order derivatives of thechange of coordinates.
O. Veblen [1927.1, pp. 14, 19] calls these objects invariants. Subse-quently, J. A. Schouten and E. R. van Kampen [1930.1] propose the lessoverworkedterm geometric object.

Veblenand Whitehead [1932.1, pp.

46-49] adopt the same term and make a few very elucidating commentson the correspondence between geometric objects and geometric structures.
In a 1934 lecture [1937.1] A. Wundheiler gave a more precise defini-tion of geometric objects than our previous references, based as always,
Research partially supported by a National Science Foundation Grant.

332

ALBERT NIJENHUIS

sisting of a group element (base point) and an element of the Lie algebra(tangent vector at the identity element e) obtained by left translating thetangent vector. More generally, let F be a manifold on which a Lie group

G acts simply transitively ; a tangent to F is identifiable with a pair consisting of its base point in F and that element of the Lie algebra of G
whose “infinitesimal’’ action on F is that tangent vector. More formally,

if p: G X F—

F determines the action of G on F, then the map p,.: T(G)X T(F) —T(F) can be restricted to the zero tangent vectors of F (1.e. toF itself) and to the tangent space to G at e to yield a map T.(G) X F —T'(F): simple transitivity implies this map is a diffeomorphism.
A special case occurs when F is affine and G the group (vector space) oftranslations of F.
Theorem 5.8. If £(X) is a natural bundle of groups, then v"&(X) (or€&(X)) is equivalent to the Whitney product of two natural bundles. Oneof these is £(X); the other d&§(X) (or Hom (7, d&)(X)), where d&(X) 1S
the natural bundle of Lie algebras of the fibers of &(X). If &'(X) is anatural bundle on which &(X) acts naturally and simply transitively, thenVE (X) (or €8(X)) is equivalent to (&' xX d&)(X) (or (&' x Hom(7, d&))(X)).

The Whitney products are semi-direct with respect to the group structures(or group actions).
Remarks.
5.15.
The bundle d&(X) is the pull-back e~'7"&(X), ite: £d(X) — £(X) is the identity section in £(X).
5.16. Theorem 5.8 implies that for sections in natural bundles of Lie‘groups or spaces on which such a bundle of groups acts simply transitively,

the Lie derivative or covariant derivative consists of two parts: the section
itself and the essential part, which is a section in a natural vector bundle.
For example, the (essential part of the) Lie derivative of a connection 1s
a tensor field; the (essential part of the) Lie derivative of a frame field (oforder 1) can be interpreted as a field of linear transformations in the tangent
bundle.

Theorem 5.9.

If &(X) is a natural bundle of groups then ¢#'€(X) isa natural bundle of groups.
The group
Proof (sketch).

properties|of&(X)

areexpressible

bydiagrams, cf. Remark 5.8. There are maps u,v and e, and relations between them expressing associativity, the fact that e is the identity section
and that v is the inverse map. The last is, for example, expresseed by thecommutativity of the diagram (4 is the diagonal map)80) —1s (€ x OX)

IgdX) —————(6 x OX)|EX)GEOMETRIC OBJECTS

333

Passing to tangent spaces we find new diagrams expressing analogous commutativities for 7 £(X). For example, the above diagram becomes
4,TEX) —>(id, v,)(TE X TEX) ——(TE xX TEX)

|2TX) ——————

TEX)

Although this one is very similar to the first, there is one property whichkeeps 7 &(X) from being a natural bundle of groups: e, is not an identity
section. Its domain is no longer X; it is T(X). If, however, in all thesenew diagrams every natural bundle of tangent spaces (use J &(X) for atypical one,

where«/(X) = (4, a, D, X)) is replacedby the bundleoflinear maps 7,(X) — T,(4), with ze a~!(x), which project back to theidentity map on T,(X), then we find (cf. Theorem 5.6) this commutative
diagram:HEX) —(F'6€ X FENX) —(Fé X F6)X)sdX) —m—mm———————

F(X)

where all the arrows denote naturally induced maps from the previous
diagram. The bundle 7 (X) has been reduced again to £d(X) since foreach xe X there is only one map 7T.(X) — T(X) which is admissible,
namely the identity map.
Theorem 5.10.
If &(X) is a natural bundle of order k, then J (X),v"6(X) and €&(X) are of order k + 1; #'€(X) is of order k + |; F*(X)is of order k. A natural bundle &(X) is of finite order k it and only if it isa differentiable fiber bundle with principal bundle the natural bundle F*(X)of frames of order k.
|Remark.
5.17.
The last statement can be re-phrased; this re-statement gives a sketch of its proof. Note, first, that £(X) and #*(X) haveessentially unique extensions to R"; n = dim X. Further, the fiber F, ofF*¥(X U R") over the origin ¢ of R" is a group (the same as G,), and it
acts on the left on E,, the corresponding fiber of £(X U R"). Also, F, actson the right on the total space F of .#*(X U R"). The principal map on
F x E, which factors out F, (cf. Steenrod [1951.1], p. 35ff) produces anatural bundle which 1s equivalent to £(X) and whose principal bundle 1sF(X).

ALBERT

NIJENHUIS

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UNIVERSITY OF PENNSYLVANIA
PHILADELPHIA,
UNA.

PA. 19104