Page 29 - Curvature and Homology

1.3. TENSOR BUNDLES 11
structure derived from the Euclidean space of the components of its
elements it becomes a differentiable manifold. Now, a topology is
defined in 9-1 by the requirement that for each U, qu maps open sets
of U x T: into open sets of F:. In this way, it can be shown that Fi
is a separable Hausdorff space. In fact, .Ti is a differentiable manifold
of class k - 1 as one sees from the equations (1.3.7).
The map gUv: U n V -t GL(c) is continuous since M is of class
k 5 1. Let P be a point in the overlap of the three coordinate neighbor-
hoods, U,V,W U n V n W # 0. Then,

and since

these maps form a topological subgroup of GL(T~. The family of
maps guv for U n V # $. where U, V, ... is a covering of M is called
the set of transition functions corresponding to the given covering.
Now, let
m:q-+M

be the projection map defined by ?r(c(P)) = P. For 1 < k, a map
of
f: M -+ c class 1 satisfying n f = identity is called a tensor field of
type (r, s) and class 1. In particular, a tensor field of type (1,O) is called
a vector field or an injinitesimal transformation. The manifold 9-i is called
the tensor bundle over the base space M with structural group GL (nr+8, R)
and j3re c. In the general theory of fibre bundles, the map f is called
a cross-section. Hence, a tensor field of type (r, s) and class 1 < k is a
cross-section of class 1 in the tensor bundle c over M.
The bundle is usually called the tangent bundle.
Since a tensor field is an assignment of a tensor over Tp for each
point P E M, the components - af/aur (i = 1, ..., n) in (1.2.8) define a
covariant vector field (that is, there is a local cross-section) called
the gradient off. We may ask whether differentiation of vector fields
gives rise to tensor fields, that is given a covariant vector field ti, for
example (the ti are the components of a tensor field of type (O,l)), do
the n2 functions agt/auj define a tensor field (of type (0,2)) over' U? We
see from (1.2.12) that the presence of the term (a2uj/atikari3fj in

yields a negative reply. However, because of the symmetry of i and k
in the second term on the right the components +jrk - +jkt define a skew-