, . . . , x
(called local coordinates) and such that when a point
belongs to two charts and has two sets of
local coordinates they
are related by a transformation (see 272).
Here the functions f{sup i}, with which
the transformation is expressed, are dealt with on the basis
of having continuous first partial derivatives with a Jacobian
determinant (denoted by det), which is not equal to zero (see
273). If all such functions
f{sup i} have continuous partial derivatives
of orders equal to or less than k, the manifold is said
to be of
class k. If all such functions have
continuous partial derivatives of all orders, the manifold is
said to be of class infinity. This definition is necessary because
a space can rarely be described by one coordinate system. The
local coordinates themselves have no geometrical meaning, and
the analytical tool for the study of manifolds must be furnished
by concepts that behave in a simple way under a change of local
coordinates.The most important among such concepts are the tensors or tensor fields. A tensor of contravariant order r and covariant order s is defined by specifying its components (see 274) in each coordinate chart. Under a change of local coordinates (in which all the indices run from 1 to n) these components follow the conventional transformation rule involving first partial derivatives of the two sets of coordinates (see 275). The tensor is called contravariant if s equals zero and covariant if r equals zero. A contravariant (respectively covariant) tensor of order one--i.e., of single order--reduces to a contravariant (respectively covariant) vector. In tensor algebra, a tensor can be multiplied by a scalar quantity and two tensors of the same orders can be added. Moreover, two tensors of contravariant orders r, p and covariant orders s, q, respectively, have a product that is a tensor of contravariant order r + p and covariant order s + q (see 276). The vanishing of a tensor (i.e., of all the components) is a condition invariant with a change of local coordinates.
A tensor is symmetric (respectively antisymmetric) in two indices if the components remain unchanged (respectively change sign) on a permutation of the indices (see 277). A covariant tensor of order s is symmetric (respectively antisymmetric) if it is symmetric (respectively antisymmetric) in every pair of indices. The symmetry or antisymmetry of a tensor is again invariant with respect to a change of local coordinates.
All the tensors of given orders at a point form a
vector space and the
collection of all of them forms a
tensor bundle of the manifold
M. In particular, all the contravariant (respectively
covariant) vectors form the tangent (respectively cotangent)
bundle of M. The tensor bundles are special cases of
a general vector bundle over M. A vector bundle
over M is a collection of vector spaces over the points
of M such that it is locally a product and that the linear
structures on the vector spaces (referred to as fibres) have
a respective representation. Two vector bundles E and
F over M can be added; their sum, E 
F, is the vector bundle the fibre of which at a point
of M is the direct sum of the fibres of E and
F, respectively. The set of all vector bundles over M,
with this addition, is a semi-group. By the introduction
of "virtual bundles" (analogous to the introduction of negative
integers in elementary arithmetic), all the finite linear combinations
of the vector bundles over M, real or virtual, are made
into a group. Through the tensor product of vector bundles the
group acquires a multiplication representation and has a ring
structure. This ring is called the
Grothendieck ring (after
the French mathematician Alexandre Grothendieck) or K-ring of
M; it is a significant global invariant of M. For
technical reasons a simpler concept is obtained by the consideration
of complex vector bundles over M, although M is
itself a real manifold.
The fibre need not be a vector space. An important example
is when it is the space of all
frames (i.e., ordered
sets of n linearly independent tangent vectors) at a
point P of M. The resulting
fibre space, the space
of all frames at all points of M, is called the principal
fibre space; its dimension is n
+ n. If M is the three-dimensional Euclidean
space and the vectors of the frame are supposed to form an orthonormal
system, then a frame is an oriented rectangular
trihedral; the corresponding
fibre space plays a basic role in the method of moving trihedrals
in the theory of curves and surfaces in Euclidean space and
in kinematics (motion without regard to mass or force). The
method of moving frames, used extensively and successfully by
Élie-Joseph Cartan, a French mathematician, in differential
geometry, is a forerunner of the notion of principal fibre space.
