Page 25 - Curvature and Homology

1.2. TENSORS 7
The invariant expression q,dui is called a linear (dzyerential) form or
1-form. Conversely, when a linear (differential) form is given, its coeffi-
cients define an element of T$. If we agree to identify T,* with the space
of 1-forms at P, the dui at P form a base of T,* dual to the base a/ad
(i = 1, ..., n) of tangent vectors at P:

where 6j is the 'Kronecker delta', that is, 6j = 1 if i = j and 8j = 0 if i # j.
We proceed to generalize the notions of contravariant and covariant
vectors at a point P E M. To this end we proceed in analogy with the
definitions of contravariant and covariant vector. Consider the triples
(P, U,, gl-irjl...j,) and (P, Up, ~l.-i~jl...j8). . They . are said to be equivalent
if P = P and if the nr+. constants ,$'1.-'rjl..,, are related to the nr+.
constants @-..irjl--j, by the formulae

An equivalence class of triples (P, U,, @...irjl..,j.) is called a tensor of type
(r, S) over Tp contravariant of order r and covariant of order s. A tensor
of type (r, 0) is called a contravariant tensor and one of type (0, s) a
covariant tensor. Clearly, the tensors of type (r, s) form a linear space-
the tensor space of tensors of type (r, s). By convention a scalar is a tensw
of type (0,O).
If the components fi-.+i*jl...j, of a tensor are all zero in one local coor-
dinate system they are zero in any other local coordinate system. This
tensor is then called a zero tensor. Again, if fi...irjl...j, is symmetric or skew-
symmetric in &, i, (or in j,, j,), ~l-irjl.,.j, has the same property. These
properties are therefore characteristic of tensors. The tensor @,'r (or
is said to be symmetric (skew-symmetric) if it is symmetric (skew-
symmetric) in every pair of indices.
The product of two tensors (P, U,, @-..irjl...j.) and (P, U,, qil...i~;l...j8,)
one of type (r, s) the other of type (r', sf) is the tensor (P, U,, @-.'rjl...j,
.
.
q'r+l...'r+r' .
j,+,*) of type (r + r', s + sf). In fact,

aa .
I,... l, vkr+~**.kr+r,
l,+l..'Z.+,~-