Page 26 - Curvature and Homology

8 I. RIEMANNIAN MANIFOLDS
It is also possible to form new tensors from a given tensor. In fact,
be a tensor of type (r, .
let . (P, . U,, ~l-irjl-..j.) The triple (P, U,,
...
,.
eel .. i,...ja:..j.) where the indices i, and j, are equal (recall that repeated
s
I
indices ~cd~cate summation from 1 to n) is a rLpesentative of a tensor
of type (r - 1, s - 1). For,

since

This operation is known as contraction and the tensor so obtained is
called the contracted tensor.
These operations may obviously be combined to yield other tensors.
A particularly important case occurs when the tensor Cij is a symmetric
covariant tensor of order 2. If qC is a contravariant vector, the quadratic
form fij qi r)j is a scalar. The property that this quadratic form be
positive definite is a property of the tensor CU and, in this case, we call
the tensor positive dejinite.
Our definition of a tensor of type (I, s) is rather artificial and is
actually the one given in classical differential geometry. An intrinsic
definition is given in the next section. But first, let v be a vector
space of dimension n over- R and let V* be the dual space of V. A
tensor of type (r, s) over V, contravariant of order r and covariant
of order s, is defined to be a multilinear map of the direct product
V x ... x V x V* X ... x V* (V:s times, V*:r times) into R. All tensors of
type (r, s) form a linear space over R with respect to the usual addition
and scalar multiplication for multilinear maps. This space will be
denoted by Ti. In particular, tensors of type (1,O) may be identified with