Page 24 - Curvature and Homology

6 I. RIEMANNIAN MANIFOLDS
(i = 1, ..., n). That the contravariant vectors form a linear space over R
is clear. In analogy with surface theory this linear space is called the
tangent space at P and will be denoted by Tp. (For a rather sophisticated
definition of tangent vector the reader is referred to $3.4.)
Let f be a differentiable function defined in a neighborhood of
P E Ua n Ug. Then,

Now, applying (1.2.6) we obtain

The equivalence class of "functions" of which the left hand member
of (1.2.8) is a representative is commonly called the directional derivative
off along the contravariant vector e. In particular, if the components
e(i = 1, ..., n) all vanish except the kth which is 1, the directional
derivative is the partial derivative with respect to uk and the corres-
ponding contravariant vector is denoted by a/auk. Evidently, these vectors
for all k = 1, ..., n form a base of Tp called the natural base. On the
other hand, the partial derivatives off in (1.2..8) are representatives of
a vector (which we denote by df) in the dual space T,* of Tp. The
elements of T,* are called covariant vectors or, simply, covectors. In the
sequel, when we speak of a covariant vector at P, we will occasionally
employ a set of representatives. Hence, if T~ is a covariant vector and e
a contravariant vector the expression is a scalar invariant or, simply
scalar, that is
and so,

are the equations of transformation defining a covariant vector. We
define the inner product of a contravariant vector v = e and a covariant
vector w* = 7, by the formula
(0, w*) = Tip. (1.2.11)
That the inner product is bilinear is clear. Now, from (1.2.10) we obtain

where the duqi = 1, ..., n) are the differentials of the functions ul, ..., un.