Page 23 - Curvature and Homology

1.2. TENSORS 5
1.2. Tensors

To every point P of a regular surface S there is associated the tangent
plane at P consistihg of the tangent vectors to the curves on S through P.
A tangent vector t may be expressed as a linear combination of the
tangent vectors Xu and X, "defining" the tangent plane:

t=tlXu+pX,,, PER, i=1,2. (1 -2.1)
At this point, we make a slight change in our notation: We put u1 = u,
u2 = v, XI = Xu and X, = X,, so that (1.2.1) becomes

Now, in the coordinates zil, zi2 where the zii are related to the d by means
of differentiable functions with non-vanishing Jacobian

where 8 = X(ul (zil, ii2), u2 (zil, 3). If we put

equation (1.2.3) becomes
t = pxj.

In classical differential geometry the vector t is called a contravariant
vector, the equations of transformation (1.2.4) determining its character.
Guided by this example we proceed to define the notion of contravariant
vector for a differentiable manifold M of dimension n. Consider the
triple (P, U,, p) consisting of a point P E M, a coordinate neighborhood
U, containing P and a set of n real numbers ti. An equivalence relation
is defined if we agree that the triples (P, U,, e) and (P, Up, p) are
equivalent if P = P and

where the u%re the coordinates of u,(P) and iii those of ue(P), P E Up Up.
An equivalence class of such triples is called a contravariant vector at P.
When there is no danger of confusion we simply speak of the
contravariant vector by choosing a particular set of representatives