Page 30 - Curvature and Homology

12 I. RIEMANNIAN MANIFOLDS

symmetric tensor field called the curl of the vector 6,. If the 6, define
a gradient vector field, that is, if there exists a real-valued function f
defined on an open subset of M such that 6, = (af/dui), the curl must
vanish. Conversely, if the curl of a (covariant) vector field vanishes, the
vector field is necessarily a (local) gradient field.

1.4. Differential forms
Let M be a differentiable manifold of dimension n. Associated to each
point P E M, there is the dual space Tp* of the tangent space Tp at P.
We have seen that T$ can be identified with the space of linear differential
forms at P. Hence, to a 1-dimensional subspace of the tangent space
there corresponds a linear differential form. We proceed to show that
to a p-dimensional subspace of Tp corresponds a skew-symmetric
covariant tensor of type (0, p), in fact, a 'differential form of degree P'.
To this end, we construct an algebra over Tp* called the Grassman or
exterior algebra:
An associative algebra A (V) (with addition denoted by + and
multiplication by A) over R containing the vector space V over R
is called a Grassman or exterior algebra if
(i) A (V) contains the unit element 1 of R,
(ii) A (V) is generated by 1 and the elements of V,
(iii) If x E V, x A x = 0,
(iv) The dimension of A (V) (as a vector space) is 2n, n =dim V.
Property (ii) means that any element of A (V) can be written as a
linear combination of 1 E R and of products of elements of V, that is
A (V) is generated from V and 1 by the three operations of the algebra.
Property (iii) implies that x A y = - y A x for any two elements
x, y E V. Select any basis (el, ..., en} of V. Then, A (V) contains all
products of the e, (i = 1, ..., n). By using the rules

we can arrange any product of the e, so that it is of the form

or else, zero. The latter case arises when the original product contains
a repeated factor. It follows that we can compute any product of two
or more vectors alel + ... + anen of V as a linear combination of the