Page 160 - Curvature and Homology

142 IV. COMPACT LIE GROUPS
tensor by Eijkl with respect to a given system of local coordinates
ul, -a, un we obtain

where the Ri,,, are the components of the Riemannian curvature tensor.
Since the Ejkl all vanish and since D,Tj," 0, it follows from the
Jacobi identity that

By virtue of the equations (4.3.1) and (4.3.10)

Hence, forming the Ricci tensor by contracting on i and 1 in (4.5.1)
we conclude that
Rjk = h,k*
It follows that G is locally an Einstein space with positive scalar curvature,
and so by theorem 3.2.1, the first betti number of G is zero.
In order to prove that b,(G) is also zero we establish the following
Lemma 4.5.1. In a coordinate naghborhood U of G with the local
coordinates (ui) (i = 1, .so, n), we have the inequalities

where the fij = - fji are functions in U defining a skew-symmetric
tensor $eld f of type (0,2) andf = t(ij, dui A duj [74.
In general, the curvature tensor defines a symmetric linear trans-
formation of the space of bivectors (cf. 1.1.). The above inequality says
it is negative definite with eigenvalues between 0 and - 4.
Since the various sides of the inequalities are scalar functions on G
the lemma may be proved by choosing a special system of local
coordinates. In fact, we fix a point 0 of G and choose (geodesic)
coordinates so that at 0, gij = 8;. Then, since

and so the 21/Z TI, (r < s, j = I, am., n) represent n orthonormal vector