Page 111 - Curvature and Homology

1
(
I j I 5 1 - 6) i#j,k*
In order to obtain estimates for the R4jk, ((i, j) # (k, I), i < j, k < I)
we consider the inequalities

for any orthonormal set of vectors {Xi, Xj, X,., X,) and a, b, c, d E R. Put

The function F may be considered as a polynomial in a, b, c and d.
As such it is of degree 4 but only of degree 2 in the a, b, c, d taken
by themselves. The polynomial

contains only terms in a2c2, a2d2, bec2, b8d2 and abed. Now, put

By employing the identities (1.10.24) and (1.9.20) in the term involving
abed then, by virtue of (3.2.18) and (3.2.19), the polynomial H may be
expressed as
H = Aa2c2 + Ba2d2 + Cb2c2 + Dbed2 + 2Eabcd 2 0 (3.2.20)
where
A = Kir + Kit - 26, B = Kjj + Kit - 26,

By a suitable choice for a and b the inequality (3.2.20) gives rise to
ACc' + (AD + BC - &?)8dB + BDd' 2 0.
Since this inequality holds for any c and d