Page 112 - Curvature and Homology
P. 112
94 111. RIEMANNIAN MANIFOLDS: CURVATURE, HOMOLOGY
Another estimate is obtained from the inequalities
by following a similar procedure. In fact,
+
I
I R~~~~ I; #(A'D')~~~ (B'c')~~~] (3.2.22)
where
A'=2-Kij-Ki,, B'=2-Kij-.Kik,
C' = 2 - Klk - Klj, D' = 2 - Kk, - Kk,.
From (3.2.21) and (3.2.22) we deduce
This estimate for the components of the curvature tensor is now
applied to (3.2.10). Indeed, for p = 2
The right hand side may be evaluated more readily by choosing an
orthonormal basis (X,, X,,), s = 1, ..a, m such that only those components
of a of the form a,,, are different from zero. (The existence of such a
basis is a - :andard fact in linear algebra.) Hence,
Consequently, since K, >= S, K,,, 2 S for all s and i we obtain by
virtue of (3.2.23)
