Page 108 - Curvature and Homology

90 III. RIEMANNIAN MANIFOLDS: CURVATURE, HOMOLOGY

of En by a subgroup of translations and is therefore locally equivalent
to ordinary affine space where there is no distinction made between
vectors and covectors.
Consider the sectional curvature determined by the plane n defined
by the orthonormal tangent vectors X = e(a/aui) and Y = ?li(a/aut)
at P. Then,

Assume that for all planes w at all points P of M there are constants
Kl and K, such that

Let {XI, ..., Xn) be an orthonormal frame at P where Xj = (&(a/a~$)
(j = 1, .--, n). Then, since

r, s = 1,2, ..., n, it follows that

the inequalities holding for arbitrary unit tangent vectors X,. Hence,
for any tangent vector X = p(a/auc)
(n - 1) Kl 5'" ti 4 Rik 5i tk 5 (a - 1) K2 5i &. (3.2.15)

It follows from 5 1.2 (by taking tensor products) that

for any tensor whose components are expressed in the given local
coordinates. In terms of the bivector

where X and Y are orthonormal tangent vectors, the inequalities (3.2.14)
become by virtue of (3.2.13)

(The curvature tensor defines a symmetric linear transformation of the
space of bivectors (cf. 1.1.1). These inequalities say that it is positive