Page 55 - Curvature and Homology

1.10. SECTIONAL CURVATURE 37

where K denotes the common value of R(P, n) for all planes n. By
(1 -8.1 1)
Sijkl = 'fh f&) ft0 (RPr gzla - gas gpr)
(1.10.5)
= K(Sjk Sit - Sik)
since the frames are orthonormal. Equation (1.10.5) may be rewritten
by virtue of the second of equations (1.8.9) as

If we assume that at every point P E M, R(P, n) is independent of
the plane section n, then, by substituting (1.10.6) into (1.9.18) and
applying (1.9.16) we get
dK A Oi A Oj = 0.
Hence, dK must be a linear combination of B, and 0, from which dK = 0
if n 2 3. This result is due to F. Schur: If the sectional curvature at
every point of a Riemannian manqold does not depend on the two-dimenst'mal
section passing through the point, then it is constant over the manyold.
Such a Riemannian manifold is said to be of constant curwature.
Assume that the constant sectional curvature K vanishes. We may
conclude then that the tensor Rfkl vanishes, and so the manifold is
locally flat. This means that there is a coordinate system with the
property that relative to it the components ck) of the Levi Civita
connection vai,iqh. For, the equations

obtained from (1.7.4) by putting rf, = 0 are completely integrable.
Hence, there is a coordinate system in which the r& vanish. It follows
that the components gjk of the fundamental tensor are constants. Thus,
we have a local isometry from the manifold to En. Conversely, if such
a map exists, then clearly Pjk, vanishes.
Let X, = ff4)(a/Chk) (i = 1, ..., n) denote n mutually orthogonal unit
vectors at a point in a Riemannian manifold with the local coordinates
ul, ..., un. Then from (1.9.1)

It follows from the equations (1.9.1 1) that