Page 70 - Curvature and Homology

52 I. RIEMANNIAN MANIFOLDS

The idea of translating, wherever possible, problems of Riemannian geometry
to problems of Euclidean geometry is due to E. Cartan [Le~ons uur la ghrndtrie
des espaces de Riernann, Gauthier-Villars (1928; 2nd edition, 194611.

He Geodesic coordinates
1. Show that at the pole of geodesic coordinates (uf) the Riemannian curvature
tensor has the components

Hence, the curvature tensor has the symmetry property (1.9.20).

I. The curvature tensor
1. The curvature tensor (which we now denote by L) of a Riemannian manifold
with metric tensor g is completely determined by the sectional curvatures.
To see this, consider L as a transformation

The relation (a) says that as a function of the first two variables L depends only
on X A Y. Thus, we may write

The metric tensor g may be extended to an inner product on Az(T) as follows:
Mll A Xl*, X*l A Xm) = det g(Xfj9 XP,*)

for any vectors Xll, XI,, X,, X, E T where i,j = 1,2; l* = 2, 2* = 1. Then,
(b) says that g(L(X A Y,Z),W) is a function of Z A W only. Hence, there is a
unique QX A Y) E AYT) such that

The relation (c) says that is a symmetric transformation of AYT).