Page 54 - Curvature and Homology

36 I. RIEMANNIAN MANIFOLDS
3 5 r 5 n) are equal to zero at P. Conversely, if the Oar vanish at P,
then from (1.9.16) and (1.9.17)

These are the equations which hold on S. Hence, the quantity - S1,,,
at a point P of a Riemannian manifold is equal to the Gaussian curvature
at P of the surface tangent to the plane spanned by the first two vectors
and which is geodesic at P.
The Gaussian curvature at a point P of the surface geodesic at P
and tangent to a plane w in the tangent space at P is called the sectional
curvatur~ at (P, W) and is denoted by R(P, w). If p, qi are two ortho-
normal vectors which span W, it follows from (1.8.1 1) that

since R$jkl = g$nhpjkl*
Let f*C, r)*i be any two vectors spanning W. Then,

where ad - bc # 0. In terms of the vectors f*i, rl*t,
R(P,r) = - (ad - bc)2 Rijkl f*i ?*j f*k 7*l,

where llad - bc: is the oriented area of the parallelogram with (*i, q*i
as adjacent sides:

If we drop the asterisks, we obtain the following formula for the sectional
curvature at (P, v):

Now, assume that R(P, W) is independent of W, that is, suppose that
the sectional curvature at (P, W) does not depend on the two-dimensional
section passing through this point. Then, from (1 .lO.3), we obtain