Page 56 - Curvature and Homology

38 I. RIEMANNIAN MANIFOLDS
The sectional curvature K,, determined by the vectors X, and X,
is given by
KT, = - &kt Sf,) &I Sf",, &. (1.10.9)
Taking the sum of both sides of this equation from s = 1 to s = n
we obtain

2 Krs = Rik St, 5:,, (1.10.10)
8-1

where we have put Rik = - gj1RU,,, that is

The tensor Rf, is called the Ricci curvature tensor or simply the Ricci
tensor. Again,

where we have put
R = gik Rik.
The scalar Rik (t,., (f,, is called the Ricci curvature with respect to the
unit tangent vector X,. The scalar R determined by equation (1.10.12)
is independent of the choice of orthonormal frame used to define it.
It is called the Ricci scalar curvature or simply the scalar curvature.
The Ricci curvature K in the direction of the tangent vector is
defined by

It follows that
(Rjk - ~8jk) fj Sk = 0.
The directions which give the extrema of K are given by

In general, there are n solutions (f, ,, ..., &, of this equation which are
mutually orthogonal. These directions are called Ricci directions. A
manifold for which the Ricci directions are indeterminate is called
an Einstein manifold. In this case, the Ricci curvature is given by