Page 57 - Curvature and Homology

1.10. SECTIONAL CURVATURE 39
If we multiply both sides of this equation by gjk, we obtain

(In the sequel, the operation of multiplying the components of a tensor
by the components of the metric tensor and contracting will be called
trattsolection.) It follows that

Now, the Bianchi identity (1.8.14), or rather (1.9.18) can be expressed as

where Df denotes covariant differentiation in terms of the Levi Civita
connection. Transvecting this identity with gim we obtain ---

which upon transvection with gjk results in

Substituting (1.10.19) into (1.10.22) and noting that

we see that for n > 2, the scalar curvature is a constant. Hence, in an
Einstkn mani~old the scalar curvature is constant (n > 2).
It should be remarked that the tensor Rjk is symmetric. In fact, from
equations (1.8.11) and (1.9.2 1) we obtain

Contracting (1.10.24) with respect to i and I gives

by virtue of the symmetry relations (1.9.19) and the definition (1.10.1 1).
Hence, the Ricci curvature tensor is symmetric.