Page 156 - Curvature and Homology

138 IV. COMPACT LIE GROUPS
Since the equations (4.1.2) may be expressed in terms of the local
coordinates (u" in the form

it is easy to check that
i P Y
T,: = g,3; 5. 6, Zk
from which we conclude that the covariant torsion tensor

Tjkl = gig Gki
is skew-symmetric. It follows from (1.9.12) that

where the u,} are the coefficients of the Levi Civita connection. Hence,
from (4.3.7)

Lemma 4.3.1. The elements of the Lie algebra L of G a'ejne translations
in G.
Indeed, from (4.3.12) and (4.3.8)

where D, is the operator of covariant differentiation with respect to the
Levi Civita connection. Multiplying these equations by and con-
tracting we obtain
e:
- f: Dk = T,:.
Again, if we multiply by 6; and contract, the result is

These equations may be rewritten in the form
Dk 6; = T,: 8

from which we conclude that 8(XB)g = 0.