1 8 4 D O C . 1 9 R E L AT I V I T Y L E C T U R E N O T E S
[91]The introduction of the curvature tensor through the consideration of a “parallel field” in a
Euclidean space follows Weyl 1918b, p. 108. The “parallel field” is constructed by taking some fixed
vector at a chosen origin and parallel transporting that vector to all other points of the manifold. This
only gives a well-defined vector field in a Euclidean space, where parallel transport from one point to
another is independent of the path chosen.
[92]The definition of the curvature tensor on this page differs by a minus sign from the def-
inition given on [p. 11].
[93]In the manuscript, the three lines below, which give the first few terms in , appear in
the left-hand margin.
[94]The expression for is equivalent to the one given at the foot of [p. 11] (after correction
of the errors mentioned in note 19) and differs by a minus sign from the expression one obtains by
computing using the expression for above.
Rμ
τν
α
gακRiα
lm
Riκ,lm
gακRiα
lm
Riα
lm