Page 51 - Curvature and Homology

1.9. RIEMANNIAN GEOMETRY 3 3

(cf. I.GS),

where the forms 4 and Oij (i< j) are linearly independent; moreover,
the functions Sijk, (cf. (1.8.9)) have the symmetry properties

Equations (1.9.16) and (1.9.17) are the restrictions to B of the cor-
responding equations (1 3.7) and (1.8.8).
Consider the bundle of frames over C(t) and denote once again the
restrictions of 8,, Oij to the submanifold over C(t) by the same symbols.
To describe this bundle we choose a family of orthonormal frames
(Al(t), ..., A,(t)) along C(t)-one for each value of t. Then, for a given
value of t the vectors Xl(t), ..., Xn(t) of a general frame are given by

The frames (Xl(t), ..., Xn(t)) can be mapped into frames in the bundle
An of frames over An so that their relative positions remain unchanged.
In particular, frames with the same origin along C(t) are mapped into
frames with the same origin in An. This follows from the fact that
under the mapping the 8, and Oij are the dual images of corresponding
differential forms in An (cf. I.F. I).
Let C(tl) and C(t2) be any two points of C(t). A vector of Tcct) is
given by xiAi(t). Consider the map which associates with a vector
xgAi(tl) E Tc(t,) the vector $Ai(t2) E TC,q defined by

where the prime denotes the image in An of the corresponding vector
with origin on C(t) and Cf(t) is the image of C(t). In this way, the
various tangent spaces along C(t) can be 'compared'. This situation may