# Differential Geometry. The Mathematical Works of J. H. C. by John Henry Constantine Whitehead

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14a n ) we have, if p = 0, p = / > + l , • • • , w. T h e s t u d y of the connection L m a y , therefore, be confined to its behavior on the sub-space Vp, of Vni given p + 1 n by # = • • • = # = 0. T h e number /> m a y be called the class of the connection* L. 14). 14a n ) assert that the connection L defines an integrable displacen ment along the curves of parameter x . If L defines an integrable displacement on each surface of the congruence given b y a a {a = 1, x = c 2), n a where c are arbitrary constants, we can discard the condition # = 0 in ( 7 .

852, also p. 1083. 22 ON LINEAR CONNECTIONS T h e associated space at each point can be identified with the sub-space spanned in the tangent space b y the vectors £«, but the essential feature which distinguishes the theory of a linear connection from that of an affine connection is retained: namely that the frame of reference m a y be changed in each associated space independently of coordinate transformations. 7. Integral sub-spaces. 1) dZ« + Z U £ < * * ' - 0. 1 n 1 m I t will be convenient to say that a set of numbers (x , • • • , x ; Z , • • • , Z ) determine, on the one hand a point x in the underlying'manifold Vn, together with a point Z in the linear space associated with x, and on the other hand a point in a space oim+n dimensions, which we shall denote b y Sn+m.

Let Hjk be the components of D in a coordinate system x, in which the i i l equations to G are x = {y ). 2) k = 0 1 which reduce to p\ for y — 0 . The equations to the path C 2 are given b y l those solutions, 4>\{y y t), to the equations = 0. 3) -= v\(y\ 0, • • • , 0 ) . * At the rth step we shall move the matrix along the path C r, which passes through PR-I in the l r direction vr, from P r~i to a point P r , whose coordinates are to be (y , • • • , y } 0. - • • , 0). ON LINEAR CONNECTIONS 15 T h e components va at P 2 are given b y the sets of solutions to the equations 0 l 2 which reduce to va(y , 0, • • • , 0) for t = 0.