test ocr 2

necessary to consider the second bundle. The curvature form of our connection is a tensorial quadratic differential form in M, of type ad(G^′) and with values in the Lie algebra L(O^′) of G^′. Since the Lie algebra L(O) of G is a subalgebra of L(G^′), there is a natural projection of L(O^′) into the quotient space L(G^′)∕L(G). The image of the cur- vature form under this proiection will be called the torsion form or the torsion tensor. If the forms π^ρ in (13) define a G-connection, the vanishing of the torsion form is expressed analytically by the con- ditions

We proceed to derive the analytical formulas for the theory of a G-connection without torsion in the tangent bundle. In general we will consider such formulas in B_G. The fact that the O-connection has no torsion simplifies (13) into the form

By taking the exterior derivative of (23) and using (18), we get

where we put

For a fixed value of k we multiply the above equation by

getting

or ΣρaρkΠρl ≡ 0, mod ω;.

Since the infinitesimal transformations X_ρ are linearly independent, this implies that

It followo that II ρ is of the form

where ϕ_j^ρ are Pfaffian forms. Substituting these expressions into (24), we get

It follows that

Since G has the property (C), the above equations imply that

OCR based on this tiff scan