The Riemann Tensor. If we have given a curve extending from the point to the point of the continuum, then a vector , given at , may, by a parallel displacement, be moved along the curve to . If the continuum is Euclidean (more generally, if by a suitable choice of co-ordinates the , are constants) then the vector obtained at as a result of this displacement does not depend upon the choice of the curve joining and . But otherwise, the result depends upon the path of the displacement. FIG.4. In this case, therefore, a vector suffers a change, (in its direction, not its magnitude), when it is carried from a [Pg 79] point of a closed curve, along the curve, and back to P. We shall now calculate this vector change: As in Stokes' theorem for the line integral of a vector around a closed curve, this problem may be reduced to the integration around a closed curve with infinitely small linear dimensions; we shall limit ourselves to this case. We have, first, by (67), In this, is the value of this quantity at the variable point of the path of integration. If we put [Pg 80] and denote the value of at by then we have, with sufficient accuracy, Let, further, be the value obtained from by a parallel displacement along the curve from to . It may now easily be proved by means of (67) that - is infinitely small of the first order, while, for a curve of infinitely small dimensions of the first order, is infinitely small of the second order. Therefore there is an error of only the second order if we put If we introduce these values of and into the integral, we obtain, neglecting all quantities of a higher order of small quantities than the second, The quantity removed from under the sign of integration refers to the point . Subtracting from the integrand, we obtain This skew-symmetrical tensor of the second rank, , characterizes the surface element bounded by the curve in magnitude and position. If the expression in the brackets in (85) were skew-symmetrical with respect to the indices and , we could [Pg 81] conclude its tensor character from (85). We can accomplish this by interchanging the summation indices and in (85) and adding the resulting equation to (85). We obtain in which The tensor character of follows from (86); this is the Riemann curvature tensor of the fourth rank, whose properties of symmetry we do not need to go into. Its vanishing is a sufficient condition (disregarding the reality of the chosen co-ordinates) that the continuum is Euclidean. By contraction of the Riemann tensor with respect to the indices , , we obtain the symmetrical tensor of the second rank, The last two terms vanish if the system of co-ordinates is so chosen that . From , we can form the scalar,