RIEMANNIAN GEOMETRY. Any n independent vari ables xi where i takes the values I to n, may be thought of as the coordinates of an n-dimensional space, or variety V in the sense that each set of values of the x's defines a point of In a space as thus defined there is not an a priori basis for the determination of magnitude nor for the comparison of directions at two different points. Riemann proposed the study of the metric properties of a general V by introducing as the basis for measurement a quadratic differential form where the g's are functions of the x's, subject to the restrictions that the determinant of the g's is not zero and that for all values of the differentials the above sum is positive. By definition the distance ds between the points of coordinates xi and xid-dxi is given by This is a generalization of the first fundamental form of a surface in ordinary space when the surface is defined in terms of two para meters, as proposed by Gauss (see DIFFERENTIAL GEOMETRY). In this case the metric on the surface is induced by the Euclidean metric of the enveloping space, whereas in a general Riemannian space the metric is assigned.
From the hypotheses concerning (I) it can be shown that at any point is less than unity for two different sets of differentials dxi and 6xi. Consequently a real angle 0 is determined by the equation by definition it is the angle between the directions at the point determined by the two sets of differentials. This is in keeping with the fact that the cosine of the angle between two tangents, at a point, to a surface in ordinary space when expressed in terms of the induced metric, is given by an equation of the form (2).
When we have n independent functions of the x's the equations define a transformation of coordinates of the space. If the g's in (I) are such, which is rarely the case, that by a suitable trans formation the form (I) is reducible to which is a generalization of the metric of ordinary space in cartesian coordinates, we say that the space is flat, or plane; otherwise it is curved. The locus of points defined by for all values of the parameter t is called a curve. When these expressions are substituted in (I), we obtain an expression of the form ds=F(t)dt, and then the length of arc of the curve is given by integration. If the result of the integration is s =OW, by
means of this equation the coordinates at points of the curve are expressible as functions of the arc s as parameter. The theory of curves involves n—e principal curvatures, which are generalizations of the curvature and torsion of a curve in ordinary space.
Using the terminology of the calculus of variations, we say that the extremals of the integral are the shortest lines, or geodesics, of the space. The geodesics are found to be the integral curves of a system of differential equations where the F's are certain functions of the g's and their first derivatives. When the space is flat and the coordinates are those for which the fundamental form is (3), all the functions F vanish identically. Consequently in the coordinate system the equations of the geodesics of the flat space are where the a's and b's are constants. Thus the geodesics of a Riemannian space are the analogues of straight lines of a Eucli dean space. Riemann showed that in a general space a coordinate system exists such that all the geodesics through a given point are defined by a is (i—i,...n) = I, but those through other points are not given by (5). In such a coordinate system the F's vanish at the given point, but not their derivatives.
Two sets of differentials and determine two directions at a point, and adxid-b6xi, where a and b are parameters, a linear pencil of directions at the point. The geodesics issuing from a point P in a linear pencil of directions constitute a surface; the Gaussian curvature (see DIFFERENTIAL GEOMETRY) of this sur face at P was taken by Riemann as the measure of curvature of the space for the given pencil. It is expressed in terms of the directions, and the components of a tensor of the fourth order, which involves the functions and their first derivatives; it is now known as the Riemannian curvature tensor (see TENSOR). Ordinarily the curvature varies with the choice of the pencil. Schur showed that, if it is the same for all pencils at each point of the space, then it has the same value at every point ; these are the spaces of constant Riemannian curvature ; when, and only when, the constant is zero, the space is flat.