From time to time important contributions to Riemannian geometry were made by Bianchi, Beltrami, Christoffel, Voss and others, and Ricci co-ordinated and extended the theory simul taneously with the development of tensor calculus. These con tributions include the study of a sub-space of a Riemannian space analogous to that of a surface in ordinary space. Such a sub-space of order r is the locus of points defined by the equations xi=iki(ul, . . . , (i= I, • • • , ii), (6) where the u's are independent parameters. When these expres sions are substituted in (I), we obtain an induced metric for the sub-space—a generalization of the first fundamental differential form of a surface (see DIFFERENTIAL GEOMETRY). There is also a generalized second fundamental form, whose coefficients enter in the relations between the curvatures of a curve in the sub space relative to the latter and the curvatures of the curve as of the enveloping space. Among the curves of the sub-space there are geodesics, lines of curvature, asymptotic lines and conjugate systems of curves, which are generalizations of these types of curves on a surface in ordinary space. Einstein based his theory of gravitation upon the assumptions that physical space and time constitute a four-dimensional continuum whose metrical character is determined by the presence of matter, and that these spaces are of a particular kind defined in invariantive form by means of the curvature tensor ; in this theory the fundamental form (I) is not positive for every choice of the differentials. This and
other physical interpretations of differential geometry of spaces have stimulated the development of the theory.
Notable among the recent contributions is the concept of parallelism of vectors in a general Riemannian space as intro duced by Levi-Civita. In such a space parallelism is not absolute, as it is in Euclidean space, but is relative to the curve joining the points of application of the vectors. Thus for a curve xi=fi(s) each set of solutions of the equations are the components of a family of vectors at the points of the curve which are parallel to one another with respect to the curves. Certain Riemannian spaces admit one or more fields of vectors, such that any two of them are parallel with respect to any curve joining their points of application. When there are n independent fields of this kind, the space is flat. In particular, the tangents to a geodesic are parallel with respect to the geodesic, dxi as follows from (7) and (4), when we put Ei = , geodesics are ds the straight lines of the space. This concept of parallelism is involved in many of the recent developments of Riemannian geometry and its generalizations have opened up new fields (see