In a previous paper we have given an intrinsic proof of the formula of Allendoerfer–Weil which generalizes to Riemannian manifolds of \( n \) dimensions the classical formula of Gauss–Bonnet for \( n=2 \). These two topological invariants have a linear combination which is the Euler–Poincaré characteristic. 0ur main result consists of two formulas, which express two topological invariants of a compact orientable differentiable manifold of four dimensions as integrals over the manifold of differential invariants constructed from a Riemannian metric previously given on the manifold. This necessitates a more careful study of spherical geometry than hitherto given in the literature, except, so far as the writer is aware, in a paper by E. Study. It is the purpose of this note to give a study of a compact orientable Riemannian manifold of four dimensions at each point of which is attached a three-dimensional spherical space. On the other hand, a knowledge of three-dimensional elliptic or spherical geometry is useful for the study of orientable Riemannian manifolds of four dimensions, because their tangent spaces possess a geometry of this kind. This last-mentioned property has no analogue for orthogonal groups in \( n \) ( \( > 4 \)) variables. A group-theoretic reason for the most important of these properties is the fact that the universal covering group of the proper orthogonal group in four variables is the direct product of the universal covering groups of two proper orthogonal groups in three variables. It is well known that in three-dimensional elliptic or spherical geometry the so-called Clifford’s parallelism or parataxy has many interesting properties. As applications of the general notions we give in the last section two formulas which are respectively generalizations of the well-known formulas of Crofton and Cauchy. In §2 we define the incidence of the geometrical elements of different fields, which plays a fundamental rôle in all subsequent discussions. In §1 we give a brief summary of Cartan’s theory of Lie’s groups with some results concerning the measures of geometrical elements. The paper is divided into three sections. The discussion is mainly based on Cartain’s theory of Lie’s groups. It is the object of this paper to give the fundamental concepts of integral geometry in a general space of Klein, by which we mean a number space of \( n \) dimensions with a transitive \( r \)-parameter group \( G_r \) of transformations. The classical results in integral geometry found by Crofton, Poincaré, Cartan, and recently developed by Blaschke and his school, are mostly restricted to Eulidean spaces.
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