Showing papers on "Ricci decomposition published in 1967"

Vertical and complete lifts from a manifold to its cotangent bundle

Abstract: Publisher Summary This chapter considers various methods by which certain types of tensor fields in M can be extended to c T (M) to give useful information about the relationships between the structures of the two manifolds. The chapter focuses on complete lifts of vector fields, tensor fields of type (1, 1) and skewsymmetric tensor fields of type (1, 2). In each of these cases, the complete lift is defined to be a tensor field of the same type as the original. In general, the vertical lift of a tensor field does not have the same type as the original; nevertheless the construction is a useful one. The methods enable to examine the structure of C T( M ) in relation to that of M. In particular, it is shown how almost complex and similar structures on M can be extended to C T(M). Lifts of affine connections in M is also examined, using the idea of a Riemann extension.

Projective-symmetric spaces

Abstract: Gy. Soos [1] and B. Gupta [2] have discussed the properties of Riemannian spaces V n ( n > 2) in which the first covariant derivative of Weyl's projective curvature tensor is everywhere zero; such spaces they call Protective-Symmetric spaces . In this paper we wish to point out that all Riemannian spaces with this property are symmetric in the sense of Cartan [3]; that is the first covariant derivative of the Riemann curvature tensor of the space vanishes. Further sections are devoted to a discussion of projective-symmetric af fine spaces A n with symmetric af fine connexion. Throughout, the geometrical quantities discussed will be as defined by Eisenhart [4] and [5].

Über das Schwarzsche Lemma und Verwandte Sätze

Abstract: Upper bounds for the Jacobian determinant by holomorphic mappings of bounded domainsD into itself were given first more then thirty years ago by Stefan Bergman by means of his theory of the kernel function ofD. In this paper a different method shall be developed and distortion theorems for holomorphic mappings of bounded domains of a Kahler manifoldM n into a Kahler manifoldM 0 shall be proved. The special casesM n =C n (unit sphere of C n ) andM n =M 0 =|C n shall also be considered. The proof depends essentially on the two Hermitian quadratic forms corresponding to the metric and to the Ricci tensor. The manifolds must be of negative Ricci curvature and fulfil two conditions given in section 4.

On Classification of Curvature Tensor

Abstract: A new method for classification of the gravitational field is presented. It is shown that this scheme leads to exactly the same types of gravitational fields as previously obtained by Petrov, Penrose, and Sachs, etc. Some physical applications are also given.

Geometrization of a Complex Scalar Field. II. Analysis

Abstract: The necessary and sufficient conditions that a Ricci tensor should represent the energy tensor of a complex scalar field are given for the general non‐null case. The complexion gradient of the field is determined only to within a sign by the Ricci tensor, unlike the case of the electromagnetic problem. The physical content of a field which corresponds to massless ``pions'' is expressed entirely in geometric terms within the Rainich scheme.