Salomon Bochner

Curvature and Betti Numbers

*Frontmatter, pg. i*Preface, pg. v*Contents, pg. vii*Chapter I. Riemannian Manifold, pg. 2*Chapter II. Harmonic and Killing Vectors, pg. 26*Chapter III. Harmonic and Killing Tensors, pg. 59*Chapter IV. Harmonic and Killing Tensors in Flat Manifolds, pg. 77*Chapter V. Deviation from Flatness, pg. 81*Chapter VI. Semi-simple Group Spaces, pg. 90*Chapter VII. Pseudo-harmonic Tensors and Pseudo-Killing Tensors in Metric Manifolds with Torsion, pg. 97*Chapter VIII. Kaehler Manifold, pg. 117*Chapter IX. Supplements, pg. 170*Bibliography, pg. 187*Backmatter, pg. 192

Almost periodic functions in groups. II

The present paper is a continuation of the article by J. von Neumann on Almost periodic functions in a group, I [l].f Its main object is to extend the theory of almost periodicity to those functions having values which are not numbers but elements of a general linear space L. For functions of a real variable this extension was begun by Bochner [2], and then applied by him, see [3], to a problem concerning partial differential equations. Bochner assumed L to be both complete and metric. In the present paper we shall admit more general linear spaces. We shall drop the metric but keep the completeness. Since the usual notion of completeness is based on the notion of metric, it was necessary to establish, for linear spaces, a notion of completeness independent of it. This was done in the preceding note of J. von Neumann [4]. The results of this note will be employed throughout, and we observe that, from the very beginning, we shall assume that L is linear with respect to arbitrary complex coefficients, see [4], Appendix I. As in [1 ], the main difficulty to overcome was the definition and the establishment of a mean. This was done in Part I. The definition of a mean remained actually the same as in [l], but the proof of the existence of a mean necessitated a more elaborate argument, although, in broad lines, the argument does not differ essentially. In Part II we deduce the existence and uniqueness of a Fourier expansion for any almost periodic function. It is worth pointing out that the representations occurring in the Fourier expansions of abstract almost periodic functions are the same as for numerical almost periodic functions, only the constant coefficients by which the representations are multiplied are abstract elements instead of numbers. (More than that, if in a linear manifold L different topologies are suitable for our purposes, then even the nature of the coefficients no longer determines the precise nature of abstractness of the almost periodic function.) Thus, roughly speaking, there are no more abstract almost periodic functions than numerical almost periodic functions. In particular, if a group admits of no other numerical almost periodic functions than the constant ones, there exists no non-constant abstract almost periodic function, no matter how general the range-space L may be.

Linear Partial Differential Equations, With Constant Coefficients

We will derive by a simple method some elementary properties of solutions of systems of linear partial differential equations with constant coefficients. In particular, we will obtain a general theorem on removable singularities. No use will be made of Greens functions or other source functions. Accordingly, our results will be stated for equations in general, although most of them will be of consequence only for equations of elliptic or similar type.

COMPLEX SPACES WITH SINGULARITIES

In the case of one complex variable if a Riemann surface A is spread out over a Riemann surface B and A has (isolated) ramification points of the familiar kind, then by adding such points to A the local Euclidean character is preserved and the analytic structure can be then adjusted so as to absorb the ramification points conformally as well. For several variables the situation is entirely different. A pertinent analog to a ramification point is a lower-dimensional singular subset as may occur in a complex locus that is defined by a system of equations

Vector fields on riemannian spaces with boundary

SummaryNon-existence of harmonic andKilling rectors on a Riemannian space with boundary. Our statements are more detailed than some previously given byK. Yano.