# H S Ruse - some of his papers

Below we list a few of Ruse's papers together with some brief comments by reviewers extracted from longer reviews:
1. Harmonic Riemannian spaces (1940).

W Feller writes: The authors study Riemannian $V_{n}$ in connection with the fundamental solution of the corresponding equation $\Delta _{2u} = 0$, where $\Delta _{2}$ stands for the second differential parameter.
2. On the line-geometry of the Riemann tensor (1944).

J L Vanderslice writes: The family of all lines through the contact point of any tangent space of differentials of a general manifold $V_{n}$ forms a projective space $S_{n-1}$ , the components of contravariant vectors being homogeneous point coordinates therein. When $V_{n}$ is Riemannian the metric tensor $g_{jk}$ and Riemann tensor $R_{ijkl}$ define in each $S_{n-1}$ a quadric and a quadratic line complex, respectively. The case $n = 4$ gives the most elegant and important results because lines are then self-dual and there are relativistic applications. ... The present author considers briefly the general case and proceeds to special results for $n = 3$ and $n = 4$.
3. Sets of vectors in a V defined by the Riemann tensor (1944).

J L Vanderslice writes: The components of the Riemann tensor at a nonsingular point of a Riemannian $V_{n}$ define a quadratic complex in the space at infinity in the affine tangent space $A_{n}$ associated with the point. The author continues his study of this configuration for $n = 4$ ...
4. A G D Watson's principal directions for a Riemannian $V_{4}$ (1945).

J L Vanderslice writes: The purpose of the present paper is to analyze Watson's theory in relation with the work of others by using methods developed by the author in his study of the Riemann complex. These methods are summarized for the reader's convenience
5. Multivectors and catalytic tensors (1947).

A Schwartz writes: Two of the results for self-dual six-vectors of null invariant in Galilean space-time which E T Whittaker demonstrated with the help of spinor theory are here demonstrated by direct geometric methods, without the use of spinor theory, and for a Riemannian V of any signature and curvature. Light is thrown on the nature of catalytic tensors, and we have an illustration of the value of geometric methods in discovering tensor formulae which involve covariant derivatives.
6. On the geometry of metrisable Lie algebras (1957).

A G Walker writes: The object of this paper is to display something of the geometrical background of metrisable Lie algebras as defined by Tsou and the reviewer. Properties of such algebras are examined in terms of the geometry of the associated projective space, and applications are confined to metrisable algebras of dimensions 3, 4 and 6.
7. Tensor extensions of metrisable local Lie groups (1959).

L Auslander writes: The author in this paper considers the following general problem. Let $L$ be a Lie algebra over the real field and let $A$ be a finite dimensional associative commutative algebra over the reals. Then $A ⊗ L$ is a Lie algebra and hence corresponds to a Lie group $G(A ⊗ L)$. Assume the Lie group $G(L)$ has a left and right invariant non-degenerate Riemann metric; when will a group with Lie algebra $A ⊗ L$ have the same property? The author shows that if we restrict ourselves to local groups, the existence of the desired Riemann metric on the local group is implied by the existence of a non-singular bilinear form on $A$.
8. Harmonic spaces (1961) (with A G Walker and T J Willmore).

J A Wolf writes: This book is an account of the local theory of harmonic Riemannian (indefinite metric allowed) spaces, in whose development the authors have been prominent. Several tantalizing problems are brought to the reader's attention as the relations are drawn between harmonic, symmetric and recurrent spaces.

9. General solutions of Laplace's equation in a simply harmonic manifold (1963).

T J Willmore writes: Explicit formulae are obtained for general solutions of Laplace's equation in a real $n$-cell equipped with a simply harmonic riemannian metric. The formulae are obtained by an analytical device involving regular functions of a complex variable.

Last Updated November 2007