
W Feller writes: The authors study Riemannian V_{n} in connection with the fundamental solution of the corresponding equation Δ_{2u} = 0, where Δ_{2} stands for the second differential parameter.
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_{n1} , 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_{n1} a quadric and a quadratic line complex, respectively. The case n = 4 gives the most elegant and important results because lines are then selfdual and there are relativistic applications. ... The present author considers briefly the general case and proceeds to special results for n = 3 and n = 4.
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 ...
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
A Schwartz writes: Two of the results for selfdual sixvectors of null invariant in Galilean spacetime 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.
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.
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 nondegenerate 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 nonsingular bilinear form on A.
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.
T J Willmore writes: Explicit formulae are obtained for general solutions of Laplace's equation in a real ncell equipped with a simply harmonic riemannian metric. The formulae are obtained by an analytical device involving regular functions of a complex variable.
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