Harmonicity of proper almost complex structures on Walker 4-manifolds

Every Walker [Formula: see text]-manifold [Formula: see text], endowed with a canonical neutral metric [Formula: see text], admits a specific almost complex structure called proper. In this paper, we find the conditions under which a proper almost complex structure is a harmonic section or a harmonic map from [Formula: see text] to its hyperbolic twistor space.

The projective invariants of four mediale

The theory of four particular linear forms, or matrices of k columns and 2k rows, occurred to me many years ago in an attempt to study the invariants of any number of compound linear forms, or subspaces within a space of n dimensions. In what follows, the invariant theory is given, and its significance for a study of the general matrix of k rows and columns is suggested. The collineation used in §4 was considered by Mr J. H. Grace, who emphasized the importance of the k cross ratios upon transversal lines of four [k−1]'s in [2k−1]. It seemed appropriate to examine these cross ratios which are irrational invariants μi, of the figure of four such spaces, and to work out their relation to the known rational invariants Xi. The main result is given in § 5 (7). In § 5 (10) it is shewn that the harmonic section of a line transversal of the four spaces exists when a linear relation holds between the invariants.