SECTIONAL CURVATURES OF GRASSMANN MANIFOLDS

(1) Introduction.-Let F be the field R of real numbers, the field C of complex numbers, or the field H of real quaternions; Fn+m a left (n + m)-dimensional Hermitian vector space over F; and Gn(FP+m) the Grassmann manifold of nplanes in F"+m provided with the invariant Riemannian metric with respect to which the distance between two points A and B in G.(Fn +m) is equal to the square root of the sum of the squares of the angles between the n-planes A and B in Ff +n (see ref. 14). Previous studies of G,(Fn +m) have not unearthed sufficiently precise information about its sectional curvatures. Although the components of the curvature tensor at a point of GC(Rn+") and Gn(Cn+") have been computed in references 5, 8, and 10, all that is so far known about the sectional curvatures of G.(F +1") seems to be that they are nonnegative but not all positive unless min(n,m) = 1, in which case they have the range of values given in Theorem 1 in §(4) (see ref. 2, p. 171; ref. 3, Theorem 7; ref. 1, p. 59; ref. 9, pp. 351 and 358; and ref. 8, Theorem 4.5). In this note, which is a continuation of reference 14 and a companion to reference 15, we shall give a complete description of the sectional curvatures of Gn(F"+m) In § § (2) and (3), we express the curvature tensor and the sectional curvature of G, (Fn+m) in terms of local coordinates and define the unitary curvature of Gn(Hn+m). In §(4), we state our main results concerning the range of values of the sectional curvature and certain characteristic properties of sections of minimum and maximum curvatures. Details of these results and similar results for the classical bounded symmetric domains will be published later. (2) The Curvature Tensor.-We know (ref. 14) that the invariant Riemannian metric on Gn(Fn+m) can be arrived at in a natural and geometric way; moreover, in a typical local coordinate system (U,Z) with neighborhood U and coordinate Z (which is an n X m matrix with elements in F), it has the explicit expression