Let (M, g) be a complete, simply connected, irreducible Riemannian spin manifold. It is well known (see [H] p. 8 and the footnote on p. 54 and also [F]) that if (M, g) admits a non-zero parallel spinor, then g is Ricci flat. Using the Berger-Simons theorem as in the footnote of [H] and some representation theory, one can state a more precise result about the dimension of the space of parallel spinors: Proposition. Let (M, g) be a complete, simply connected, irreducible Riemannian spin manifold of dimension n. Let N denote the dimension of the space of parallel spinors. If (M, g) is non-flat and N > O, then one of the following holds: (a) n = 2m, m > 2, the holonomy representation is (SU(m), [lm]R), and N = 2, (b) n = 4m, m > 2, the holonomy representation is (Sp(m), [V2m]R), and N = m + 1, (c) n = 8, the holonomy representation is (Spin(7)*, d7), and N = 1, (d) n = 7, the holonomy representation is (G 2 , (p 7 ), and N = 1. Conversely, if the holonomy representation is one of the above, then N must assume the value given. (i, is the vector representation of SU(m) on Cm, v2m is the vector representation of Sp(m) on C 2 m, 47 is the spin representation, and p,7 is the 7-dimensional irreducible representation of G 2 .)The manifolds in (a) are Kahler with vanishing first Chern class, those in (b) are hyperkahler, and by the work of E. Bonan [Bo], there is a unique parallel 4-form on manifolds in (c), while there is a unique parallel 3-form and a unique parallel 4-form on manifolds in (d). In this note we give a uniform description of these special structures in terms of the parallel spinors and bilinear invariants of the spin representations. This is an illustration of the general philosophy that on a Riemannian spin manifold the spinor bundle is a refinement of the exterior bundle and hence is more basic. Thus, the apparently different special structures on manifolds with SU(m), Sp(m), Spin(7)', or G 2 holonomy all arise from the same basic object in the spin case.We shall need an explicit construction of the bilinear invariants for spin representations, which we present in § 2. While the material is not new (compare [R], [L], and the appendix of [PR II]), our description uses only the explicit construction of spin representations * The author acknowledges partial support from the Natural Sciences and Engineering Research Council of Canada.

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The notion of curvature on isotropic two-planes on a Riemannian manifold of dimension > 4 was introduced in [MMr]. It is a notion that arises very naturally in the study of the second variation of area of minimal surfaces, just as sectional curvature arises in the study of geodesics. In this paper, we shall refer to "curvature on isotropic two-planes" as "isotropic curvature", whose definition we now recall.For any real vector space V with inner product g(., .), let V V (R) denote the complexification of V, let O(', ") denote also the complex bilinear extension of 9(', ") to V, and let (., .) denote the Hermitian extension of 9(', ") to V which is complex linear in the first argument and conjugate linear in the second. A subspace W c V is isotropic if 9(w, w) 0 for all w W. Now let : A 2 TM A2TM denote the curvature operator and also its complex linear extension to AZTM (R) (E. A Riemannian manifold is said to have positive isotropic curvature if ((v^w), (v^w)) > 0 whenever span {v, w} is a two-dimensional isotropic subspace of TcM. The notion of positive isotropic curvature generalizes classical curvature conditions such as strict pointwise quarter pinching and positive curvature operator. Indeed, the Morse theory of minimal two-spheres in a Riemannian manifold of positive isotropic curvature yields the following extension of the classical sphere theorem. THEOREM [MMr]. Let M be a compact n-dimensional Riemannian manifold without boundary, n > 4. If M has positive isotropic curvature, then Ir.i(M {0} for 2 < In/2] where Ix] denotes the integer part of x .I n particular, if M is also simply connected, then M is homeomorphic to a sphere. An interesting example of a nonsimply connected manifold with positive isotropic curvature is S x S", n > 3, with the product of the metric on S and the constant curvature + metric on S" [-MMr p. 205]. Other instructive examples are the conformally flat four-manifolds with positive scalar curvature [MWo, Prop. 2.1]. These examples show that positive isotropic curvatue does not imply positive, or even nonnegative, Ricci tensor. It does, however, imply positive scalar curvature (Prop. 2.5 below). The examples also suggest that the fundamental group of a manifold of positive isotropic curvature may be quite large. In the first section of this paper we show that this is indeed the case by proving the following theorem.

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We analyse some properties of the cohomogeneity one Ricci soliton equations, and use Ansätze of cohomogeneity one to produce new explicit examples of complete Kähler Ricci solitons of expanding, steady and shrinking types. These solitons are foliated by hypersurfaces which are circle bundles over a product of Fano Kähler-Einstein manifolds or over coadjoint orbits of a compactly connected semisimple Lie group.