CHAPTER IV THE REAL GRASSMANNIANS §1. The real Grassmannians Let m 1, n 0 be given integers and let F be a real vector space of dimension m + n endowed with a positive definite scalar product. Let X be the real Grassmannian G R (F ) of all oriented m-planes in F . m Let V = V X be the canonical vector bundle (of rank m) over X whose fiber at x X is the subspace of F determined by the oriented m-plane x. We denote by W = W X the vector bundle of rank n over X whose fiber at x X is the orthogonal complement W x of V x in F . Then we have a natural isomorphism of vector bundles (4.1) V W T m over X. We may view X as a submanifold of F . In fact, the point m x X corresponds to the vector v 1 · · · v m of F , where {v 1 , . . . , v m } is a positively oriented orthonormal basis of the oriented m-plane x. The isomorphism (4.1) sends an element (V W ) x into the tangent vector dx t /dt| t=0 to X at x, where x t is the point of X corresponding to the vector (v 1 + t(v 1 )) · · · (v m + t(v m )) F , for t R. Since the vector bundles V and W are sub-bundles of the trivial vec- tor bundle over X whose fiber is F , the scalar product on F induces by restriction positive definite scalar products g 1 and g 2 on the vector bun- dles V and W , respectively. If we identify the vector bundle V with V by means of the scalar product g 1 , the isomorphism (4.1) gives rise to a natural isomorphism of (4.2) V W T m of vector bundles over X, which allows us to identify these two vector 2 2 2 bundles and the vector bundle T with V W . In fact, if 2 2 2 2 1 V , 2 W , we identify the element 1 2 of V W 2 with the element u of T determined by u(v 1 w 1 , v 2 w 2 ) = 1 (v 1 , v 2 ) · 2 (w 1 , w 2 ), for v 1 , v 2 V and w 1 , w 2 W . The scalar product g on T induced by the scalar product g 1 g 2 on V W is a Riemannian metric on X.