CHAPTER I SYMMETRIC SPACES AND EINSTEIN MANIFOLDS §1. Riemannian manifolds Let X be a differentiable manifold of dimension n, whose tangent and cotangent bundles we denote by T = T X and T = T X , respectively. Let k E, S l E, C (X) be the space of complex-valued functions on X. By j E, we shall mean the k-th tensor product, the l-th symmetric product and the j-th exterior product of a vector bundle E over X, respectively. k k We shall identify S k T and T with sub-bundles of T by means of the injective mappings S k T k T , k T k T , sending the symmetric product 1 · . . . · k into (1) · · · (k) S k and the exterior product 1 · · · k into