On Betti numbers of compact, locally sysmmetric Riemannian manifolds
YOZÔ MATSUSHIMA · Institutional Repositories DataBase (IRDB) · 1962
This is a continuation of our paper [9].In [9] we have studied the vanishing of the first Betti number of compact, locally symmetric Riemannian manifolds.We shall study in this paper the ^-th Betti number of these manifolds from a somewhat different point of view.Let X be a simply connected, symmetric Riemannian manifold, all of whose irreducible components are non-euclidean and non-compact.Let G be the identity component of the group of all isometries of X and let Γ be a discrete subgroup of G with compact quotient space G/Γ and without element of finite order different from the identity.The group Γ acts on X discontinuously and the quotient space M=X/Γ is a compact, locally symmetric Riemannian manifold.Let A p be the vector space of all G-invariant ^-forms on X.By a well-known theorem of E. Cartan, the covariant derivatives of each form in A p vanish (see [10]).Since ΓC^G, each &>eA p is Γ-invariant and hence there exists a ^-form η on M such that ω = ηop y p denoting the projection of X onto M. Since p is a locally isometric mapping and the covariant derivatives of ω vanish, the covariant derivatives of η also vanish.In particular η is a harmonic ^-form.Hence the mapping ω-*η defines an injection of A p into the vetcor space fy p of all harmonic p-forms on M. The purpose of this paper is to study when A p can be isomrphic to § p .Let x 0 6 X and let K be the subgroup of G of all elements which leave fixed the point x 0 .It is well-known that K is a maximal compact subgroup of G and X is identified with the quotient space K\G.Let g denote the Lie algebra of G and let ϊ be the subalgebra of g corresponding to K. Denote by m the orthogonal complement of ϊ in Q with respect to the Killing form of g.We have then g = m -f ϊ, [m, m] C[ I, [ΐ, m] C m .Let g c be the complexification of g and let G c be the complex Lie group,