Curvature Estimates For Symmetric Spaces With Applications

According to the classifications of Riemannian symmetric spaces, we transform the Cartan subalgebras with maximal compact part into the Cartan subalgebras with maximal noncompact part by using the Cayley transformation. We give the bounds of sectional curvature for all irreducible Riemannian symmetric spaces.From the classifications of the maximal subsystems of abstract root systems, we calculate the s-values of partial positivity for all irreducible symmetric spaces.As an application we give the following Liouville type theorem:Theorem 0.1. Let M be one of the irreducible symmetric spaces of noncompact type in the following cases,SL(n,R)/S0(n),n≥4; SU*(2n)/Sp(n); SU(p,q)/S(U_p×U_q),p + q≥4; SO₀(p, q)/SO(p)× SO(q),for r = 1,p + q≥4, for r > 1,p + q≥ 6, here r = min(p,q); SO*(2n)/U(n),n≥3; Sp(n,R)/U(n),n≥3; Sp(p,q)/Sp(p)×Sp(q); EI, EII, EIII, EIV, EV, EVI, EVII, EVIII, EIX, FI, FII and G.Let f be a harmonic map from M into any Riemannian manifold with moderate divergent energy, then f has to be constant.For exceptional geometry, we give the following result on the vanishing of integral homology groups. Theorem 0.2. Let M be one of the following irreducible Riemannian symmetric spacesof compact type, then Hi(M, Z) = 0, for i ≥ s,EI, s = 26; EII,s = 19; EIII,s = 11;q EIV,s = 10; EV,s = 43; EVI,s = 31; EVII,s = 27; EVIII, s = 71; EIX, s = 55; FI,s = 13; FII,s = 1;G,s=3.