Geometrical Properties Of Compact Riemannian Symmetric Spaces And Their Applications

In the thesis, we shall study several global properties of compact Riemannian symmetric spaces and give their applications. The thesis consists of two parts. In the first part (Chapter 2 and Chapter 3), with the aid of the theory of Lie groups and Lie algebras, we explore the relationship between the cut locus of an arbitrary irreducible Riemannian symmetric space of compact type and the Cartan polyhedron of corresponding restricted root system and obtainTheorem 1. [61][62] Let (u,θ, < ,>) be a reduced, compact and irreducible orthogonal symmetric Lie algebra, (?) be the simply connected Riemannian symmetric space associated with (u,θ, < ,>), (?) = (?)/Γbe a Clifford-Klein form of M, whereΓis a subgroup of Z_?((?)), then (?)(o) = Ad(K)(Ï€(?)P_Γ) and C(o) = Ad(K)(Ï€(?)P'_Γ).Here Z_?((?)) denote the set of all fix points of (?) under the left action of (?); on it there exists a natural group structure. There is a one-to-one correspondence between every subgroup of Z_?((?)) and each Clifford-Klein form of (?); hence every irreducible Riemannian symmetric space M of compact type could be written in the form M = (?)/Γ. The polyhedron P_Γand P'_Γ. corresponding toΓis defined byP_Γ= {x∈△: (x,e_i)≤1/2(e_i,e_i) for every (?)};P'_Γ= {x∈P_Γ:(x,ψ) = 1 or (x, e_j) = 1/2(e_j, e_j) for some j such that (?)}.whereâ–³is the Cartan polyhedron, e₁,…, e_l denotes the vertices of A, and (?) denotes the exponential mapping of (?).On the basis, we compute injectivity radius and diameter for every type of irreducible Riemannian symmetric spaces of compact type, and list the results in Table 3.4.1 and Table 3.4.2. Especially we haveTheorem 2. [61] Let M be a simply-connected, irreducible Riemannian symmetric space of compact type,κbe the maximum of the sectional curvatures of M, then i(M) =πκ^-1/2. In the second part (Chapter 4 and Chapter 5), the object we shall study is a special Riemannian symmetric space: (real) Grassmannian manifold. J. Jost and Y.L. Xin [26] found the largest geodesic convex domain B_jx(P₀) of G_n,m; we shall construct convex functions v and u on B_jx(P₀) and give estimates of the Hessian of them:Theorem 3. [60] v,u are convex functions on B_jx(P₀) (?) U (?) G_n,m. On {P∈U : v(P)≤2}, we have the estimateon {P∈U : u(P)≤2}, we have the estimateHere p = min(n, m).As applications, we derive curvature estimates for minimal submanifolds in Euclidean space for arbitrary dimension and codimension via Gauss map. With aid of the estimates, we obtain a series of geometric conclusions, including the following Bernstein type theorems :Theorem 4. [59] Let M be an n-dimensional complete minimal submanifold of R^n+m with n≤6 and m≥2. If the Gauss image of M is contained in a geodesic ball of G_n,m of radius (?)/4Ï€, then M has to be an affine linear subspace.n,mTheorem 5. [60] Let M = {(x, f(x)) : x∈Rⁿ} be an n-dimensional minimal graph given by m functions f^α(x¹,…,xⁿ). Here m≥2,n≤4. IfThen f^αhas to be affine linear, hence M has to be an affine linear subspace.Theorem 6. [59] Let M be an n-dimensional complete minimal submanifold of R^n+m with m≥2. If the Gauss image of M is contained in a geodesic ball of G_n,m centered at P₀ and of radius (?)/4Ï€, and ((?)/4Ï€-ρoγ)^-1 has growth:where p denotes the distance function from P₀ on G_n,m, R is the Euclidean distance from any point on M. Then M has to be an affine linear subspace. Theorem 7. [60] Let M - {(x, f(x)) : x∈Rⁿ} be an n-dimensional minimal graph given by m functions f^α(x¹,…,xⁿ). Here m≥2.Ifand(2-â–³_f)^-1=0(R^4/3),where R² = |x|² +|f|². Then f^αhas to be affine linear, hence M has to be an affine linear subspace.