Isometric imbedding of Riemannian manifolds

1.确定了许多黎曼对称空间G/K的内蕴不变量p(G/K)的值，并得到了G/K可以局部等距浸入的欧氏空间的维数的估计。特别地，对于空间Sp(M)/U(M)和Sp(M)，确定了最低维欧氏空间。证明了对称空间SU(3)/SO(3)及其非紧对偶空间允许余维为5的Gauss方程的解，也允许余维为4.3的几乎解.我们确定了DIMM[小于等于]9时由高斯方程定义的二次映射的秩.这一结果表明在M^9⊂R^&lt；23&gt；4的情况下存在局部等距嵌入的障碍.给出了外代数中高斯方程的一种新形式，并说明了它与原方程的关系。知道多项式环在类曲率张量空间上的GL(V)-不可约分解是非常重要的。这是表象理论中出现的一种“过生症”。给出了一些特殊多项式的分解公式。证明了四元数射影平面可以局部等距浸入的最低维欧氏空间是R^&lt；14&gt；

In this research, we obtained the following results on isometric imbeddings of Riemannian manifolds :1. We determine the value of the intrinsic invariant p(G/K) for many Riemannian symmetric spaces G/K, and obtain the estimates on the dimension of the Euclidean space into which G/K can be locally isometrically immersed. In particular, for the spaces Sp(m)/U(m) and Sp(m), the least dimensional Euclidean spaces are determined.2. We show that the symmetric space SU(3)/SO(3) and its non-compact dual space admit solutions of the Gauss equation in codimension 5, and also admit almost solutions in codimension 4.3. We determine the rank of the quadratic map defined by the Gauss equation for the case dim M【less than or equal】9. This result shows the existence an obstruction of local isometric imbeddings for the case M^9⊂R^<23>.4. We give a new formulation of the Gauss equation in the exterior algebra, and state the relation to the original equation.5. It is quite important to know the GL(V)-irreducible decomposition of the polynomial ring on the space of curvature like tensors. This is a sort of "plethysm" appeared in the representation theory. We give some decomposition formulas of special plethysms.6. We show that the least dimensional Euclidean space into which the quaternion projective plane can be locally isometrically immersed is R^<14>.