Isometric imbedding of Riemannian manifolds

在本研究中，我们得到了黎曼流形等距嵌套的以下结果：我们确定了许多黎曼对称空间G/K的固有不变量p（G/K）的值，并得到了G/K可以局部等距浸入的欧几里德空间的维数估计。特别是对于空间Sp(m)/U(m)和Sp(m)，确定了最小维欧几里德空间。证明了对称空间SU(3)/SO(3)及其非紧对偶空间允许余维数为5的高斯方程的解，并允许余维数为4.3的高斯方程的几乎解。在M小于等于9的情况下，我们确定了由高斯方程定义的二次映射的秩。这一结果表明，对于M^9∧R^<23>.4的情况，存在局部等距嵌入的阻碍。给出了高斯方程外代数的一种新形式，并说明了它与原方程的关系。认识类曲率张量空间上多项式环的GL(V)-不可约分解是十分重要的。这是表象理论中出现的一种“过剩”现象。给出了一些特殊体积的分解公式。我们证明了四元数投影平面可以局部等距浸入的最小维欧几里德空间是R^<14>。

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>.