二次型判别曲面的几何
Geometry of the Discriminant Surface for Quadratic Forms
摘要：
研究了n > 1元的具有重特征值的（实）二次型流形$\cal{M}$.除了已知的事实外，我们还证明了：1）$\cal{M}$是不可约的; 2）在n = 3的情况下，标量矩阵是$\cal{M}$上的奇点.对于$n = 3$，$\cal{M}$也被描述为$\cal{M}$ $_0$上的直圆柱，其中$\cal{M}$ $_0$是对角矩阵$\diag(1,1,-2)$通过坐标的正交变化的轨道上的圆锥。我们分析了这个轨道的某些性质，它发生在射影平面的一个单纯像上。
英文摘要：
We investigate the manifold $\cal{M}$ of (real) quadratic forms in n > 1 variables having a multiple eigenvalue. In addition to known facts, we prove that 1) $\cal{M}$ is irreducible, 2) in the case of n = 3, scalar matrices and only them are singular points on $\cal{M}$. For $n = 3$, $\cal{M}$ is also described as the straight cylinder over $\cal{M}$$_0$, where $\cal{M}$$_0$ is the cone over the orbit of the diagonal matrix $\diag(1,1,-2)$ by the orthogonal changes of coordinates. We analyze certain properties of this orbit, which occurs a diffeomorphic image of the projective plane.
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