曲面上二秩向量丛上一阶对称符号特征值函数的奇异性
Singularities of the eigenvalue functions for first order symmetric symbols on rank two vector bundles over surfaces
摘要：
对于曲面$X$上的秩为2的丛$F$，研究了$F$上一阶微分算子对应的对称符号$σ$的特征值函数的奇点集$M_σ$.我们证明了这些奇点的存在如下从拓扑考虑。我们定义的程度的一组奇点，并表明，它可以被用来计数的数量在特殊的光学现象发生的方向。对于$F=TX$的情况下，我们计算的程度的公式在$X$和$N_σ$的欧拉特征，其中$N_σ\subset X$是一个流形与边界相关的$σ$。
英文摘要：
For a rank two bundle $F$ over a surface $X$, we study the set of singularities $M_σ$ of the eigenvalue functions of symmetric symbols $σ$ associated to first order differential operators on $F$. We prove that the existence of these singularities follows from topological considerations. We define the degree of the set of singularities and show that it can be used to count the number of directions at which special optical phenomena occur. For the case when $F=TX$ we compute a formula for the degree in terms of the Euler characteristics of $X$ and $N_σ$, where $N_σ\subset X$ is a manifold with boundary associated to $σ$.
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