A natural way to obtain a system of partial differential equations on a manifold is to vary a suitably defined sesquilinear form. The sesquilinear forms we study are Hermitian forms acting on sections of the trivial [Formula: see text]-bundle over a smooth [Formula: see text]-dimensional manifold without boundary. More specifically, we are concerned with first order sesquilinear forms, namely, those generating first order systems. Our goal is to classify such forms up to [Formula: see text] gauge equivalence. We achieve this classification in the special case of [Formula: see text] and [Formula: see text] by means of geometric and topological invariants (e.g., Lorentzian metric, spin/spinc structure, electromagnetic covector potential) naturally contained within the sesquilinear form — a purely analytic object. Essential to our approach is the interplay of techniques from analysis, geometry, and topology.

中文翻译：

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一阶倍半线性形式的分类

获得流形上的偏微分方程系统的一种自然方法是改变适当定义的倍半线性形式。我们研究的倍半线性形式是 Hermitian 形式，作用于平凡的 [公式：参见文本]-束在平滑的 [公式：参见文本] 维流形上，没有边界。更具体地说，我们关注一阶倍半线性形式，即那些生成一阶系统的形式。我们的目标是将这些形式分类为 [公式：见文本] 量规等价。我们通过几何和拓扑不变量（例如，洛伦兹度量，自旋/自旋C结构，电磁协矢量势）自然包含在倍半线性形式中——一个纯粹的分析对象。我们方法的关键是分析、几何和拓扑技术的相互作用。