Local normal form theorems for smooth equivariant maps between infinite-dimensional manifolds are established. These normal form results are new even in finite dimensions. The proof is inspired by the Lyapunov–Schmidt reduction for dynamical systems and by the Kuranishi method for moduli spaces. It uses a slice theorem for Fréchet manifolds as the main technical tool. As a consequence, the abstract moduli space obtained by factorizing a level set of the equivariant map with respect to the group action carries the structure of a Kuranishi space, i.e., such moduli spaces are locally modeled on the quotient by a compact group of the zero set of a smooth map. The general results are applied to the moduli space of anti-self-dual instantons, the Seiberg–Witten moduli space and the moduli space of pseudoholomorphic curves.

建立了无限维流形之间光滑等变映射的局部范式定理。即使在有限维度中，这些标准形式的结果也是新的。该证明的灵感来自于动力系统的 Lyapunov-Schmidt 约简和模空间的 Kuranishi 方法。它使用 Fréchet 流形的切片定理作为主要技术工具。因此，通过对群作用的等变映射的水平集进行因式分解而获得的抽象模空间带有 Kuranishi 空间的结构，即，这种模空间通过零的紧致群局部建模在商上。一套平滑的地图。一般结果应用于反自对偶瞬子的模空间、Seiberg-Witten 模空间和伪全纯曲线的模空间。