This work studies operators mapping vector and scalar fields defined over a manifold $\mathcal{M}$, and which commute with its group of diffeomorphisms $\text{Diff}(\mathcal{M})$. We prove that in the case of scalar fields $L^p_\omega(\mathcal{M,\mathbb{R}})$, those operators correspond to point-wise non-linearities, recovering and extending known results on $\mathbb{R}^d$. In the context of Neural Networks defined over $\mathcal{M}$, it indicates that point-wise non-linear operators are the only universal family that commutes with any group of symmetries, and justifies their systematic use in combination with dedicated linear operators commuting with specific symmetries. In the case of vector fields $L^p_\omega(\mathcal{M},T\mathcal{M})$, we show that those operators are solely the scalar multiplication. It indicates that $\text{Diff}(\mathcal{M})$ is too rich and that there is no universal class of non-linear operators to motivate the design of Neural Networks over the symmetries of $\mathcal{M}$.

翻译：这项工作研究运算符映射由 $$\ mathcal{M} $ 定义的矢量和 标度域, 并且与它的二异形网络组 $\ text{Diff} (\ mathcal{M} $ 。 我们证明, 在 scal 字段 $L\ p ⁇ omega (\ mathcal{M} $ ) 的情况下, 这些运算符与点非线性域对应, 恢复并扩展$\ mthb{R} $ 的已知结果。 在 由 $\ mathcal{M} 定义的神经网络组中, 点向非通用的非直线性域组操作员。 $\ t{ m} 系统化操作员与特定对齐的线性操作员连接使用。 在 矢量域域域域域域域域域域域域域域域域域域域域域中, 我们显示这些操作员仅仅是 缩数的多运算数 。