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：本研究探讨了定义在流形 $\mathcal{M}$ 上的向量场与标量场映射算子，并证明其与微分同胚群 $\text{Diff}(\mathcal{M})$ 交换。我们证明：对于标量场 $L^p_\omega(\mathcal{M,\mathbb{R}})$，此类算子对应于逐点非线性函数，从而恢复并推广了 $\mathbb{R}^d$ 上的已知结论。在定义于 $\mathcal{M}$ 的神经网络语境中，该结果表明逐点非线性算子是唯一与任意对称群交换的通用算子族，并验证了其与特定对称性交换的专用线性算子协同使用的系统性方法。对于向量场 $L^p_\omega(\mathcal{M},T\mathcal{M})$，我们证明此类算子仅为标量乘法。这表明 $\text{Diff}(\mathcal{M})$ 结构过于丰富，不存在可驱动基于 $\mathcal{M}$ 对称性进行神经网络设计的通用非线性算子类。