来自普适标量过程的热几何
Heat Geometry from a Universal Scalar Process
AI总结：
本文证明一个普适标量过程可完全确定闭Riemann流形、紧致RCD空间及带联络和势的Euclidean丛，重构热生成元，热包函子全忠实，张量自同构恰为几何自同构。
AI中文摘要：
我们证明，在有限Wiener混沌中，一个光滑标量过程的分布完全决定了任意实可分Hilbert空间上对称张量的有限族，直至同时正交等价。高斯图特征恢复所有收缩，而一个内蕴的迹类Gram算子将问题简化为有限维不变量理论。将该原理应用于规范二次和四次热包在单个固定正时刻的情形，可重构热生成元，并迫使每个重构的酉算子都是空间的。因此，一个普适标量过程决定了闭Riemann流形、有限维紧致RCD空间，以及带有度量联络和自伴势的Euclidean向量丛。热包函子是 fully faithful 的，其张量自同构恰好是几何自同构。
英文摘要：
We prove that the law of one smooth scalar process in a finite Wiener chaos completely determines finite families of symmetric tensors on arbitrary real separable Hilbert spaces, up to simultaneous orthogonal equivalence. Gaussian graph characters recover all contractions, while an intrinsic trace class Gram operator reduces the problem to finite dimensional invariant theory. Applied at a single fixed positive time to the canonical quadratic and quartic heat packet, this principle reconstructs the heat generator and forces every reconstructed unitary to be spatial. Consequently, one universal scalar process determines closed Riemannian manifolds, compact $\operatorname{RCD}$ spaces of finite dimension, and Euclidean bundles with metric connection and self adjoint potential. The heat packet functor is fully faithful, and its tensor automorphisms are precisely the geometric ones.
