The concept of ergodicity--convergence of the temporal averages of observables to their ensemble averages--is the cornerstone of thermodynamics. The transition from a predictable, integrable behavior to ergodicity is one of the most difficult physical phenomena to treat; the celebrated KAM theorem is the prime example. This Letter is founded on an observation that for many classical and quantum observables, the sum of the ensemble variance of the temporal average and the ensemble average of temporal variance remains approximately constant across the integrability-ergodicity transition. We show that this property induces a particular geometry of quantum observables--Frobenius or Hilbert-Schmidt one--that naturally encodes all the phenomena associated with the emergence of ergodicity: the Eigenstate Thermalization effect, decrease in the inverse participation ratio, and the disappearance of the integrals of motion. As an application, we use this geometry to solve a known problem of optimization of the set of integrals of motion needed to describe the steady state of an integrable or near-integrable system. Finally, we show how the invariance of the "sum of two variances" can manifest itself empirically in dynamic of far-from-equilibrium one-dimensional harmonically trapped bosons.
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