  We classify all subsets $S$ of the projective Hilbert space with the
following property: for every point $\pm s_0\in S$, the spherical projection of
$S\backslash\{\pm s_0\}$ to the hyperplane orthogonal to $\pm s_0$ is isometric
to $S\backslash\{\pm s_0\}$. In probabilistic terms, this means that we
characterize all zero-mean Gaussian processes $Z=(Z(t))_{t\in T}$ with the
property that for every $s_0\in T$ the conditional distribution of
$(Z(t))_{t\in T}$ given that $Z(s_0)=0$ coincides with the distribution of
$(\varphi(t; s_0) Z(t))_{t\in T}$ for some function $\varphi(t;s_0)$. A basic
example of such process is the stationary zero-mean Gaussian process
$(X(t))_{t\in\mathbb R}$ with covariance function $\mathbb E [X(s) X(t)] =
1/\cosh (t-s)$. We show that, in general, the process $Z$ can be decomposed
into a union of mutually independent processes of two types: (i) processes of
the form $(a(t) X(\psi(t)))_{t\in T}$, with $a: T\to \mathbb R$, $\psi(t): T\to
\mathbb R$, and (ii) certain exceptional Gaussian processes defined on
four-point index sets. The above problem is reduced to the classification of
metric spaces in which in every triangle the largest side equals the sum of the
remaining two sides.
