  In quantum mechanics, the Schrieffer--Wolff (SW) transformation (also called
quasi-degenerate perturbation theory) is known as an approximative method to
reduce the dimension of the Hamiltonian. We present a geometric interpretation
of the SW transformation: We prove that it induces a local coordinate chart in
the space of Hermitian matrices near a $k$-fold degeneracy submanifold.
Inspired by this result, we establish a `distance theorem': we show that the
standard deviation of $k$ neighboring eigenvalues of a Hamiltonian equals the
distance of this Hamiltonian from the corresponding $k$-fold degeneracy
submanifold, divided by $\sqrt{k}$. Furthermore, we investigate one-parameter
perturbations of a degenerate Hamiltonian, and prove that the standard
deviation and the pairwise differences of the eigenvalues lead to the same
order of splitting of the energy eigenvalues, which in turn is the same as the
order of distancing from the degeneracy submanifold. As applications, we prove
the `protection' of Weyl points using the transversality theorem, and infer
geometrical properties of certain degeneracy submanifolds based on results from
quantum error correction and topological order.
