  We provide a geometric characterization of $k$-dimensional
$\mathbb{F}_{q^m}$-linear sum-rank metric codes as tuples of
$\mathbb{F}_q$-subspaces of $\mathbb{F}_{q^m}^k$. We then use this
characterization to study one-weight codes in the sum-rank metric. This leads
us to extend the family of linearized Reed-Solomon codes in order to obtain a
doubly-extended version of them. We prove that these codes are still maximum
sum-rank distance (MSRD) codes and, when $k=2$, they are one-weight, as in the
Hamming-metric case. We then focus on constant rank-profile codes in the
sum-rank metric, which are a special family of one weight-codes, and derive
constraints on their parameters with the aid of an associated Hamming-metric
code. Furthermore, we introduce the $n$-simplex codes in the sum-rank metric,
which are obtained as the orbit of a Singer subgroup of $\mathrm{GL}(k,q^m)$.
They turn out to be constant rank-profile - and hence one-weight - and
generalize the simplex codes in both the Hamming and the rank metric. Finally,
we focus on $2$-dimensional one-weight codes, deriving constraints on the
parameters of those which are also MSRD, and we find a new construction of
one-weight MSRD codes when $q=2$.
