  We show that bounded self-contracted curves are rectifiable in metric spaces
with weak lower curvature bound in a sense we introduce in this article. This
class of spaces is wide and includes, for example, finite-dimensional
Alexandrov spaces of curvature bounded below and Berwald spaces of nonnegative
flag curvature. (To be more precise, our condition is regarded as a
strengthened doubling condition and holds also for a certain class of metric
spaces with upper curvature bound.) We also provide the non-embeddability of
large snowflakes into (balls in) metric spaces in the same class. We follow the
strategy of the last author's previous paper based on the small rough angle
condition, where spaces with upper curvature bound are considered. The results
in this article show that such a strategy applies to spaces with lower
curvature bound as well.
