  We consider stochastic versions of Euler--Arnold equations using the
infinite-dimensional geometric approach as pioneered by Ebin and Marsden. For
the Euler equation on a compact manifold (possibly with smooth boundary) we
establish local existence and uniqueness of a strong solution (in the
stochastic sense) in spaces of Sobolev mappings (of high enough regularity).
Our approach combines techniques from stochastic analysis and
infinite-dimensional geometry and provides a novel toolbox to establish local
well-posedness of stochastic Euler--Arnold equations.
