  In this article we investigate hypocoercivity of Langevin-type dynamics in
nonlinear smooth geometries. The main result stating exponential decay to an
equilibrium state with explicitly computable rate of convergence is rooted in
an appealing Hilbert space strategy by Dolbeault, Mouhot and Schmeiser. This
strategy was extended in [GS14] to Kolmogorov backward evolution equations in
contrast to the dual Fokker-Planck framework. We use this mathematically
complete elaboration to investigate wide ranging classes of Langevin-type SDEs
in an abstract manifold setting, i.e. (at least) the position variables obey
certain smooth side conditions. Such equations occur e.g. as fibre lay-down
processes in industrial applications. We contribute the Lagrangian-type
formulation of such geometric Langevin dynamics in terms of (semi-)sprays and
point to the necessity of fibre bundle measure spaces to specify the model
Hilbert space.
