  We introduce Geomstats, an open-source Python toolbox for computations and
statistics on nonlinear manifolds, such as hyperbolic spaces, spaces of
symmetric positive definite matrices, Lie groups of transformations, and many
more. We provide object-oriented and extensively unit-tested implementations.
Among others, manifolds come equipped with families of Riemannian metrics, with
associated exponential and logarithmic maps, geodesics and parallel transport.
Statistics and learning algorithms provide methods for estimation, clustering
and dimension reduction on manifolds. All associated operations are vectorized
for batch computation and provide support for different execution backends,
namely NumPy, PyTorch and TensorFlow, enabling GPU acceleration. This paper
presents the package, compares it with related libraries and provides relevant
code examples. We show that Geomstats provides reliable building blocks to
foster research in differential geometry and statistics, and to democratize the
use of Riemannian geometry in machine learning applications. The source code is
freely available under the MIT license at \url{geomstats.ai}.
