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We present the Julia package Manifolds.jl, providing a fast and easy-to-uselibrary of Riemannian manifolds and Lie groups. This package enables workingwith data defined on a Riemannian manifold, such as the circle, the sphere,symmetric positive definite matrices, or one of the models for hyperbolicspaces. We introduce a common interface, available in ManifoldsBase.jl, withwhich new manifolds, applications, and algorithms can be implemented. Wedemonstrate the utility of Manifolds.jl using B\'ezier splines, an optimizationtask on manifolds, and principal component analysis on nonlinear data. In abenchmark, Manifolds.jl outperforms all comparable packages for low-dimensionalmanifolds in speed; over Python and Matlab packages, the improvement is oftenseveral orders of magnitude, while over C/C++ packages, the improvement istwo-fold. For high-dimensional manifolds, it outperforms all packages exceptfor Tensorflow-Riemopt, which is specifically tailored for high-dimensionalmanifolds.

Manifolds.jl: An Extensible Julia Framework for Data Analysis on Manifolds