  Sampling is a fundamental and arguably very important task with numerous
applications in Machine Learning. One approach to sample from a high
dimensional distribution $e^{-f}$ for some function $f$ is the Langevin
Algorithm (LA). Recently, there has been a lot of progress in showing fast
convergence of LA even in cases where $f$ is non-convex, notably [53], [39] in
which the former paper focuses on functions $f$ defined in $\mathbb{R}^n$ and
the latter paper focuses on functions with symmetries (like matrix completion
type objectives) with manifold structure. Our work generalizes the results of
[53] where $f$ is defined on a manifold $M$ rather than $\mathbb{R}^n$. From
technical point of view, we show that KL decreases in a geometric rate whenever
the distribution $e^{-f}$ satisfies a log-Sobolev inequality on $M$.
