  We specify the conditions when a manifold M embedded in an inner product
space E is an invariant manifold of a stochastic differential equation (SDE) on
E, linking it with the notion of second-order differential operators on M. When
M is given a Riemannian metric, we derive a simple formula for the
Laplace-Beltrami operator in terms of the gradient and Hessian on E and
construct the Riemannian Brownian motions on M as solutions of conservative
Stratonovich and Ito SDEs on E. We derive explicitly the SDE for Brownian
motions on several important manifolds in applications, including
left-invariant matrix Lie groups using embedded coordinates. Numerically, we
propose three simulation schemes to solve SDEs on manifolds. In addition to the
stochastic projection method, to simulate Riemannian Brownian motions, we
construct a second-order tangent retraction of the Levi-Civita connection using
a given E-tubular retraction. We also propose the retractive Euler-Maruyama
method to solve a SDE, taking into account the second-order term of a tangent
retraction. We provide software to implement the methods in the paper,
including Brownian motions of the manifolds discussed. We verify numerically
that on several compact Riemannian manifolds, the long-term limit of Brownian
simulation converges to the uniform distributions, suggesting a method to
sample Riemannian uniform distributions
