Optimization on Lipschitz manifold

Question

I'm fairly new to the topic of manifold, but I found it very useful, such as the Stiefel manifold can be used to characterize orthornormal matrices. I know that there is a Lipschitz manifold, but I cannot find relevant materials on it. For instance, for a function $f(x)=Ax$, the Lipschitz constant is given by the spectral norm of $A$. I wonder if the concept of Lipschitz manifold can be used here to constrain such matrices, and how to optimize on such manifold?