Sublinear functions on a Riemannian manifold

Question

I would like to know if there is any notion of sublinear function or subadditive function for Riemannian manifolds.

Thank you!

Answer 1

If such functions were defined on an abstract Riemannian manifold, I believe we would have to define them on the tangent space (or cotangent space), since in general we have no notion of additivity on the actual manifold. But then if we did this, we're pretty much just using the already existent definition for vector spaces given in the links you provided. So I guess in that sense the answer is yes, the tangent space is a vector space so we can just use the already existing definition for a function defined on the tangent space. For the actual manifold, unless you have some notion of additivity explicitly defined, I don't see an obvious way to extend the definition.