Global Norms on compact Smooth manifolds

Question

Suppose an open subset $U$ of the euclidean space $R ^{m}$ is given and let $f$ be a smooth function on $U$ in which $f$ and all of its first order partial derivatives vanish at infinity. In this case define the norm of $f$ by $\|f\|=Sup_{x \in U} (| f(x) | + | \frac{\partial f(x)}{\partial x_1}|+...+|\frac{\partial f(x)}{\partial x_m}|)$. My question is that how i can generalize this norm in case of smooth functions on Riemanian manifolds? more precisely let $M$ be a smooth manifold endowed with a Riemanian metric $g$. Given a smooth complex valued function $f$ on $M$ how can i generalize the aformentioned norm in this case?

Answer 1

Replace $\mathbb R^n$ by a Riemannian manifold $(M, g)$, and consider an open set $U \subset M$ and a smooth function $f : U \to \mathbb R$. The metric $g$ defines a metric on the cotangent bundle of $M$, so you could try working with (the supremum over $U$ of) $$ \|f(x)\|^2 = |f(x)|^2 + |df_x|^2 $$ instead of your original norm.

The problem in generalizing your norm word-for-word is that you use the supremum of the pointwise $L^1$ norm of the gradient of $f$. On a Riemannian manifold, you only get the $L^2$ norm induced by the Riemannian metric.