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?

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    $\begingroup$ Welcome to Math.SE! Can you add something to your question about what application you have in mind for your generalization? In the absence of further information, you might take the supremum of $|f| + \|df\|$ over all functions for which this quantity vanishes at infinity (as defined by exhaustion by compact sets), but the precise choice of "suitable" norm is usually dictated by the details of the analytic problem you want to attack. $\endgroup$ – Andrew D. Hwang Mar 14 '17 at 13:30
  • $\begingroup$ Thank you so much. I need a global norm for this case which generalizes exactly the previous norm for functions on open sets. $\endgroup$ – Arya Jamshidi Mar 14 '17 at 13:57

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.

  • $\begingroup$ Thank you, Is there a natural Finsler metric that generalizes my norm on a Finsler manifold( instead of Riemaniann metric)? $\endgroup$ – Arya Jamshidi Mar 15 '17 at 8:14
  • $\begingroup$ You should be able to do exactly the same thing on a Finsler manifold. You could also use exactly your norm (if $M$ is compact) by splitting $M$ up into coordinate patches by a partition of unity, calculating your norm on each one, and summing them up via the partition. This would depend on the partition and the coordinate patches chosen, but maybe you don't care. $\endgroup$ – Gunnar Þór Magnússon Mar 15 '17 at 10:37

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