Let $(\mathcal{M}, g)$ be a Riemannian $n$-manifold. We know that
$$
\Vert v \Vert_g := \langle v, v \rangle_g^\frac{1}{2}
$$
defines a norm on the tangent space $T_p \mathcal{M}$. Can we derive the norm $\Vert \cdot \Vert_*$ of $T_p^* \mathcal{M}$ from $\Vert \cdot \Vert_g$? Would simply defining $\Vert \omega \Vert_* := \Vert \omega^\sharp \Vert_g$ works?
In particular, I'm interested in knowing whether Hölder's inequality holds on $T_p \mathcal{M}$, i.e.
$$
\sum_{k=1}^n \vert v_k w_k \vert \leq \Vert v \Vert_g \Vert w \Vert_* \, ,
$$
for any $v, w \in T_p \mathcal{M}$. (Though when $\Vert \cdot \Vert_*$ is defined to be $T_p^* \mathcal{M} \to \mathbb{R}$, that doesn't make sense.)
Is there any other definition of dual norm on tangent space of Riemannian manifold, such that Hölder's inequality holds?
Best Answer
I don't know what the $\sharp$ symbol means, but any scalar product $g(\cdot, \cdot) $on a finite dimensional vector space $X$ induces a natural isomorphism from $X$ to $X^*$ by mapping $v$ to $\omega_v$ by letting $\omega_v(w):= g(v, w)$.
You can request this to be an isometry, which induces a norm and a scalar product on $X^*$ in a natural way (so $||\omega_v||= ||v||$)
It's not difficult to see that then $g^*(\omega_v,\omega_w) = g(v,w)$.
These definitions easily carry over naturally and continuously to the tangent bundle construction.
(If that matches the inverse of your $\sharp$ map then your pertaining question is answered by yes :-)
I'm not sure what kind of Hölder inequality you want, but clearly $$||\omega_v(w)||= ||g(v,w)||\le ||v||\,||w|| = ||\omega_v||\,||w||$$
by the Cauchy Schwartz inequality for $g$ and using the fact that the map in question is an isometry.