Can we expect that certain inequalities hold in tangent spaces of any Riemannian $N$-manifold if it holds in $\mathbb{R}^N$?
For example, take the Cauchy-Schwarz inequality, for $x,y\in\mathbb{R}^N$ we have $$\langle x,y\rangle\leq|x||y|$$ Now if we consider some Riemannian $N$-manifold, say $(M,g)$, we can define the norm of a vector field $X\in T_pM$ as $|X|_g=\sqrt{\langle X,X\rangle_g}$, where $\langle\cdot,\cdot\rangle_g$ is the inner product induced by the metric $g$ in $T_pM$. So it is natural to expect the inequality would hold in $M$ as $$\langle X,Y\rangle_g\leq|X|_g|Y|_g$$ for all $X,Y\in T_pM$, since we also have $T_pM\cong\mathbb{R}^N$ for all $p\in M$.
Is this a correct argument? If it is correct, can we apply the same for any arbitrary kind of inequality available in $\mathbb{R}^N$? Sorry if it is a very basic question. Thanks in advance.