Let $f:N \to \mathbb{R}^2$ be a differentiable map of smooth manifolds. Let $\mathbb{R}^2$ be decomposed as a direct sum of line bundles, i.e. $\mathbb{R}^2=E(x) \oplus F(x)$, where $F(x)$ and $E(x)$ are not constant. We also denote $\alpha_{F}(x, y):=|\sin \angle(F(x), F(y))|$.
Let $P(x)$ be the oblique projection onto $E(x)$ along $F(x)$, and $Q(x)=\operatorname{Id}-P(x)$ be the oblique projection onto $F(x)$ along $E(x)$.
I want to know whether there is a relation between $\|P(x)-P(y)\|$ (or$\|Q(x)-Q(y)\|$ ) and $\alpha_{F}(x,y)$ or not. For instance, do we have $\|P(x)-P(y)\|=\|Q(x)-Q(y)\|\leq C \alpha_{F}(x, y)$?
Remark: I know that how to show that $\|\pi(x)-\pi(y)\|\leq C \alpha_{F}(x, y)$, where $\pi$ is orthogonal projection.
