Finsler geometry and actions of the $p$-Schatten unitary groups

Let $p$ be an even positive integer and $U_p(\mathcal{H})$ the Banach-Lie group of unitary operators $u$ which verify that $u-1$ belongs to the $p$-Schatten ideal $\mathcal{B}_p(\mathcal{H})$. Let $\mathcal{O}$ be a smooth manifold on which $U_p(\mathcal{H})$ acts transitively and smoothly. Then one can endow $\mathcal{O}$ with a natural Finsler metric in terms of the $p$-Schatten norm and the action of $U_p(\mathcal{H})$. Our main result establishes that for any pair of given initial conditions

$$x\in\mathcal{O}\hbox{ and } X\in(T \mathcal{O})_x$$

there exists a curve $\delta(t)=e^{tz}\cdot x$ in $\mathcal{O}$, with $z$ a skew-hermitian element in the $p$-Schatten class, such that

$$\delta(0)=x \hbox{ and } \dot{\delta}(0)=X,$$

which remains minimal as long as $t\Vert z\Vert _p\le \pi/4$. Moreover, $\delta$ is unique with these properties. We also show that the metric space $(\mathcal{O},d)$ (where $d$ is the rectifiable distance) is complete. In the process we establish minimality results in the groups $U_p(\mathcal{H})$ and a convexity property for the rectifiable distance. As an example of these spaces, we treat the case of the unitary orbit

$$\mathcal{O}=\{uAu^*: u\in U_p(\mathcal{H})\}$$

of a self-adjoint operator $A\in\mathcal{B}(\mathcal{H})$.