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In this paper, we prove global in time Strichartz estimates for thefractional Schr\"odinger operators, namely $e^{-it\Lambda_g^\sigma}$ with$\sigma \in (0,\infty)\backslash \{1\}$ and $\Lambda_g:=\sqrt{-\Delta_g}$ where$\Delta_g$ is the Laplace-Beltrami operator on asymptotically Euclideanmanifolds $(\mathbb{R}^d,g)$. Let $f_0\in C^\infty_0(\mathbb{R})$ be a smoothcutoff equal 1 near zero. We firstly show that the high frequency part$(1-f_0)(P)e^{-it\Lambda_g^\sigma}$ satisfies global in time Strichartzestimates as on $\mathbb{R}^d$ of dimension $d\geq 2$ inside a compact setunder non-trapping condition. On the other hand, under the moderate trappingassumption, the high frequency part also satisfies the global in timeStrichartz estimates outside a compact set. We next prove that the lowfrequency part $f_0(P)e^{-it\Lambda_g^\sigma}$ satisfies global in timeStrichartz estimates as on $\mathbb{R}^d$ of dimension $d\geq 3$ without usingany geometric assumption on $g$. As a byproduct, we prove global in timeStrichartz estimates for the fractional Schr\"odinger and wave equations on$(\mathbb{R}^d, g), d\geq 3$ under non-trapping condition.

Global in time Strichartz estimates for the fractional Schr\"odinger equations on asymptotically Euclidean manifolds