Null test for cosmic curvature using Gaussian process

The cosmic curvature \begin{document}$ \Omega_{K,0} $\end{document}, which determines the spatial geometry of the universe, is an important parameter in modern cosmology. Any deviation from \begin{document}$ \Omega_{K,0}=0 $\end{document} would have a profound impact on the primordial inflation paradigm and fundamental physics. In this work, we adopt a cosmological model-independent method to test whether \begin{document}$ \Omega_{K,0} $\end{document} deviates from zero. We use the Gaussian process to reconstruct the reduced Hubble parameter \begin{document}$ E(z) $\end{document} and the derivative of the distance \begin{document}$ D'(z) $\end{document} from observational data and then determine \begin{document}$ \Omega_{K,0} $\end{document} with a null test relation. The cosmic chronometer (CC) Hubble data, baryon acoustic oscillation (BAO) Hubble data, and supernovae Pantheon sample are considered. Our result is consistent with a spatially flat universe within the domain of reconstruction \begin{document}$ 0<z<2.3 $\end{document}, at the \begin{document}$ 1\sigma $\end{document} confidence level. In the redshift interval \begin{document}$ 0<z<1 $\end{document}, the result favors a flat universe, while at \begin{document}$ z>1 $\end{document}, it tends to favor a closed universe. In this sense, there is still a possibility for a closed universe. We also carry out the null test of the cosmic curvature at \begin{document}$ 0<z<4.5 $\end{document} using the simulated gravitational wave standard sirens, CC+BAO, and redshift drift Hubble data. The result indicates that in the future, with the synergy of multiple high-quality observations, we can tightly constrain the spatial geometry or exclude the flat universe.