Null test of the cosmic curvature using $H(z)$ and supernovae data

We introduce a model-independent approach to the null test of the cosmic curvature which is geometrically related to the Hubble parameter $H(z)$ and luminosity distance $d_L(z)$. Combining the independent observations of $H(z)$ and $d_L(z)$, we use the model-independent smoothing technique, Gaussian processes, to reconstruct them and determine the cosmic curvature $\Omega_K^{(0)}$ in the null test relation. The null test is totally geometrical and without assuming any cosmological model. We show that the cosmic curvature $\Omega_K^{(0)}=0$ is consistent with current observational data sets, falling within the $1\sigma$ limit. To demonstrate the effect on the precision of the null test, we produce a series of simulated data of the models with different $\Omega_K^{(0)}$. Future observations in better quality can provide a greater improvement to constrain or refute the flat universe with $\Omega_K^{(0)}=0$.