Constraining cosmic curvature by using age of galaxies and gravitational
  lenses
  We use two model-independent methods to constrain the curvature of the
universe. In the first method, we study the evolution of the curvature
parameter ($\Omega_k^0$) with redshift by using the observations of the Hubble
parameter and transverse comoving distances obtained from the age of galaxies.
Secondly, we also use an indirect method based on the mean image separation
statistics of gravitationally lensed quasars. The basis of this methodology is
that the average image separation of lensed images will show a positive,
negative or zero correlation with the source redshift in a closed, open or flat
universe respectively. In order to smoothen the datasets used in both the
methods, we use a non-parametric method namely, Gaussian Process (GP). Finally
from first method we obtain $\Omega_k^0= 0.025\pm0.57$ for a presumed flat
universe while the cosmic curvature remains constant throughout the redshift
region $0<z<1.37$ which indicates that the universe may be homogeneous.
Moreover, the combined result from both the methods suggests that the universe
is marginally closed. However, a flat universe can be incorporated at $3\sigma$
level.
