  Let M be a complete non-compact connected Riemannian n-dimensional manifold.
We first prove that, for any fixed point p in M, the radial Ricci curvature of
M at p is bounded from below by the radial curvature function of some
non-compact n-dimensional model. Moreover, we then prove, without the pointed
Gromov-Hausdorff convergence theory, that, if model volume growth is
sufficiently close to 1, then M is diffeomorphic to Euclidean n-dimensional
space. Hence, our main theorem has various advantages of the Cheeger-Colding
diffeomorphism theorem via the Euclidean volume growth. Our main theorem also
contains a result of do Carmo and Changyu as a special case.
