  We study the rigidity results for self-shrinkers in Euclidean space by
restriction of the image under the Gauss map. The geometric properties of the
target manifolds carry into effect. In the self-shrinking hypersurface
situation Theorem 3.1 and Theorem 3.2 not only improve the previous results,
but also are optimal. In higher codimensional case, using geometric properties
of the Grassmanian manifolds (the target manifolds of the Gauss map) we give a
rigidity theorem for self-shrinking graphs.
