  From the viewpoint of differential geometry and topology, we investigate the
characterization of the shadows in a Kerr spacetime. Two new quantities, the
length of the shadow boundary and the local curvature radius are introduced.
Each shadow can be uniquely determined by these two quantities. For the black
hole case, the result shows that we can constrain the black hole spin and the
angular coordinate of the observer only by measuring the maximum and minimum of
the curvature radius. While for the naked singularity case, we adopt the length
parameter and the maximum of the curvature radius. This technique is completely
independent of the coordinate system and the location of the shadow, and is
expected to uniquely determine the parameters of the spacetime. Moreover, we
propose a topological covariant quantity to measure and distinguish different
topological structures of the shadows.
