  We prove Cheng's eigenvalue comparison theorems for geodesic balls within the
cut locus under weaker geometric hypothesis, and we also show that there are
certain geometric rigidity in case of equality of the eigenvalues. This
rigidity becomes isometric rigidity under upper sectional curvature bounds or
lower Ricci curvature bounds. We construct examples of smooth metrics showing
that our results are true extensions of Cheng's theorem. We also construct a
family of complete smooth metrics on the Euclidean space non-isometric to the
constant sectional curvature k metrics of the simply connected space forms of
constant sectional curvature k such that the geodesic balls of radius r have
the same first eigenvalue and the geodesic spheres have the same mean
curvatures. In the end we construct examples of Riemannian manifolds M with
arbitrary topology with positive fundamental tone positive that generalize
Veeravalli's examples.
