  We prove comparison, uniqueness and existence results for viscosity solutions
to a wide class of fully nonlinear second order partial differential equations
$F(x, u, du, d^{2}u)=0$ defined on a finite-dimensional Riemannian manifold
$M$. Finest results (with hypothesis that require the function $F$ to be
degenerate elliptic, that is nonincreasing in the second order derivative
variable, and uniformly continuous with respect to the variable $x$) are
obtained under the assumption that $M$ has nonnegative sectional curvature,
while, if one additionally requires $F$ to depend on $d^{2}u$ in a uniformly
continuous manner, then comparison results are established with no restrictive
assumptions on curvature.
