  We observe, utilize dualities in differential equations and differential
inequalities, dualities between comparison theorems in differential equations,
and obtain dualities in "swapping" comparison theorems in differential
equations. These dualities generate comparison theorems on differential
equations of mixed types I and II and lead to comparison theorems in Riemannian
geometry with analytic, geometric, P.D.E.'s and physical applications. In
particular, we prove Hessian comparison theorems and Laplacian comparison
theorem under varied radial Ricci curvature or radial curvature assumptions,
generalizing and extending the work of Han-Li-Ren-Wei, and Wei. We also extend
the notion of function or differential form growth to bundle-valued
differential form growth of various types and discuss their interrelationship.
These provide tools in extending the notion, integrability and decomposition of
generalized harmonic forms to those of bundle-valued generalized harmonic
forms, introducing Condition W for bundle-valued differential forms, and
proving duality theorem and unity theorem, generalizing the work of Andreotti
and Vesentini, and Wei. We then apply Hessian and Laplacian comparison theorems
to obtain comparison theorems in mean curvature, generalized sharp
Caffarelli-Kohn-Nirenberg type inequalities, embedding theorem for weighted
Sobolev spaces, geometric differential-integral inequalities, generalized sharp
Hardy type inequalities on Riemannian manifolds, monotonicity formulas and
vanishing theorems for differential forms of degree $k$ with values in vector
bundles, such as $F$-Yang Mills fields (when $F$ is the identity map, they are
Yang-Mills fields), generalized Yang-Mills-Born-Infeld fields on manifolds,
Liouville type theorems for $F$-harmonic maps, and Dirichlet problems on
starlike domains for vector bundle valued differential $1$-forms and
$F$-harminic maps, etc.
