Mutual Invariance between Geometric Structures and Toplogical Structures on Manifolds

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We have observed the ralation between geometric and topological structures on smooth manifolds. H.Weyl has introduced the notion of conformal structure from the viewpoint of the Gauge theory. And he found an invariant on conformal structure, which is now called Weyl Conformal Curvature Tensor. It is the fundamental result in differential geometry that the Weyl conformal curvature tensor of an n-dimensional Riemannian manifold M^n (n>3) vanishes if and only if M^n is locally conformally equivalent to the flat euclidean space. Along this line but only Riemannian geometry, we have examined the invariance of conformal sturusture to other geometric structures. More precisely, as a geometric structure and a conformal invariant to even (resp. odd) dimensional manifolds, we brougth into focus Kahler manifolds for which the Bochner curvature tensor has been defined and CR-manifolds for which the Chern-Moser-Webster curvature tensor has been defined respectively. We shall define a conformal equivalence to the given geometric structure, and then construct a conformal invariant (tensor) on it.When that invariant vanishes, we observed what kind of new (or classical) geometry (G,X) comes out. Similtaneously, we have obtained a classification theorerm that such a manifold with vanishing invariant tensor can be uniformized with respect to the model space (G,X).