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In this paper a geometrical link between partial differential equations (PDE) and special coordinate nets on two-dimensional smooth manifolds with the a priori given curvature is suggested. The notion of G-class (the Gauss class) of differential equations admitting such an interpretation is introduced. The perspective of this approach is the possibility of applying the instruments and methods of non-Euclidean geometry to the investigation of differential equations. The equations generated by the...
For a vector bundle functor with the point property we prove that is product preserving if and only if for any and there is an -natural operator transforming connections on -dimensional fibered manifolds into connections on . For a bundle functor with some weak conditions we prove non-existence of -natural operators transforming connections on -dimensional fibered manifolds into connections on .
We extend a construction by K. Yamato [Ya] to obtain new explicit examples of Riemannian 3-manifolds as in the title. Some of these examples have an interesting geometrical interpretation.
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