Leading local manifold

It is possible, however, to avoid any violation of these fundamental properties, and derive a result on the local electron densities of non-zero volume subsystems of boundaryless electron densities of complete molecules [159-161]. A four-dimensional representation of molecular electron densities is constructed by taking the first three dimensions as those corresponding to the ordinary three-space E3 and the fourth dimension as that representing the electron density values p(r). Using a compactifi-cation method, all points of the ordinary three- dimensional space E3 can be mapped to a manifold S3 embedded in a four- dimensional Euclidean space E4, where the addition of a single point leads to a compact manifold representation of the entire, boundaryless molecular electron density. 

In summary, the principle of local invariance in a curved Riemannian manifold leads to the appearance of compensating fields. The electromagnetic field is the compensating field of local phase transformation and the gravitational field is the compensating field of local Lorentz transformations. 

In the next section, these local stability considerations will be shown to lead to corresponding global results. For this analysis, it will be important to approximate the one-dimensional unstable manifold of Ei when both El and "2 exist and (3.10) holds. To this end, we provide information on an eigenvector corresponding to the eigenvalue A] of f. Let x = (xi,Qi,X2,02) denote such an eigenvector. We find that... 

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