Abstract
In this paper the following three goals are addressed. The first goal is to study some strong partial differential equations (PDEs) that imply curvature-flatness, in the cases of both symmetric and non-symmetric connection. Although the curvature-flatness idea is classic for symmetric connection, our main theorems about flatness solutions are completely new, leaving for a while the point of view of differential geometry and entering that of PDEs. The second goal is to introduce and study some strong partial differential relations associated to curvature-flatness. The third goal is to introduce and analyze some vector spaces of exotic objects that change the meaning of a generalized Kronecker delta projection operator, in order to discover new PDEs implying curvature-flatness. Significant examples clarify some ideas.
Highlights
Introduction and ContributionsDifferential geometry is often considered “the art of manipulating partial differential equations (PDEs)”
This point of view can be found in the papers [1,2,3,4,5,6,7,8,9,10,11,12], which develop the following topics: general theory of PDEs, symmetries and overdetermined systems of PDEs, fully nonlinear equations on Riemannian manifolds with negative curvature, basic evolution PDEs in Riemannian geometry, foundations of differential geometry, affine differential geometry, overdetermined systems of linear PDEs, geometric dynamics on Riemannian manifolds, the role of PDEs in differential geometry, the Dirichlet problem for first-order PDEs, and differential inclusions
If a symmetric connection Γijk is a solution of curvature-flatness PDEs, i.e., Ri jkl = 0, the connection Γij0 k0 obtained by an arbitrary changing of coordinates xi = xi ( xi ) is a solution of Ri j0 k0 l 0 = 0, since Ri jkl = 0 is a tensorial equation with the unknown Γijk
Summary
Differential geometry is often considered “the art of manipulating partial differential equations (PDEs)”. It is our hope that this paper, taken as a whole, will provide a broad overview of geometry and its relationship to PDEs. The subjects of the constructive Sections are as follows: (2) curvature-flatness connections (symmetric connection case, vector spaces of exotic objects, strong Riccati PDEs, non-symmetric connection case), (3) curvature-flat. Riemannian manifolds (properties of a projection operator; strong Riccati PDE system; giving the curvature tensor field, finding the Riemannian metric), (4) invariant version for curvature-flatness (symmetric connection case, non-symmetric connection case), (5) conclusions. The vector spaces of exotic objects introduced by us to give another meaning to the generalized Kronecker-type projector are a novelty worthy of use in the future
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